Set Theory and Logic

Recursive Inaccessibility and the Inductive Hypothesis as to Countable Sets in ZFC

Authors: Amel Mara
Comments: 32 Pages.

The significance of the Inductive Hypothesis is examined with respect to the Principle of Mathematical Induction. A few relevant theorems that involve functions in set theory are specified with respect to the Inductive Hypothesis.The countability of rational numbers is reviewed, as to Cantor’s "intuition" (i.e., the "zig-zag" method of enumerating rational numbers) and constructive formulas that would map the set of natural numbers to a subset of the rational numbers (e.g., a multiplicative inverse function, a divisive function).Von Neumann and Zermelo ordinals are introduced to support the definition of a non-dense, well-ordered set of numbers. It is determined that for a specific transfinite set of ordinals with a maximal element that is a limit ordinal, the set must contain at least one successor ordinal that cannot be recursively accessed in a finite number of steps from a specified base ordinal.
Category: Set Theory and Logic

Understanding Universal Disjunction Better

Authors: Garry Goodwin
Comments: 11 Pages. (Note by viXra Admin: An abstract in the article is required)

The asymmetry that distinguishes the direction of entailment of the two forms of universal disjunction is a tenet of Quantificational Logic but it is also poorly understood. The paper offers a semantics that sheds light on this asymmetry.
Category: Set Theory and Logic

A Pathology of the Von Neumann Ordinals

Authors: David L. Selke
Comments: 4 Pages.

We show that the definition of the Von Neumann ordinal $omega_2$ fails to have the properties of an ordinal. In the notation that uses braces to denote ordinals, the ``depth" or ``height" of nesting of these braces becomes infinite leading to objects which have no top level of braces and which therefore have their elements undefined, leading to the conclusion that they are not sets at all, let alone ordinals. The least of these objects occurs before (that is, within) $omega_2$.
Category: Set Theory and Logic

Formalizing Mechanical Analysis Using Sweeping Net Methods

Authors: Parker Emmerson
Comments: 12 Pages.

We present a formal mechanical analysis using sweeping net methods to approximate surfacingsingularities of saddle maps. By constructing densified sweeping subnets for individual vertices and integrating them, we create a comprehensive approximation of singularities. This approach utilizes geometric concepts, analytical methods, and theorems that demonstrate the robustness and stabilityof the nets under perturbations. Through detailed proofs and visualizations, we provide a new perspective on singularities and their approximations in analytic geometry.
Category: Set Theory and Logic

On the Time Complexity of Problem and Solution Spaces

Authors: Theophilus Agama
Comments: 6 Pages. This paper introduces the notion of time complexity.

We continue with the development of the theory of problems and their solutions spaces cite{agama2023topology} and cite{agama2022theory}. We introduce and study the notion of verification and resolution time complexity of solutions and problem spaces.
Category: Set Theory and Logic

The Geometry of Gödelian Categorical Singularities: A Refined Mathematical Framework for Incompleteness Phenomena (Part 2: Extending the Topological and Geometric Aspects)

Authors: Paul Chun-Kit Lee
Comments: 115 Pages.

This paper, building upon our previous work on Gödel category singularities (https://vixra.org/abs/2407.0164),presents a comprehensive geometric theory of Gödelian phenomena. By recasting logical structures as intricatemathematical landscapes, we offer a novel perspective on the nature of incompleteness and undecidability. Our approachsynthesizes concepts from category theory, algebraic topology, differential geometry, and dynamical systemsto create a rich, multidimensional view of logical spaces. We introduce the concept of Gödelian manifolds, wherestatements in formal systems are represented as points in a vast terrain. The elevations and contours of this landscapecorrespond to logical complexity and provability, with Gödelian singularities emerging as profound chasmsor peaks. This geometric framework allows us to apply tools from various mathematical disciplines to analyze thestructure of incompleteness. Our approach enables a nuanced analysis of different types of logical complexity. Wedevelop theoretical constructs to explore the nature of self-referential paradoxes, non-self-referential undecidability,and the characteristics of difficult but provable statements within our geometric model. This provides new mathematicalinsights into the structure of formal systems and the limits of provability. To illustrate the conceptualpower of this approach, we draw an analogy to the alleged "Gödel loophole" in the U.S. Constitution. While nota direct application, this metaphorical exploration demonstrates how our abstract framework can provide intuitiveunderstanding of complex logical structures, offering an accessible entry point for non-specialists to grasp theseintricate ideas.
Category: Set Theory and Logic

The Continuum Theorem II

Authors: David Selke
Comments: 8 Pages.

An injection from the countable ordinals to the paths in the binary tree leads to a bijection between all paths in the binary tree and only those in the $aleph_1$-sized injection. Since there are $2^{aleph_0}$ many paths in the binary tree, this proves the Continuum Hypothesis, $2^{aleph_0} = aleph_1$.
Category: Set Theory and Logic

Higher Categorical Structures in Gödelian Incompleteness: Towards a Topos-Theoretic Model of Metamathematical Limitations

Authors: Paul Chun-Kit Lee
Comments: 18 Pages. (Note by viXra Admin: AI assisted/generated contents/results are in general not acceptable)

This paper presents a novel approach to Gödelian incompleteness using higher category theory and topos theory. We construct a hierarchy of (∞,1)-categories modeling increasingly powerful formal systems, and prove a generalized incompleteness theorem in this context. Using techniques from homotopy type theory, we develop a topos-theoretic model of metamathematical reasoning that captures subtle aspects of incompleteness phenomena. Our results have implications for the foundations of mathematics and theoretical computer science. Furthermore, we explore how this framework offers new perspectives on fundamental questions in physics and cognitive science. We discuss potential implications for theories of computational physics, the nature of mathematical insight, and the limits of formal models of physical reality. By situating Gödelian phenomena within the rich context of higher category theory, we open new avenues for understanding the nature of mathematical truth, the limits of formal reasoning, and the connections between metamathematics, theoretical physics, and human understanding.
Category: Set Theory and Logic

The Problem P=NP (in French)

Authors: Thierry Delort
Comments: 20 Pages.

In this article, we are going to solve the problem P=NP for a particular kind of problems called basic problems of numerical determination. We are going to propose 3 fundamental Axioms permitting to solve the problem P=NP for basic problems of numerical determination, those Axioms can also be considered as pure logical assertions, intuitively evident and never contradicted, permitting to understand the solution of the problem P=NP for basic problems of numerical determination. We will see that those Axioms imply that the problem P=NP in undecidable for basic problems of numerical determination. Nonetheless we will see that it is possible to give a theoretical justification (which is not a classical proof) of the proposition "P≠NP". We will then study a 2nd problem, named "PN=DPN problem" analogous to the problem P=NP but which is fundamental in mathematics.

Dans cet article, nous allons résoudre le problème P=NP pour un cas particulier de problèmes appelés problèmes de détermination numérique basiques. Nous allons proposer 3 Axiomes fondamentaux permettant de résoudre le problème considéré pour les problèmes de détermination numérique basiques, ces Axiomes pouvant aussi être considérés comme des assertions de logique pure évidentes intuitivement et jamais contredites permettant de comprendre la solution du problème considéré. On verra que ces Axiomes entraînent l’indécidabilité du problème P=NP pour les problèmes de détermination numérique basiques. On montrera cependant qu’on peut donner une justification théorique (qui n’est pas une démonstration classique) de P≠NP. Nous étudierons ensuite un 2nd problème, appelé problème « PN=DPN », analogue au problème P=NP mais ayant une importance fondamentale en mathématique.
Category: Set Theory and Logic

Fractal Partitioning and Subconvexity

Authors: Parker Emmerson
Comments: 18 Pages.

This document presents a comprehensive study of fractal partitioning and its application to subconvexity generalizations across various mathematical contexts. By utilizing a combination of advanced equations andinequalities, the paper develops robust models for partitioning sets into subsets of varying sizes, measuring the similarity and complexity within these partitions, and ensuring consistent interactions across boundaries. Special attention is given to computing the norm of differences betweensubsets and assessing their similarity, along with complexity measurements utilizing tensor equations and sums. These calculations provideinsights into the partitions’ fractal behavior and their probabilistic interactions.The document also delves into task scheduling algorithms based on SRPT, round-robin, and deadline-driven protocols, highlighting practical implications of fractal partitioning in optimizing resource management and minimizing distortions in dynamic systems. An emphasis is placedon ensuring the robustness and efficiency of fractal partitions through rigorousmathematical proofs and algorithmic implementations. By applyingthese models to data compression and analysis, the study demonstrates how fractal partitioning can efficiently represent complex data sets, expose hidden patterns, and identify anomalies in various domains such as finance and natural systems. Furthermore, the paper explores the concept of subconvexity in higher powers of the Riemann zeta function, establishing stronger forms of subconvexity conditions for different mathematical functions. This includesgeneralizations for cubic and higher powers of zeta functions, providing substantial evidence in support of hypotheses like the Riemann Hypothesis. The comprehensive approach combines theoretical constructs with practical algorithms, offering a powerful framework for analyzing and understanding complex mathematical and natural phenomena through fractalpartitioning and subconvexity measures.
Category: Set Theory and Logic

Non-Parametric Identification of Feedback System Without Excitation

Authors: Chung Sung Jang, Yu Sung Kim, Myong Hyok Sin, Nam Ho Kim
Comments: 7 Pages.

This paper describes the non-parametric identification of feedback system by two different controllers without exterior excitation. The proposed method doesn’t necessarily require any prior information for processing and, furthermore, it can assume time delay and modeling degree with accuracy. Its efficiency is proved by simulation.
Category: Set Theory and Logic

Beyond Gödel

Authors: Jim Rock
Comments: 2 Pages.

In 1930 Gödel wrote a landmark paper showing that in any formal system there will always bestatements that cannot be proven. But the deficiency of formal systems goes much deeper. The same logically valid statement can be used in conjunction with two different sets simultaneously proving a true statement and a false statement. This result is profound. It explains why people can use the same sound argument to prove two contradictory statements. It is no wonder the most lucid arguments still sometimes result in hung juries and earnest people can disagree on the most fundamental issues. Truth is a much deeper concept than logical validity.
Category: Set Theory and Logic

About Countability Sets

Authors: Tomasz Soltysiak
Comments: 6 Pages. (Note by viXra Admin: Please cite and list scientific references)

Dispute Cantor's theorem about power sets for infinite sets. Proof of the equivalence of sets of natural and real numbers. Theorem about countable of all sets.
Category: Set Theory and Logic

Proof of the Generalized Continuum Hypothesis

Authors: David L. Selke
Comments: 6 Pages.

The Continuum Hypothesis has recently been proven in a form that might have been accepted had it appeared before ZFC but after Hilbert’s challenge in 1900. This work will develop the same technique to prove the Generalized Continuum Hypothesis by induction on aleph and beth subscripts.
Category: Set Theory and Logic

Why Transfinite Set-Theory is Wrong and an Alternative Theory of Numeric Structures

Authors: David Peralta
Comments: 55 Pages. (Abstract added to Article by viXra Admin as required - Please conform in the future)

Set Theory is a formalization of the existence and fundamental properties of mathematical objects as collections of elements and/or elements included in collections. Its formulation is so basic and comprehensive that it has been postulated as the foundation of all mathematics. Perhaps, the major achievement of Set Theory is that, after being criticized by many reputable mathematicians and philosophers since its appearance, it is now commonly accepted as the primary explanation of the most basic components of mathematics: numbers; and not only the numbers we have needed or we may ever need but all the numbers that could potentially exist. In Set Theory, an infinite sequence of numbers exists not as the mere projection of a construction algorithm but as a complete and self-identical mathematical object: a set. In Set Theory the words infinite and infinity do not refer to the property of growing endlessly (potential infinity) but to a definite magnitude; a number; the actual infinity. As a result of such a conception, Set Theory arrives at the conclusion that there exist infinitely many infinities, each one with a different value. The set of postulates, proofs and theorems used to justify the existence of such infinities is commonly known as Transfinite Set-Theory.The first part of this work shows how some of the properties and theorems applied to infinite sets, in Set Theory, necessarily lead to internal and fundamental contradictions under classical logic, even when the idea of actual infinity is accepted. Throughout the second part, motivated by the necessity of an alternative to Transfinite Set-Theory, due to the incapacity of such a theory to explain some of the findings shown in the first part (especially the proof of the existence of as many rational as irrational numbers), the author develops a theory to provide a better understanding of infinite sequences of numbers.
Category: Set Theory and Logic

The Continuum Theorem

Authors: David L. Selke
Comments: 4 Pages. (Note by viXra Admin: Please list scientific references in future submissions)

An injection from omega-1 to the paths in the binary tree leads to a bijection between the omega-length binary strings and an aleph-1 sized set. Since there are 2^aleph-null many omega-length binary strings, this proves the Continuum Hypothesis, 2^aleph-null = aleph-1. The technique may be extended to show the Generalized Continuum Hypothesis, which however will be reported separately.
Category: Set Theory and Logic

First Case of Mistake in Cantor’s Proof: Pure Double Contradiction Technique

Authors: Juan Carlos Caso Alonso
Comments: 39 Pages. In English and Spanish. Contact: recursos.clja@gmail.com

The proof of Cantor's Theorem ( |A| < |P(A)| ), changes depending to which mathematician you ask. There are two versions of it. The one I call "Creating an extern element", and the other one I call "Pure double contradiction".This document will try to explain the difference between them, and after that show that the second one should not be never used, for being "not reliable".In case you are interested, you have an extra chapter, with informal data about the first one.
Category: Set Theory and Logic

Equality Hypothesis in Diagonal Argument

Authors: Hongyi Li
Comments: 5 Pages.

In the Diagonal Argument, there are two independent hypotheses, the countability hypothesis and the equality hypothesis, which violate the basic principle of proof by contradiction, so the Diagonal Argument cannot be held. Closer analysis shows that the Diagonal Argument only falsifies the equality hypothesis and has nothing to do with the countability hypothesis. Since the equality hypothesis is inherently wrong and does not need to be falsified at all, the Diagonal Argument actually proves nothing. The transfinite number theory and Continuum hypothesis therefore become meaningless. Once again calls on the education department to suspend the teaching of infinite sets, so as not to mislead people.
Category: Set Theory and Logic

Stringy Motivic Spectra II: Higher Koszul Duality

Authors: Ryan J. Buchanan
Comments: 11 Pages.

This is a rendition of [2]. We study stringy motivic structures. This builds upon work dealing with $mathbb{F}_p$-modives for a suitable prime p. In our case, we let p be a long exact sequence spanning a path in a pre-geometric space. We superize a nerve from our previous study.
Category: Set Theory and Logic

Method of Listing Finite and Infinite Decimals Simultaneously: the Simplest Proof of Countability of Real Numbers

Authors: Hongyi Li
Comments: 5 Pages.

The finite and infinite decimals of binary are listed one by one at the same time, thus completely proving that the real numbers are countable. This paper will completely rewrite the history of mathematics. Because there are lots of errors in Cantor's theory, and the uncountability of real numbers is only one of them, it is necessary to launch a campaign to crack down on false and correct errors, lest these errors continue to destroy the normal capacity of human thinking and continue to mislead people.
Category: Set Theory and Logic

Fixed Point Properties of Precompletely and Positively Numbered Sets

Authors: Marat Faizrahmanov
Comments: 14 Pages.

In this paper, we prove a joint generalization of Arslanov’s completenesscriterion and Visser’s ADN theorem for precomplete numberings. Then we considerthe properties of completeness and precompleteness of numberings in the context ofthe positivity property. We show that the completions of positive numberings are nottheir minimal covers and that the Turing completeness of any set A is equivalent to theexistence of a positive precomplete A-computable numbering of any infinite family withpositive A-computable numbering.
Category: Set Theory and Logic

Seven Internal Contradictions of Set Theory

Authors: Wolfgang Mückenheim
Comments: 3 Pages. (Correction made by viXra Admin to conform with scholarly norm)

Seven internal contradictions of set theory are discussed.
Category: Set Theory and Logic

On Non-Principal Arithmetical Numberings and Families

Authors: Marat Faizrahmanov
Comments: 13 Pages.

The paper studies Σ-0-n-computable families (n ⩾ 2) and their numberings. It is proved that any non-trivial Σ-0-n-computable family has a complete with respect to any of its elements Σ-0-n-computable non-principal numbering. It is established that if a Σ-0-n-computable family is not principal, then any of its Σ-0-n-computable numberings has a minimal cover.
Category: Set Theory and Logic

Several Questions About Infinity

Authors: Jincheng Zhang
Comments: 15 Pages.

What is infinity? What principles should be adhered to when researching infinity? For a predicate on a natural number system, is it true that finite holds, and infinite holds? This paper reexamines the nature of infinity and proposes two opposite infinite axioms (δ+1=δ or δ+1≠δ). Based on these two infinite axioms, the "infinite induction" of the identity formula is proved; it is found that the infinite axioms in the ZF system do not satisfy the equality axiom, and there are many contradictions in the reasoning of the Cantor ordinal number. The ordinal theory of set theory ZF system is not strict. It is hoped that the mathematics community will pay attention to these questions and give a convincing answer.
Category: Set Theory and Logic

The Non-uniqueness of the Set of Natural Numbers

Authors: Hongyi Li
Comments: 18 Pages.

Some counterexamples to the uniqueness of the set of natural numbers were given and the non-uniqueness of the set of natural numbers was proved on the basis of strict definition of the number of elements of any set. It was proved that the number of elements in any infinite set is more than that in its proper subset and any one-to-one correspondence cannot be established between two sets with different number of elements according to the definition of bijection. As a result, an infinite set cannot correspond one-to-one to its proper subset. So, there is no infinite hotel paradox and Galilean paradox. The set of natural numbers that corresponds one-to-one to the rational numbers set Q is not the proper set of Q, but another set of natural numbers. The part does not equal the whole. The number of digits after the decimal point for infinite decimals may also be different for infinite decimals. When the number is the same, there is no one-to-one correspondence between different dimensional spaces and there is no one-to-one correspondence between different length segments. The real numbers are countable. There is no uncountable set. Almost all of Cantor's counterintuitive theories are wrong in their logic, so the conflict between intuition and logic is not normal and intuitionism and logicism must be united. It is possible to reach limit but the infinite process can never be completed. There is no Zeno paradox. The fact that there are so many errors in the most rigorous mathematics and that they remain uncorrected for so long shows that human logical thinking ability needs to be greatly improved and no one can be sure that he is correct. Therefore, it is foolish to impose one's own views, including ideology and values, on others, even by force, to push mankind towards self-destruction in nuclear and biological wars.
Category: Set Theory and Logic

On the Mechanics of Quasi-Quanta Realization

Authors: Ryan J. Buchanan, Parker Emmerson, Oliver Hancock
Comments: 7 Pages.

We model an absolute reference frame using a pullback on a certain locally trivial line bundle. We demonstrate that this pullback is unrealizable in $mathbb{R}^4$. We devote section 3 to process-based thinking.
Category: Set Theory and Logic

The Anisotropy of Factions

Authors: Ryan J. Buchanan
Comments: 3 Pages. (Abstract added by viXra Admin - Please conform!)

Factions, or agent-based secret-sharing networks, are discussed. Convergence of multiple belief systems is described as the anisotropic propagation of truth by factions.
Category: Set Theory and Logic

On V-Categories

Authors: Ryan J. Buchanan
Comments: 3 Pages.

Lawvere introduced a deceptively simple category, V, which is complete, symmetric, and monoidal closed. Here, we extend this construction to describe a rather general notion of localization called ��-truncation. We show that this procedure produces tame, realizable n-cells in a standard Grothendieck universe, ��. Finally, we clarify our notion of smallness for objects of stable rings in ��.
Category: Set Theory and Logic

The Diagonal Sections of Bivariate Archimedean Copulas and the Estimation of Parameters

Authors: Kum-Chol Son, Song Jin-A, Ro Song-Chol
Comments: 20 Pages.

We introduce new concepts-a generator of degree and a diagonal section of degree for any real number . A diagonal section of degree is the one of the bivariate Archimedean copula with a generator of degree . Generators of many well-known parametric families of bivariate Archimedean copulas, including those of Clayton, Frank and Gumbel-Hougaard, are of degree . In this article, we show that each bivariate Archimedean copula with a generator of degree is uniquely determined by its diagonal section. An asymptotic representation of these copulas in terms of corresponding diagonal sections is obtained. We also provide a sufficient condition to be a diagonal section of degree . These results allow us to construct several statistical inference procedures for bivariate Archimedean copulas. Since diagonal sections of copulas are absolutely continuous, we suggest a parametric estimation procedure for bivariate Archimedean copulas based on the likelihood of a full sample from the diagonal section.
Category: Set Theory and Logic

A Proof of a Conjecture of lu Concerning Inequality for the Gamma Function

Authors: CholMyong Song, TaeHyon Mun, YongIl Han, WonChol Kim, HyonChol Kim
Comments: 6 Pages.

Dawei Lu [Dawei Lu, A generated approximation related to Burnside’s formula, Journal of Number Theory 136 (2014) 414—422; http://dx.doi.org/10.1016/j.jnt.2013.10.016] proposed a conjecture: for every real number k>0, there exists depending k, such that for every, it holds: He guessed that it is suitable for taking = 0.5. In this paper, we prove the conjecture of Dawei Lu.
Category: Set Theory and Logic

One Third Crucial Theorem for the Refoundation of Elementary Set Theory and the Teaching of that Discipline to Future Generations

Authors: Nhat-Anh Phan
Comments: 22 Pages. In French - Copyright All rights reserved

For a given infinite countable set A, we demonstrate that A is an infinite countableset if and only if A is equal to an infinite countable set indexed to the infinity. Saidotherwise we demonstrate that A is an infinite countable set iff there exists an infinite numberof non-empty, distinct elements a_i ǂ ∅, i ∈ N∗, ∀i, j ∈ N∗, i ǂ j, a_i ǂ a_j such thatA = U_{+∞}_{i=1} {a_i}. At this occasion, for infinite countable sets constituted by the union of two giveninfinite countable sets A and B, Au2032 = [A ∪ B]P(Au2032), we introduce the notion of undeterminedinfinite countable set in order to designate infinite countable sets for whichan explicit indexation is not determined meanwhile such indexation must necessarilyexist.
Category: Set Theory and Logic

Categorifying Connections

Authors: Ryan J. Buchanan
Comments: 5 Pages.

The notion of a connection from differential geometry is employed in a category-theoretic context. We discuss the properties of holonomy from a tangent ∞-category perspective.
Category: Set Theory and Logic

Some Notes on Shadows

Authors: Ryan J. Buchanan
Comments: 4 Pages. (Abstract added to Article by viXra Admin - Please conform!)

These notes were taken whilst thinking about the monotone-light factorization, which lead to the productive idea of a shadow category, or a certain kind of category with a preordered structure.
Category: Set Theory and Logic

Letter Theory, Creating and Describing Mathematical Axioms of a System for Letters which Creates Most Simplified Language with Provable Sentences

Authors: Iman Mosleh
Comments: 16 Pages.

This paper is a cross field work between theology, linguistics and mathematics (abstract algebra, topology, set theory, fractal geometry and number theory) which tries to establish a letter system to define word meanings by letter meanings and to restart mathematical theology on this planet which we think had been started by Abraham the prophet.We try to answer this question that can we create a meaning system which every alphabet letter has an specific meaning in, and when we combine them to create words we have a clear system to define every word meaning just by its letters meaning and order of that letters in the word?We will show that if we have a fractal system we can establish a letter system for it.The other question which i try to answer is that can we produce a letter system which provide us provable sentences?
Category: Set Theory and Logic

Any Formal System That Contains Sets Arithmetic and Rational Numbers is Inconsistent

Authors: Jim Rock
Comments: 1 Page.

Gӧdel proved that any formal system containing arithmetic is incomplete. We show that any such formal system is inconsistent. We establish a collection of nested sets of rational numbers in a descending hierarchy. The sets higher in the descending hierarchy contain element(s) that are not in the sets below them in the hierarchy. Given such a descending set hierarchy, it is easy to develop two arguments that contradict each other. The conclusion of Argument#2 is false. But, Argument#2 is a valid argument.
Category: Set Theory and Logic

The Method by the Definition of Sets to Compare the Relative Number of Elements in Infinite Sets

Authors: Hongyi Li
Comments: 5 Pages.

In mathematics, the derivation from the definition is reliable. Therefore, the method proposed in this paper to compare the relative number of elements of infinite sets from the definition of sets is highly reliable and accurate, far more reliable than the traditional one-to-one correspondence method, and will not cause any paradox. This paper also eliminates the paradoxes of Galileo, the infinite hotel, the whole equals the part and so on.
Category: Set Theory and Logic

A Easy Approach for the Sinc Integral

Authors: SungJin Kim, HyonChol Kim, IlSu Choe
Comments: 5 Pages.

In this paper, by using simple mathematical method we established a generalized formula. In fact, the calculation method of this integral is introduced in several papers by using some advanced analysis knowledge like Fourier transform, Poisson summation and so on. But we used only very general mathematical knowledge.
Category: Set Theory and Logic

My ZFC Inconsistency Odyssey

Authors: Jim Rock
Comments: 1 Page. (Note by viXra Admin: Please refrain from repetitions and repeated withdrawls and resubmissions - They will not be accepted)

My progress in understanding ZFC’s inconsistency is chronicled through the papers reviewed in this document.
Category: Set Theory and Logic

MultiSet-Venn Analysis

Authors: Robert Lloyd Jackson
Comments: 1 Page.

This paper defines the union and intersection of multiple sets.
Category: Set Theory and Logic

Shortest Proof of Dark Numbers

Authors: W. Mückenheim
Comments: 1 Page.

Shortest Proof of Dark Numbers
Category: Set Theory and Logic

Cardinal Numbers and Cantor’s Continuum Hypothesis

Authors: Jincheng Zhang
Comments: 10 Pages.

:In this paper, the cardinal number problem is discussed separately in the axiom systems for set theory SZF+ and SZF-. It can be proved that:ordinal numbers and cardinal numbers are unified. There is no uncountable cardinal number; in Cantor’s set theory, definitions, theorems and propositions based on uncountable cardinal numbers and ordinal numbers are all false. In fact, Cantor’s continuum hypothesis is the cardinal number of a set of natural numbers, and whether there are other cardinal numbers between cardinal numbers of set of natural numbers |N| and cardinal numbers of power set |P (N)|, different interpretations are given in different axiom systems.
Category: Set Theory and Logic

First Principles of Mathematics

Authors: Forrest C. Taylor
Comments: 100 Pages. CC BY-NC-ND 4.0 International license

A dependent type theory is proposed as the foundation of mathematics. The formalism preserves the structure of mathematical thought, making it natural to use. The logical calculus of the type theory is proved to be syntactically complete. Therefore it does not suffer from the limitations imposed by Gödel’s incompleteness theorems. In particular, the concept of mathematical truth can be defined in terms of provability.
Category: Set Theory and Logic

Contradiction and Reconstruction of Axiom of ZF System

Authors: Jincheng Zhang
Comments: 19 Pages.

In set theory ZF system, there are specious strange conclusions when set of infinite sets are interpreted into numbers. Axiom of infinity in ZF system confuses the difference between δ+1＝δ and δ+1≠δ, Error of Cantor's set theory is the error of axiom of infinity. We use (δ + 1 = δ or δ + 1 ≠ δ) to establish two opposite sets of axioms SZF + and SZF-, which are contradictory systems, similar to Euclidean geometry and non-Euclidean geometry axioms.In set theory SZF system, after correcting the axiom of infinity, we can prove that "for any definable predicate P (n), if it is true to finite n, then for any infinite number a, P(a) is also true." Power set axiom, axiom of separation, axiom of substitution, axiom of choice, axiom of foundation these axioms have become provable propositions.In fact, the four axioms and several definitions of "axiom of empty set, set axioms, union axioms, and infinite axioms" are sufficient for the set theory system.In Cantor’s set theory, there are ordinal numbers and uncountable ordinal numbers. There are no uncountable ordinal numbers, so proof of uncountable ordinal number is false. The transfinite induction in Cantor’s set theory is also false; in fact, transfinite induction is very simple, we can prove that: for any definable predicate P (n), if it is true to finite n, then for any infinite number a, P(a) is also true. In classical set theory system, all difficulties are from infinite difficulties. After we have solved the axiom of infinity, infinity is as simple as finite.
Category: Set Theory and Logic

ZFC Inconsistency Explained and Defended

Authors: Jim Rock
Comments: 1 Page.

There is a class of sets that can be constructed within ZFC that both have and do not have a largest element. Two contradictory arguments creating these sets are explained and defended.
Category: Set Theory and Logic

Terms Constructed by the "Diagonal Method" Are Unclosed Terms

Authors: Jincheng Zhang
Comments: 9 Pages.

Nature of a proposition constructed by diagonal method of proof is a paradox, so it is an unclosed term and an extra-field proposition. There are two kinds of infinities, standard infinity and non-standard infinity, and we will explore the diagonal problem in each of the two kinds of infinities below.We conclude that: (1) In standard infinity, Cantor's diagonal number can metamorphose into real number and the contradiction vanishes. (2) In nonstandard infinity, Cantor's diagonal numbers become hyperreal number . Essentially both are unclosed terms of the calculation.Therefore, Cantor's diagonal method proves that "the real numbers are not countable" is wrong.
Category: Set Theory and Logic

Natural Sets and the Axiom of Comprehension

Authors: Albert Henrik Preiser
Comments: 12 Pages.

A natural set never contains itself. The existence or non-existence of a natural set is decided using a modified "Axiom of Comprehension". This modified axiom keeps everything that contains itself out of set formation. This also includes the property of not containing itself, although every natural set has this property. If it were possible to form a set with this property, then the antinomy "contains itself" and "does not contain itself", named after Bertrand Russell, would apply to this set. While this is paradoxical, it is still a corollary when trying to form a set with the property "does not contain itself". The modified "Axiom of Comprehension" takes this fact into account and decides on the non-existence of a set to be formed with the property "does not contain itself" because it would have the property "contains itself". There can therefore be no antinomy like that of Bertrand Russell in the case of natural sets. The modified "Axiom of Comprehension" states that a natural set does not contain itself. This means that the set to be formed cannot have the property or condition required for the set to be formed. This, together with the elimination of Russell's antinomy, provides an existence criterion for natural sets. With this, essential statements about the natural sets can be proved.
Category: Set Theory and Logic

An Inconsistent Hierarchy of Sets in [0, 1]

Authors: Jim Rock
Comments: 1 Page.

Two contradictory arguments are developed from a hierarchy of sets in [0, 1]. One argument is a proof by contradiction and its conclusion is true. The other argument is an existence argument and while its conclusion is not true, it follows logically from the a valid assumption followed by three true statements that precede the conclusion.
Category: Set Theory and Logic

On the Topology of Problems and Their Solutions

Authors: Theophilus Agama
Comments: 6 Pages.

In this paper, we study the topology of problems and their solution spaces developed introduced in our first paper. We introduce and study the notion of separability and quotient problem and solution spaces. This notions will form a basic underpinning for further studies on this topic.
Category: Set Theory and Logic

Infinity and Infinite Induction

Authors: Jincheng Zhang
Comments: 15 Pages.

What is infinity? What principles should be adhered to when researching infinity? For a predicate on a natural number system, is it true that finite holds, and infinite holds? This paper reexamines the nature of infinity and proposes two opposite infinite axioms (δ+1=δ or δ+1≠δ). Based on these two infinite axioms, the "infinite induction" of the identity formula is proved; it is found that the infinite axioms in the ZF system do not satisfy the equality axiom, and there are many contradictions in the reasoning of the Cantor ordinal number. The ordinal theory of set theory ZF system is not strict. It is hoped that the mathematics community will pay attention to these questions and give a convincing answer.
Category: Set Theory and Logic

Inconsistency Defended

Authors: Jim Rock
Comments: 1 Page.

Here two contradictory arguments are defended. They can be developed in any formal system containing sets, arithmetic, and relations between the rational numbers.
Category: Set Theory and Logic

Logical Proof of the Millennium Puzzle P-vs-NP

Authors: Yanhong Yang
Comments: 3 Pages.

The existence of the premise: S=1，Then there is at least one P Problem(Computer-table problems)equal to NP problem (The solution is a computer-verifiable problem) then: P←→NP(P=NP). Thus, For all the class P and class NP problems，There are P=NP，also have P≠NP.
Category: Set Theory and Logic

The Paradox and Uncloseness of Calculation Theorem

Authors: Jincheng Zhang
Comments: 18 Pages.

For a long time, there is a "diagonal method of proof" dominating the mathematics field; with it, Russel finds the paradox of set theory; with it, Cantor proves that "the power set of natural numbers is uncountable" and " the set of real numbers is uncountable"; with it , Gödel proves that " natural number system PA is incomplete"; with it , Turning proves that " halting problem" is undecidable and proves that " there is non-recursive sets on sets of natural numbers" in recursion theory and so on; proofs of these significant propositions all apply the same mathematic method which is praised as " a golden diagonal". On the basis of analyzing paradoxes, the paper finds that paradoxes are unclosed terms on closed calculus (that is extra-field term). Classical logic system cannot handle such extra-field terms, so it is transformed to the logic systems SL, SK that may handle unclosed calculus. It can be found that "diagonal proof method" is to construct paradoxes in nature through further analysis, and it is an unclosed proof method, which can prove that real numbers constructed by Cantor’s "diagonal proof method are extra-field terms which will not affect count-ability of sets of real numbers; The Gödel’s undeterminable proposition is an extra-field term, which will not affect completeness of system PA. The undeterminable Turing machine in the Turing halt problem is also an extra-field term. So, the proof that real number is uncountable is wrong; the proof of Gödel’s incomplete theorem and diagonal method of proof, all of them are wrong, should be completely corrected.
Category: Set Theory and Logic

Why is the Gödel Self-Referential Equation Unsolvable?

Authors: Jincheng Zhang
Comments: 13 Pages.

There exists a Gödel number for each formula of the system N of natural numbers. The Gödel undecidable proposition, which is also a formula of the system N, also exists a Gödel number p; at the same time, the Gödel undecidable proposition is a self-referential proposition u([p]) substituted into its own Gödel number, and the self-referential proposition u([p]) Gödel number is also p, i.e., there is, u([p])=p. It can be This equation has no solution.The traditional view is that the Gödel undecidable proposition u([p]) is a closed formula and is a natural number proposition; we here transform the Gödel self-referential proposition into a self-referential equation and find that this equation has no solution and the Gödel undecidable proposition u([p]) is not a natural number proposition. u([p]) is an unclosed term (out-of-domain term) that evolves on the set of natural numbers and u([p]) is not a closed formula.
Category: Set Theory and Logic

Some Remarks on Accessibility and Extended Modal Realism

Authors: Ryan J. Buchanan
Comments: 6 Pages.

This paper includes a formulation of extended modal realism and a theorem concerning its equivalence with a certain mathematical universe.
Category: Set Theory and Logic

��-Small Subobjects and Universal Categories

Authors: Ryan J. Buchanan
Comments: 5 Pages.

This paper develops the notion of ��-smallness as proposed by Barwick and Haine. This allows the author to investigate pointlike topological spaces from a category-theoretic perspective, by considering manifolds of negative dimension as cardinally inaccessible k-subobjects.
Category: Set Theory and Logic

Cardinality Between Natural and Real Numbers

Authors: Federico Gabriel
Comments: 1 Page.

Is the set of real numbers not denumerable?
Category: Set Theory and Logic

A Logical Problem

Authors: Elmar Guseinov
Comments: 1 Page. In Russian

Alpaca (A), Bull (B), Cat (C), Dog (D), Elephant (E) and Fox (F) inhabit each of three planets. Each of sixty people visited each of the planets and made friends there with exactly two inhabitants. It's known that the new friends of a person formed the whole set {A,B,C,D,E,F}. Also, any two inhabitants of different species living on different planets have exactly eight mutual human friends. For each person, determine with which inhabitants of each of the planets he is friends.
Category: Set Theory and Logic

The 3rd Incompleteness Theorem Disproving Riemann Hypothesis

Authors: Rayd Majeed Al-Shammari
Comments: 5 Pages.

There is a hidden limits in our mathematics itself that's make us cannot keep counting to infinity not because our human species incompetent but because in reality our numbers by itself are finite, in fact we will never have infinite numbers not just because we are incapable of counting to infinity but because in our mathematics there is no such thing. Numbering is not just counting, numbering is counting that’s holds a definitive value but infinity is undefined so no number could be a represent for infinity. Infinity of numbers cannot exist, because any number you think of no matter how big it’s in the end it will have value then it will be define but infinity is undefined. If we have infinite numbers then there summation will give us a well definitive value and that's would be closer to zero than to infinity and by this infinity just cannot be exist and this what I will prove in this work and to certify this theory I will use it to disprove Riemann hypothesis among other.
Category: Set Theory and Logic

A Bipolar Fuzzy Extension of the MULTIMOORA Method

Authors: Dragisa Stanujkic, Assia Bakali, Darjan Karabasevic, Edmundas Kazimieras Zavadskas, Florentin Smarandache, Willem K.m. Brauers
Comments: 18 Pages.

The aim of this paper is to make a proposal for a new extension of the MULTIMOORA method extended to deal with bipolar fuzzy sets. Bipolar fuzzy sets are proposed as an extension of classical fuzzy sets in order to enable solving a particular class of decision-making problems. Unlike other extensions of the fuzzy set of theory, bipolar fuzzy sets introduce a positive membership function, which denotes the satisfaction degree of the element x to the property corresponding to the bipolar-valued fuzzy set, and the negative membership function, which denotes the degree of the satisfaction of the element x to some implicit counter-property corresponding to the bipolar-valued fuzzy set. By using single-valued bipolar fuzzy numbers, the MULTIMOORA method can be more efficient for solving some specific problems whose solving requires assessment and prediction. The suitability of the proposed approach is presented through an example.
Category: Set Theory and Logic

About Nonstandard Neutrosophic Logic (Answers to Imamura’s "Note on the Definition of Neutrosophic Logic")

Authors: Florentin Smarandache
Comments: 16 Pages.

In order to more accurately situate and fit the neutrosophic logic into the framework of nonstandard analysis, we present the neutrosophic inequalities, neutrosophic equality, neutrosophic infimum and supremum, neutrosophic standard intervals, including the cases when the neutrosophic logic standard and nonstandard components T, I, F get values outside of the classical unit interval [0, 1], and a brief evolution of neutrosophic operators. The paper intends to answer Imamura’s criticism that we found benefic in better understanding the nonstandard neutrosophic logic — although the nonstandard neutrosophic logic was never used in practical applications.
Category: Set Theory and Logic

Introduction to the Complex Refined Neutrosophic Set

Authors: Florentin Smarandache
Comments: 5 Pages.

In this paper, one extends the single-valued complex neutrosophic set to the subsetvalued complex neutrosophic set, and afterwards to the subset-valued complex refined neutrosophic set
Category: Set Theory and Logic

Conjunto Plitogénico, Una Extensión de Los Conjuntos Crisp, Difusos, Conjuntos Difusos Intuicionistas Y Neutrosóficos Revisitado

Authors: Florentin Smarandache
Comments: 19 Pages. Spanish

En el presente artículo, introducimos el conjunto plitogénico (como generalización de conjuntos nítidos, borrosos, intuicionistas, borrosos y neutrosóficos), que es un conjunto cuyos elementos se caracterizan por los valores de muchos atributos. Un valor de atributo v tiene un grado correspondiente (difuso, intuicionista difuso o neutrosófico) de pertenencia d (x, v) del elemento x, al conjunto P, con respecto a algunos criterios dados.
Category: Set Theory and Logic

Extension of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set

Authors: Florentin Smarandache
Comments: 3 Pages.

In this paper, we generalize the soft set to the hypersoft set by transforming the function F into a multi-attribute function. Then we introduce the hybrids of Crisp, Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Hypersoft Set.
Category: Set Theory and Logic

Grado de Dependencia e Independencia de Los (Sub) Componentes de Conjuntos Borrosos Y Neutrosóficos

Authors: Florentin Smarandache
Comments: 6 Pages. Spanish

La introducción del grado de dependencia (y en consecuencia el grado de independencia) entre los componentes del conjunto difuso, y también entre los componentes del conjunto neutrosófico, se introduce por primera vez en la quinta edición del libro de Neutrosofía en el año 2006, basado en los elementos descritos en dicha edición del libro, se comienza a conocer conceptos de conjuntos neutrosóficos de los componentes borrosos así como los grados de dependencia e independencia.
Category: Set Theory and Logic

Neutrosophic Actions, Prevalence Order, Refinement of Neutrosophic Entities, and Neutrosophic Literal Logical Operators

Authors: Florentin Smarandache
Comments: 6 Pages.

In this paper, we define for the first time three neutrosophic actions and their properties. We then introduce the prevalence order on {T, I, F} with respect to a given neutrosophic operator "o", which may be subjective - as defined by the neutrosophic experts; and the refinement of neutrosophic entities. Then we extend the classical logical operators to neutrosophic literal logical operators and to refined literal logical operators, and we define the refinement neutrosophic literal space.
Category: Set Theory and Logic

Neutrosophic Systems and Neutrosophic Dynamic Systems

Authors: Florentin Smarandache
Comments: 19 Pages.

In this paper, we introduce for the first time the neutrosophic system and neutrosophic dynamic system that represent new per-spectives in science. A neutrosophic system is a quasi- or (��, ��, ��)—classical system, in the sense that the neutrosophic system deals with quasi-terms/concepts/attributes, etc. [or (��, ��, ��) − terms/ concepts/attributes], which are approximations of the classical terms/concepts/attributes, i.e. they are partially true/membership/probable (t%), partially indeterminate (i%), and partially false/nonmember-ship/improbable (f%), where ��, ��, �� are subsets of the unitary interval [0,1]. {We recall that ‘quasi’ means relative(ly), approximate(ly), almost, near, partial(ly), etc. or mathematically ‘quasi’ means (��, ��, ��) in a neutrophic way.}
Category: Set Theory and Logic

Operadores Con Conjunto Neutrosóficos de Valor único Oversets, Undersets Y Offset

Authors: Florentin Smarandache
Comments: 4 Pages. Spanish

Neutrosophic Over-/Under-/Off-Set and Logic were defined for the first time in 1995 and published in 2007. During 1995-2016 was presented them to various national and international conferences and seminars. These new notions are totally different from other sets/logics/probabilities. We extended the neutrosophic set respectively to Neutrosophic Overset {when some neutrosophic component is > 1}, to Neutrosophic Underset {when some neutrosophic component is < 0}, and to Neutrosophic Offset {when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component > 1 and other neutrosophic component < 0}.
Category: Set Theory and Logic

Plithogenic Set, an Extension of Crisp, Fuzzy, Intuitionistic Fuzzy, and Neutrosophic Sets — Revisited

Authors: Florentin Smarandache
Comments: 14 Pages.

In this paper, we introduce the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), which is a set whose elements are characterized by many attributes (parameters)’ values. An attribute value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x,v) of the element x, to the set P, with respect to some given criteria. In order to obtain a better accuracy for the plithogenic aggregation operators in the plithogenic set, and for a more exact inclusion (partial order), a (fuzzy, intuitionistic fuzzy, or neutrosophic) contradiction (dissimilarity) degree is defined be-tween each attribute value and the dominant (most important) attribute value. The plithogenic intersection and union are linear combinations of the fuzzy operators tnorm and tconorm, while the plithogenic complement, inclusion (inequality), equality are influenced by the attribute values contradiction (dissimilarity) degrees.
Category: Set Theory and Logic

Practical Applications of the Independent Neutrosophic Components and of the Neutrosophic Offset Components

Authors: Florentin Smarandache
Comments: 15 Pages.

The newly introduced theories, proposed as extensions of the fuzzy theory, such as the Neutrosophic, Pythagorean, Spherical, Picture, Cubic theories, and their numerous hybrid forms, are criticized by the authors of [1]. In this paper we respond to their critics with respect to the neutrosophic theories and show that the DST, that they want to replace the A-IFS with, has many flaws.
Category: Set Theory and Logic

Soft Set Product Extended to HyperSoft Set and IndetermSoft Set Product Extended to IndetermHyperSoft Set (updated)

Authors: Florentin Smarandache
Comments: 4 Pages.

In this paper we define the Soft Set Product as a product of many soft sets and afterwards we extend it to the HyperSoft Set. Similarly, the IndetermSoft Product is extended to the IndetermHyperSoft Set. We also present several applications of the Soft Set Product to Fuzzy (and fuzzy-extensions) Soft Set Product and to IndetermSoft Set and IndetermHyperSoft Set.
Category: Set Theory and Logic

The Score, Accuracy, and Certainty Functions Determine a Total Order on the Set of Neutrosophic Triplets (T, I, F)

Authors: Florentin Smarandache
Comments: 14 Pages.

In this paper we prove that the Single-Valued (and respectively Interval-Valued, as well as Subset-Valued) Score, Accuracy, and Certainty Functions determine a total order on the set of neutrosophic triplets (T, I, F). This total order is needed in the neutrosophic decision-making applications.
Category: Set Theory and Logic

Tripleta de Estructura Neutrosófica Y Tripleta de Estructura Neutrosófica Extendida

Authors: Florentin Smarandache
Comments: 7 Pages.

En el presente estudio se realiza una revisión de las tripletas de estructura neutrosófica y tripleta de estructura neutrosófica extendida, con el fin de introducir nuevos conceptos a emplear en trabajos futuros.
Category: Set Theory and Logic

Exploration of the Factors Causing Autoimmune Diseases using Fuzzy Cognitive Maps with Concentric Neutrosophic Hypergraphic Approach

Authors: Nivetha Martin; Florentin Smarandache; I.Pradeepa; N.Ramila Gandhi; P.Pandiamma
Comments: 7 Pages.

Neutrosophic sets are comprehensively used in decision making environment. The manifestation of neutrosophic sets in concentric hypergraphs is proposed in this research work. The intention of developing a decision making model using the combination of Fuzzy Cognitive Maps and concentric neutrosophic hypergraph is to rank the core factors of decision making problem and find the inter relational impacts. This proposed model is validated with the exploration of the causative factors of autoimmune diseases. The proposed model is highly compatible as it assists in determining the core factors and their inter association. This model will certainly benefit the decision maker at all managerial levels to design optimal decisions.
Category: Set Theory and Logic

Introduction to the IndetermSoft Set and IndetermHyperSoft Set

Authors: Florentin Smarandache
Comments: 22 Pages.

In this paper one introduces for the first time the IndetermSoft Set, as extension of the classical (determinate) Soft Set, that deals with indeterminate data, and similarly the HyperSoft Set extended to IndetermHyperSoft Set, where ‘Indeterm’ stands for ‘Indeterminate’ (uncertain, conflicting, not unique outcome). They are built on an IndetermSoft Algebra that is an algebra dealing with IndetermSoft Operators resulted from our real world. Afterwards, the corresponding Fuzzy / Intuitionistic Fuzzy / Neutrosophic / and other fuzzy-extension IndetermSoft Set & IndetermHyperSoft Set are presented together with their applications.
Category: Set Theory and Logic

Introducción a Los Conjuntos Indetermsoft e Indetermhypersoft
Introduction to the Indetermsoft and Indetermhypersoft Sets

Authors: Florentin Smarandache
Comments: 22 Pages. In Spanish

This paper presents for the first time the IndetermSoft Set, as an extension of the classical (determinate) Soft Set, which operates on indeterminate data, and similarly the HyperSoft Set extended to the IndetermHyperSoft Set, where 'Indeterm' means 'Indeterminate' (uncertain, conflicting, non-unique result). They are built on an IndetermSoft Algebra which is an algebra dealing with IndetermSoft Operators resulting from our real world. Subsequently, the IndetermSoft and IndetermHyperSoft Sets and their Fuzzy/Fuzzy Intuitionistic/Neutrosophic and other fuzzy extensions and their applications are presented.
Category: Set Theory and Logic

Introduction to Plithogenic Logic as Generalization of Multivariate Logic

Authors: Florentin Smarandache
Comments: 7 Pages.

A Plithogenic Logical proposition P is a proposition that is characterized by many degrees of truth-values with respect to many corresponding attribute-values (or random variables) that characterize P. Each degree of truth-value may be classical, fuzzy, intuitionistic fuzzy, neutrosophic, or other fuzzy extension type logic. At the end, a cumulative truth of P is computed.
Category: Set Theory and Logic

Supervisory Control of Discrete Event Systems with State-dependent Observability

Authors: Peng Wang, Xiang-Yun Wang, Kai-Yuan Cai
Comments: 9 Pages.

This paper considers a new class of discrete event systems under partial observations. The problem is presented within the background of a manufacturing process where workpieces are loaded and transported, and this process is controlled with the partial information collected by sensors. The model extracted is novel because the observation of an event does not only depend on an event itself, but also the state where the system stays. Two standard problems are discussed in this paper: supervisor existence problem and supervisor synthesis problem. With a natural revision of observable languages, a necessary and sufficient condition is given for the existence of a supervisor. For supervisor synthesis problem, two algorithms are developed: one algorithm is to check the properties of a control specification given by a regular language,and the other one is to synthesize a supervisor if the properties hold. Within the background of manufacturing systems, an example is illustrated to show how the algorithms are applied to practical computing.
Category: Set Theory and Logic

Improved Definition of Non-Standard Neutrosophic Logic and Introduction to Neutrosophic Hyperreals

Authors: Florentin Smarandache
Comments: 20 Pages. In Spanish

In the fifth version of our reply article [26] to Imamura's critique, we recall that Neutrosophic Non-Standard Logic was never used by the neutrosophic community in any application, that the quarter-century old (1995-1998) neutrosophic operators criticized by Imamura were never used as they were improved soon after, but omits to talk about their development, and that in real-world applications we need to convert/approximate the hyperreals, monads and bi-nads of Non-Standard Analysis to tiny intervals with the desired precision; otherwise they would be inapplicable. We pointed out several errors and false statements by Imamura [21] regarding the inf/sup of nonstandard subsets, also Imamura's "rigorous definition of neutrosophic logic" is incorrect, as is his definition of nonstandard unit interval, and we showed that there is no total order in Neutrosophic Computing and Machine Learning , Vol. 23, 2022 Florentin Smarandache, Definición mejorada de la lógica neutrosófica no estándar e introducción a los hiperreales neutrosóficos (Quinta versión) 2 the set of hyperreals (due to the recently introduced Neutrosophic Hyperreals which are indeterminate), so the Transfer Principle from R to R* is questionable. After his critique, several reply posts on non-standard theoretical neutrosophy followed in 2018-2022. As such, I extended the Nonstandard Analysis by adding the right-closed left monad, the left-closed right monad, the punctured binad (which we introduced in 1998), and the nonpunctured binad - all in order to close the newly extended nonstandard space (R*) under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations [23, 24]. Improved definitions of the Nonstandard Unitary Interval and Nonstandard Neutrosophic Logic are presented, along with Nonstandard Neutrosophic Operators.
Category: Set Theory and Logic

Practical Applications of Indetermsoft Set and Indetermhypersoft Set and Introduction to Treesoft Set as an Extension of the Multisoft Set

Authors: Florentin Smarandache
Comments: 9 Pages.

The IndetermSoft Set is as an extension of the Soft Set, because the data, or the function, or the sets involved in the definition of the soft set have indeterminacy - as in our everyday life, and we still need to deal with such situations. And similarly, IndetermHyperSoft Set as extension of the HyperSoft Set, when there is indeterminate data, or indeterminate functions, or indeterminate sets. Herein, ‘Indeterm’ stands for ‘Indeterminate’ (uncertain, conflicting, incomplete, not unique outcome). We now introduce for the first time the TreeSoft Set as extension of the MultiSoft Set. Several applications are presented for each type of soft set.
Category: Set Theory and Logic

Proof of the Existence of Dark Numbers

Authors: Wolfgang Mückenheim
Comments: 8 Pages. In English and German

We will prove by means of Cantor's mapping between natural numbers and positive fractions that his approach to actual infinity implies the existence of numbers which cannot be applied as defined individuals. We will call them dark numbers.
Category: Set Theory and Logic

One Second Crucial Theorem for the Refoundation of Elementary Set Theory and the Teaching of that Discipline to Future Generations

Authors: Nhat-Anh Phan
Comments: 15 Pages. Copyright. All rights reserved.

For a given infinite countable set constituted by the union of infinitely many non-empty, finite or infinite countable, disjoint sub-sets A = {i∈N*}[A_i]_P(A) , ∀i ∈ N* , A_i ≠ ∅, ∀i, j ∈ N* , i ≠ j, A_i ∩ A_j = ∅, we demonstrate that it is legitimate to partition A in a finite or infinite number of sub-sets that are themselves constituted by a finite or infinite number of sub-sets of A when the initial order of indexation of the sub-sets of A maintains a strictly increasing order in each sub-set.
Category: Set Theory and Logic

On the Theory of Problems and Their Solution Spaces

Authors: Theophilus Agama
Comments: 6 Pages.

We introduce and develop the logic of existence of solution to problems. We use this theory to answer the question of Florentin Smarandache in logic. We answer this question in the negative.
Category: Set Theory and Logic

Etude on Nontrivial Collatz Cycles

Authors: Savinov Sergey
Comments: 1 Page.

The study explores the features of non-trivial Collatz cycles.
Category: Set Theory and Logic

Rebuilding Set Theory

Authors: Hongyi Li
Comments: 12 Pages.

This paper gives a new definition of set and introduces the concept of elastic set with variable extension. Without any unprovable or unreliable proposition, hypothesis or conjecture, it eliminates Russell's paradox, Cantor's paradox, Galileo's paradox and Infinite Hotel's paradox. Reveals the essence of infinityand gives the expression of the number of elements of infinite set, Corrects the error that an infinite set can correspond to its proper subset one by one. At the same time, the hypothesis that there is an unique natural numbers set that has included all natural numbers has been refuted. It is pointed that the diagonal Argument and Cantor theorem have not proved that real numbers can not be one-to-one corresponded to natural numbers. Since the set is variable, we can also use the variable things such as variables or functions as the elements of the set.There are so many mistakes in the most strict mathematics, let alone the humanities? It shows that the wisdom of modern Westerners, who lack the gene of cultural diversity, is far less than that of the ancient Greek era, or even degenerated to the barbaric era. No wonder a good world would fight for some differences that may not be meaningful, and even foolishly walked to the edge of nuclear war!
Category: Set Theory and Logic

One First Crucial Theorem for the Re-foundation of Elementary Set Theory and the Teaching of That Discipline to Future Generations

Authors: Nhat-Anh Phan
Comments: 14 Pages. Copyright. All rights reserved.

For a given infinite countable set A = U_{i∈N∗}{a_i }, ∀i ∈ N∗ , a_i≠∅, we demonstrate that it is legitimate to partition A in a finite or infinite number of infinite countable sub-sets when the initial order of indexation of the elements of A maintains a strictly increasing order in each sub-set. At this occasion we introduce two new formalism allowing to signify the fact that it is A that has been partitioned and the fact that A and the latter partition of A belong to the same class comprising infinitely many infinite countable sets constituted by the same infinite countable sub-sets that constitute the partition of A.
Category: Set Theory and Logic

Presentation of the Unlimited Transference of Pairs Method: an Alternative to Bijections

Authors: Juan Carlos Caso Alonso, Francisco Mario Cruz Almeida
Comments: 61 Pages. In Spanish - I can be contacted at recursos.clja@gmail.com

First of all, each point of this project, has been checked as OK, unofficially, for at least, two different persons that said they were mathematicians. The problem is that they are not the same group of persons, and each one believes the mistake is in a different point, while others consider that concrete point, as totally correct. Each comment, each guessing you can made, probably have been taken into account. The unique problem of this project is its crazy goal and the incapacity to put in the same room all the people I have talked sometime.We are going to present an alternative technic to compare infinite cardinalities. Different from bijections or injections, and not totally equivalent in potential, able to work when the previous ones can't. Applied to an obvious case we are going to create. After that, show the equivalents points between the obvious example and the real example we really want to study: P(N) vs N. To show how P(N) has not a cardinality bigger than N.I would like to remember now the first paragraph.We can apply the technic to different sets, with different natures and transfinite cardinalities. Always compared with N.The injectivity concept says to us that is impossible to create a relation r: A -> B, where |B| < |A|, and r being injective. In every possible relation, we must have a minimum quantity of pairs of elements from AXA, with the same image. At least one pair. The core of the idea of the TPI technic (Unlimited Transference of Pairs) is to create a mathematically correct process that shows that minimum is not bigger than 0. Considering the case of 0 pairs with same image, the case of a perfect injection. Remember: It will not be the same as an injective relation.The technic must be considered as valid. And after that, other documents will be recovered, each one per each equivalence, between the technic I will show in this document, and the real case we want to study really. Each equivalence must be considered as valid too. Finally, we will apply the same conclussion we obtained in the obvious example, to the real case.
Category: Set Theory and Logic

A Formal Grammar of Propositional Calculus

Authors: Elmar Guseinov
Comments: 7 Pages.

Практическое значение синтаксического определения теоремы как элемента множества, порождаемого дедуктивной системой, заключается в возможности доказательства любой теоремы компьютером с неограниченным временем работы. Одним из способов непосредственной реализации данной идеи является построение формальной грамматики дедуктивной системы. Так, при автоматическом доказательстве теорем методами машинного обучения в качестве обучающей выборки мы могли бы указать множество пар вида (X,Y), где X — теорема, а Y — порождающая X последовательность правил вывода формальной грамматики. В данной статье мы ограничимся построением грамматики исчисления высказываний. Полученная грамматика типа 0 иерархии Хомского порождает язык, словами которого являются все тавтологии.

Defined syntactically, a theorem is a word generated by a deductive system. In practice, this means that each theorem can be proven by a sufficiently long working computer. A possible implementation of this idea is based on the construction of a formal grammar of the deductive system. In automated theorem proving based on the machine learning, we then can use pairs (X,Y) as the training data, where X stands for a theorem and Y is a sequence of inference rules of the formal grammar corresponding to X. This article provides the formal grammar of propositional calculus, namely, type 0 grammar producing all tautologies.
Category: Set Theory and Logic

A Formal Proof of De Morgan's Laws for Boolean Algebras

Authors: Elmar Guseinov
Comments: 5 Pages.

В статье приведено формальное доказательство законов де Моргана для булевых алгебр.

The article provides a formal proof of de Morgan’s laws for Boolean algebras.
Category: Set Theory and Logic

High School Set Theory Using Technology

Authors: Timothy W. Jones
Comments: 9 Pages.

Simple set theory problems of finding unions, intersections, and complements of simple sets consisting of integers and single letters is not a trivial exercise to automate with technology. It can be done using TI-Basic programming in the case of sets consisting of positive integers. As a TI-84 doesn't have associative arrays or just arrays for that matter doing the same for single letters of the alphabet is just about impossible. But the task can be done in Python pretty elegantly. Why bother? Well the next evolution is to see the need and power of a database and its structured query language (SQL). This tie-in of basic set theory and databases seems to never be made. It is an important connection.
Category: Set Theory and Logic

Correct Setting of Mapping Function in One-to-One Correspondence

Authors: Hongyi Li
Comments: 3 Pages.

In order to strictly discuss the one-to-one correspondence between the elements of sets, the number of elements that can participate in the correspondence was first discussed, and then the necessary conditions for the formation of injection and bijection were discussed. According to these conditions, it wss found that some mapping functions do not satisfy these conditions. For example, it is impossible to obtain a mapping function satisfying these conditions between any infinite set and its any proper subset.
Category: Set Theory and Logic

Correct Definition of the Set of Natural Numbers

Authors: Hongyi Li
Comments: 2 Pages.

It was proved that a set that already contains all natural numbers does not exist in real mathematical world because the concept of all natural numbers does not exist in the world. The correct definition of the set of natural numbers is a set that contains infinite natural numbers 1,2,3.... but can never contain all natural numbers. There are many sets of infinite natural numbers with different sizes. Thus, it is true that the subsets of N cannot correspond one-to-one with N, but it is false that the subsets cannot correspond one-to-one with any set of natural numbers.
Category: Set Theory and Logic

A Method to Check the Reliability of One-to-One Correspondence

Authors: Hongyi Li
Comments: 2 Pages.

The method of checking whether the established one-to-one correspondence is correct was given. It is proved by the verification that any infinite set cannot be in one-to-one correspondence with any of its proper subsets.Key words: mathematic fundation; infinite set; proper subset; one-to-one correspondence; check method
Category: Set Theory and Logic

Cantor Diagonal Argument for Real Numbers

Authors: Richard L. Hudson
Comments: 4 Pages.

This analysis shows Cantor's diagonal argument published in 1891 cannot form a sequence that is not a member of a complete set.
Category: Set Theory and Logic

New Approach to the Hilbert's Hotel Paradox

Authors: Juan Elias Millas Vera
Comments: 2 Pages.

In this short article I wanted to expand the Hilbert’s Grand Hotel Paradox. Building on the first two statements, I add a third, possibly necessary, statement.
Category: Set Theory and Logic

Sampling of Infinite Set by J-sequence: Solving Bertrand's (Chord) Paradox

Authors: Jeonghoon Lee
Comments: 20 Pages.

In an infinite set, probabilities are defined on the structure of the set rather than on individual elements. We should take into account the property of a σ-algebra where probabilities are defined. A σ-algebra is closed under ‘only countable’ unions, and the axioms of probability assume σ-additivity. If this is overlooked, something bizarre could be happened as the proposed three solutions of Bertrand's problem. Bertrand's problem is not a paradox, but well defined(posed). The suggested three solutions have the common problem of dividing the sample space into an uncountably infinite number of sets and treating them equally. If a set is divided into equal(treated) and uncountable infinity, all the divided sets have probability 0, so calculating conditional probabilities with these sets or comparing them with each other becomes meaningless. In a sample space composed of an uncountably infinite number of elements such as [0,1], after calculating the number of cases using the sets(J-sequence m-collection cover) generated by equally dividing the sample space into finite numbers, the probability of an event can be calculated with its limit value(as m becomes infinite). The answer of Bertrand's problem is 1/3.
Category: Set Theory and Logic

The Arrew Theorem Prover

Authors: Boro Sitnikovski
Comments: 1 Page.

Arrew (Arrow Rewriter) is a mathematical system (theorem prover) that allows expressing and working with formal systems. It relies on a simple substitution rule and set equality to derive theorems.
Category: Set Theory and Logic

Against Finitism a Criticism of Norman J Wildberger

Authors: Commie Cantor
Comments: 5 Pages.

If you study mathematics you are probably aware of the foundational crises that mathematics went through at the beginning of the 20th century. The three broad schools of thought namely constructivism, intuitionism and formalism collided and judging by the approach used today by most mathematicians, we can easily say that formalism emerged victorious in some sense. However while debates regarding the foundations of mathematics have subsided over the years, they aren’t dead. One such school of mathematics which still sees considerable traffic is finitism. In this article, we will be analysing the criticism of a finitist named Norman J Wildberger and trying to defend the current axiomatic mathematical systems against them.
Category: Set Theory and Logic

Prime Numbers on Sets

Authors: Mar Detic
Comments: 2 Pages.

All natural numbers are the universal set and all composite numbers are the subsets thus prime numbers only exist in universal set but not in any subset.
Category: Set Theory and Logic

Quantity Theory and the Continuum Hypothesis

Authors: Ron Ragusa
Comments: 12 Pages.

How far away is the nearest galaxy to the Milky Way? How long is the distance from Boston to New York? How deep is the Marianas Trench? How high is the tallest building? How many marbles are in that bag? The above questions all require measuring quantities to arrive at an answer. Measuring the number of light-years from the Milky Way to the other galaxies in the local group will reveal which is closest. Measuring the number of miles from Boston to New York will reveal the distance between them. The Marianas Trench will contain a measurable number of feet that will be its depth. The tallest building will contain a measurable number of feet that will be its height. Counting the marbles will yield the quantity of marbles contained in the bag. While the units of measure of quantities and methods of obtaining measurements vary, all measurable quantities have one thing in common; they can all be expressed as real numbers.
Category: Set Theory and Logic

On the Need to Generalize the Theory of Algorithms

Authors: Aleksey A. Demidov
Comments: 13 Pages. In Russian (Correction made by viXra Admin to conform with the rules of viXra.org)

Traditionally, the concept of an algorithm is introduced into the theory through a sequence of elementary steps leading to the solution of a problem, and parallel algorithms are considered as a technical solution external to the Theory of Algorithms, which allows speeding up the execution process. However, a number of physical processes currently used for computing, such as quantum computing, do not fit into the framework of the predictions of the Theory of Algorithms, in particular --- in terms of computational complexity, which suggests that our understanding of parallel computing processes, limited by the framework of the classical Theory of Algorithms, may not be complete. A qualitative leap in the Theory of Computability is possible if parallel algorithms are understood as a generalization of the classical ones within the framework of the hypothetical Theory of Parallel Algorithms. In this paper, pre-quantum physical processes are considered, which are already beyond the scope of the classical Theory of Algorithms. Conceptual primitives suitable for the analysis of parallel flows are proposed.
Category: Set Theory and Logic

Remarks on the Axiom of Comprehension

Authors: Albert Henrik Preiser
Comments: 2 Pages.

Remarks on the axiom of comprehension.
Category: Set Theory and Logic

Bayes Theorem

Authors: James Edwin Rock
Comments: 1 Page.

A simple explanation of the use of Bayes Theorem with AIDS testing results.
Category: Set Theory and Logic

Internal Set Theory Ist^{}based on Gyper Infinitary Logic with Restricted Modus Ponens Rule

Authors: Jaykov Foukzon
Comments: 20 Pages.

In this paper we deal with set theory NC_{∞}^{} based on gyper infinitary logic with Restricted Modus Ponens Rule.Nonconservative extensions of the canonical internal set theories IST and HST are proposed.
Category: Set Theory and Logic

Inconsistent Number Systems,nowel Development.

Authors: Jaykov Foukzon
Comments: 60 Pages.

In this paper we dealin using paraconsistent first order logic LP_{ω}^{#}=∪_{n<ω}LP_{n}^{#} with restricted modus ponens rule and infinite levels of a contradiction [1]-[4], where LP_{n}^{#} is an paraconsistent first order logic with n levels of a contradiction.
Category: Set Theory and Logic

A Paradox of “Adjacent” Real Points and Beyond

Authors: Ke Zhang
Comments: 5 pages. Email: alspa@163.com

We reveal adjacent real points in the real set using a concise logical reference. This raises a paradox, as the real set is believed as existing and complete. However, we prove each element in a totally ordered set has adjacent element(s); there is no densely ordered set. Furthermore, since the natural numbers can also be densely ordered under certain ordering, the set of natural numbers, which is involved with each infinite set in ZFC set theory, does not exist itself.
Category: Set Theory and Logic

A Basis for Set Theory Without the Use of an Axiom.

Authors: Albert Henrik Preiser
Comments: 2 Pages.

A basis for set theory without the use of an axiom.
Category: Set Theory and Logic

Demonstration of a Technique to Construct a One-to-One Correspondence Between N and the Infinite Binary Decimals in (0,1)

Authors: Ron Ragusa
Comments: 8 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

In this paper we will see how by varying the initial conditions of the Cantor's demonstration we can use the Diagonal Method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the open interval (0, 1).
Category: Set Theory and Logic

Ip(in)=in

Authors: Viola Maria Grazia
Comments: 1 Page.

In this page I will prove that IP(IN)=IN using the Zermelo’s natural Numbers costruction.
Category: Set Theory and Logic

Natural Sets Embedded in a System of Axioms.

Authors: Albert Henrik Preiser
Comments: 78 Pages.

To this day, the prevailing view is that in set theory, the selection of things based on their properties leads to contradictions. If the formation and existence of sets is based on consistent compliance with the requirements that exist when using the all-quanor, no contradictions can be identified. Consistent compliance with these requirements is ensured in this thesis with the help of a system of axioms. As a result, we have a basis for set theory and the recognition of ideas inherent in so-called "naive set theory".
Category: Set Theory and Logic

Interval Quadripartitioned Neutrosophic Sets

Authors: Surapati Pramanik
Comments: 9 Pages.

Quadripartitioned neutrosophic set is a mathematical tool, which is the extension of neutrosophic set and n-valued neutrosophic refined logic for dealing with real-life problems. A generalization of the notion of quadripartitioned neutrosophic set is introduced. The new notion is called Interval Quadripartitioned Neutrosophic set (IQNS). Quadripartitioned neutrosophic set is developed by combining the Quadripartitioned neutrosophic set and interval neutrosophic set. Several set theoretic operations of IQNSs, namely, inclusion, complement, and intersection are defined. Various properties of set-theoretic operators of IQNS are established.
Category: Set Theory and Logic

Asymmetry in the Real Number Line and: A Proof that \pi + e is an Irrational Number

Authors: Alexander C Sarich
Comments: 7 Pages.

The set of all Real numbers, R, consists of all Rational numbers, Q, being any ratio of two Integer numbers that does not divide by 0. All other Real numbers that are not a Rational number are contained in the set of Irrational numbers, R/Q. These two subsets comprising all of the Real numbers are known to have distinct cardinalities of differing magnitudes of infinity[2]. When a consecutive ordering of all Rational numbers is established, whereby any unique Rational number can be shown to be disconnected from all other Rational numbers[3], a theorem regarding asymmetry on the Real number line is established. This theorem simplifies the necessary requirements to prove that the summation of two known Irrational numbers is Rational or Irrational.
Category: Set Theory and Logic

Logic and Intuition

Authors: Bertrand Wong
Comments: 3 Pages.

This paper aims to inspire thinking on the capabilities and potential of the human brain. The brain apparently has great potential for development and great untapped capabilities. Practically everyone is keen on improving his mental capacity, especially the capability of logical reasoning, that is, the ability in utilising logic to achieve the desired outcomes. It appears that logic is equated with intelligence and is regarded as the most important aspect of thinking by many (though emotional intelligence is now the new kid in the block which seems to be gaining traction). The author here looks at reasoning or logic, as well as intuition, from a different and unique perspective.
Category: Set Theory and Logic

Some Inconsistence in Logic

Authors: Bertrand Wong
Comments: 5 Pages.

This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Godel’s incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have pride of place. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, et al. Perhaps, by Godel’s incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Godel’s incompleteness theorems will have ramifications in other areas involving logic. The paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics.
Category: Set Theory and Logic

Invalidating Cantor’s Continuum Hypothesis and Solving Hilbert’s #1 Problem

Authors: Stephane H. Maes
Comments: 4 Pages. Discussions at https://shmaes.wordpress.com/2021/07/18/invalidating-cantors-continuum-hypothesis-and-solving-hilberts-1-problem/

In this short paper, we provide a mathematical proof that in set theory, developed in a mathematical universe following the ZFC axioms, Cantor’s continuum hypothesis does not hold and Gödel had the right hunch: the cardinality of the infinity of the set of all reals is א2, and not א1, i.e., two infinity orders away from the cardinality of the infinite set of naturals, א0. The proof is derived from combinatorics, relying on ZFC solely for the model of Cantor and Gödel defining א0. It provides input to the still unresolved first of Hilbert famous 23 math problems of interest. This paper, resolves the first of the 23 Hilbert problems with invalidation of the continuum hypothesis.
Category: Set Theory and Logic

Modification of Cantor Sets With Potential Infinities

Authors: Anders Lindman
Comments: 3 Pages.

In continuous Euclidean space all lines have an infinite number of points, e.g. a line A = 10 cm has the same number of points as a line B = 5 cm. In this paper a new set theory (MST for modified set theory) is defined so that lines of different lengths always contain different numbers of points. Instead of allowing several actual infinities only one actual infinity is defined. All other sets are either finite or have potential infinite cardinality. This makes the logic of sets more straightforward than with Georg Cantor’s transfinite sets.
Category: Set Theory and Logic

One Fundamental Theorem Concerning Infinite Countable Sets

Authors: Nhat-Anh Phan
Comments: 22 Pages. Copyright : All Rights Reserved

For A an infinite countable set containing infinitely many distinct natural integers and B an infinite countable set containing infinitely many distinct natural integers such that ∀n ∈ A,n ∈ B and ∀m ∈ B,m ∈ A, we demonstrate that it is possible that A is not equal to B by exposing infinitely many counter-examples in which, for each counterexemple, A and B are respectively two sample spaces of two probability spaces having different probabilities for similar events. We thus prove that the axiom of extensionality is false for infinite countable sets.
Category: Set Theory and Logic

Set Theoretic Proof ~~X ≠ X

Authors: Salvatore Gerard Micheal
Comments: 2 Pages. [Corrections are made by viXra Admin to comply with the rules of viXra.org]

We prove that ~~X ≠ X where ~ = "not" in a logical/set-theoretic context (ALL mathematics and logic), X represents ANY logical statement equivalent to a set of associated facts (which many times is countably infinite or more), ~X, read "not X", represents the logical / set-theoretic complement of X, which is comprised of the complementary set of associated facts with respect to X. We give a proof by contradiction and the solitary exception to the rule regarding phi, the null/empty set.
Category: Set Theory and Logic

A Rigorous Examination on Cantors Diagonal Argument

Authors: Hongyi Li
Comments: 5 Pages.

It was proved in this paper that diagonal argument actually only proves that the real numbers can not be listed as complete, which has nothing to do with if the real numbers are countable or not. The confusion of the two different concepts - Countable and “complete listing” - is the reason why the diagonal argument fails. It was proved that real numbers are countable.
Category: Set Theory and Logic

The Fog Covering Cantor's Paradise: Some Paradoxes on Infinity and Continuum

Authors: Ke Zhang
Comments: 18 pages. Email: alspa@163.com

We challenge Georg Cantor's theory about infinity. By attacking the concept of “countable/uncountable” and diagonal argument, we reveal the uncertainty, which is obscured by the lack of clarity. The problem arises from the basic understandings of infinity and continuum. We perform many thought experiments to refute current standard views. The results support the opinion that no potential infinity leads to an actual infinity, nor is there any continuum composed of indivisibles statically, nor is Cantor's theory consistent in itself.
Category: Set Theory and Logic

A New Approach to Set Theory, Version 1.5

Authors: Thomas Limberg
Comments: 9 pages; language: German; for receiving the program "Demonstration of the Limberg calculus", version 1.2, email me!

We introduce a simple logical calculus called Limberg Calculus and two attached derivation sequences. The derivation sequences contain a proof, that empty and universal set exist (under the Limberg Calculus).
Category: Set Theory and Logic

Infinity and Numerical Magic

Authors: Antonio Leon
Comments: 7 Pages.

This chapter analyzes a supertask that makes it disappear numbers from a table that contains the list of natural numbers in their natural order of precedence.
Category: Set Theory and Logic

A Critique of Self-Reference: What Gödel Theorem Really Proves

Authors: Antonio Leon
Comments: 9 Pages.

This article introduces a new perspective for the analysis of self-referential sentences, and proves the conditions under which they are inconsistent. The Liar Paradox, Grelling-Nelson Paradox, Russell's Predicate Paradox, Russell's Set Paradox and Richard Paradox are proved to meet such conditions. The same is proved of the ordinary language interpretation of Gödel's undecidable formula if the corresponding formal calculus is complete. In consequence, Gödel's Theorem VI only holds if that calculus is not complete, which makes the theorem unnecessary. All proofs and arguments in this article are developed within the framework of a simplified system of ordinary logic also defined in this paper.
Category: Set Theory and Logic

Proving the Axiom of Choice of Subsets

Authors: Ralf Donau
Comments: 1 Page.

A proof of The Axiom of Choice of Subsets proposed by Antoine Balan.
Category: Set Theory and Logic

Set Theory Inc# # Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.ii) Hyper Inductive Definitions

Authors: Jaykov Foukzon
Comments: 25 Pages.

In this paper intuitionistic set theory INC# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.
Category: Set Theory and Logic

The Axiom of Choice of Subsets

Authors: Antoine Balan
Comments: 1 page, written in french

We propose a new axiom, stronger than the axiom of choice, called the axiom of choice of subsets.
Category: Set Theory and Logic

New Definitions of the Classification of Numbers

Authors: Juan Elias Millas Vera
Comments: 2 Pages.

Using definitions and properties of sets I made a complete classification of all possible existing numbers. I used different variables and define them as known sets.
Category: Set Theory and Logic

Cantor's Paradox

Authors: Antonio Leon
Comments: 11 Pages.

This paper discusses Cantor’s paradox of the set all cardinals, and proves that in Cantor’s set theory every set of cardinal C originates at least 2C inconsistent infinite sets.
Category: Set Theory and Logic

Infinity Put to the Test: Towards a Discrete Revolution in the Mathematics of the XXI Century

Authors: Antonio Leon
Comments: 404 Pages. [Corrections made by viXra Admin to conform with the requirements on the Submission Form]

From different areas of mathematics, such as set theory, geometry, transfinite arithmetic or supertask theory, in this book more than forty arguments are developed about the inconsistency of the hypothesis of the actual infinity in contemporary mathematics. A hypothesis according to which the uncompletable lists, as the list of the natural numbers, exist as completed lists. The inconsistency of this hypothesis would have an enormous impact on physics, forcing us to change the continuum space-time for a discrete model, with indivisible units (atoms) of space and time. The discrete model would be a great simplification of physical theories, including relativity and quantum mechanics. It would also suppose the solution of the old problem of change, posed by the pre-Socratics philosophers twenty-seven centuries ago.
Category: Set Theory and Logic

Set Theory Inc# Based on Intuitionistic Logic with Restricted Modus Ponens Rule.

Authors: Jaykov Foukzon
Comments: 7 Pages.

In this article Russell’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.
Category: Set Theory and Logic

Sense Theory [Part 5]: Sense Space

Authors: Egger Mielberg
Comments: 23 Pages. [Note by viXra Admin: Title must be consistant on the Submission Form and in the article]

A purely sense-to-sense connection between two arbitrary objects would allow trillions of other objects to be associated with each other by their specific sense. The main benefit of the sense connection is still lost if an object of certain nature does not still have a property's association to even a single another object (-s). We propose a new paradigm of a mathematical space that is sense-centered and AI-focused in its nature. One of the main purposes of this space are to create an informational sphere for a a massive number of heterogeneous objects by the existed sense relationship between them. It may be extremely useful if a life event (behavior) in the future of one person, say buying a car could be described by another life event (behavior) in the past of another person, say the baby's a birthday celebration or the properties of physical phenomena in the past will trigger some actions in the future.
Category: Set Theory and Logic

There Are Questions with Hidden Answer, But Without a Way to Find the Answer

Authors: Dmitri Martila
Comments: 3 Pages. Rejected by many top journals without review

Re-proof of Gödel incompleteness theorem. I use no mathematical expressions.
Category: Set Theory and Logic

Cantor Diagonal Argument

Authors: Antonio Leon
Comments: 9 Pages.

This paper proves a result on the decimal expansion of the rational numbers in the open rational interval (0, 1), which is subsequently used to discuss a reordering of the rows of a table T that is assumed to contain all rational numbers within (0, 1), in such a way that the diagonal of the reordered table T could be a rational number from which different rational antidiagonals (elements of (0, 1) that cannot be in T ) could be defined. If that were the case, and for the same reason as in Cantor’s diagonal argument, the open rational interval (0, 1) would be non-denumerable, and we would have a contradiction in set theory, because Cantor also proved the set of rational numbers is denumerable.
Category: Set Theory and Logic

Hilbert Machine

Authors: Antonio Leon
Comments: 7 Pages.

Inspired by the emblematic Hilbert Hotel, Hilbert machine is a conceptual super-machine whose functioning questions the consistency of the actual infinity hypothesis subsumed into the Axiom of Infinity.
Category: Set Theory and Logic

Zeno Dichotomies

Authors: Antonio Leon
Comments: 10 Pages.

This chapter introduces a formalized version of Zeno’s Dichotomy in its two variants (here referred to as Dichotomy I and II) based on the discreteness and separation of ω-order (Dichotomy I) and of ω∗-order (Dichotomy II) defined below in this section. Each of these formalized versions leads to a contradiction pointing to the inconsistency of the hypothesis of the actual infinity.
Category: Set Theory and Logic

Thomson Lamp

Authors: Antonio Leon
Comments: 13 Pages.

The argument of Thomson lamp and Benacerraf’s critique are reexamined from the perspective of the w-order legitimated by the hypothesis of the actual infinity subsumed into the Axiom of Infinite. The conclusions point to the inconsistency of that hypothesis.
Category: Set Theory and Logic

Thomson Lamp Formalized

Authors: Antonio Leon
Comments: 3 Pages.

The discussions on Thomson’s lamp analyzed in the precedent chapter can be formalized (at least up to a certain point) by introducing a simple symbolic notation that allows to define the lamp and its functioning in abstract terms. The symbolic definition can then be used to develop formulas that represent the functioning laws of the lamp. Being independent of the number of times the lamp is turned on/off, these laws represent the universal attributes and the universal behaviour of a Thomson’s lamp. As we will see, some of those laws are not compatible with the assumption that a Thomson’s lamp can be switched infinitely many times during a finite interval of time. This conclusion proves that, as its author defended, Thomson supertask is inconsistent.
Category: Set Theory and Logic

Supertasks, Physics and the Axiom of Infinity

Authors: Antonio Leon
Comments: 34 Pages.

It seems reasonable to assume that mathematical infinity was not the objective of Zeno’s Dichotomy (in any of its variants), however, a sort of mathematical infinity was already present in these celebrated arguments. Aristotle proposed a first solution to Zeno’s Dichotomy by introducing what we now call one-to-one correspondences, the key instrument of modern infinitist mathematics. But Aristotle, more naturalist than platonic, finally rejected the method of pairing the elements of two infinite collections (in this case of points and instants) and introduced instead the distinction between actual and potential infinities. Aristotle’s distinction served to define, gross modo, two opposite positions on the nature of infinity for more than twenty centuries. The actual infinity was finally mathematized through set theory in the first years of the XX century and the discussions on its potential or actual nature almost vanished. But, as we will see here, things still remain to be said on this issue.
Category: Set Theory and Logic

Sense Theory (Part 6): Sense Diagrams

Authors: Egger Mielberg
Comments: 12 Pages.

Simple and readable diagrams of a complex set of any kind would allow a million of connections between elements of that set to be formulated and understood clearly. Current diagrams provide the visual solution for the first four-five sets mostly, but it is unreadable for the number of sets in more than five. We propose a solution, sense diagrams, for visualization of multimillion sets using sense-to-sense paradigm [1]. The nature of the set can be any. The nature of the elements of a set may differ from each other. The main criterion of the use of this diagrams is the presence of qualitative or/and quantitative properties of an element of the set.
Category: Set Theory and Logic

About Structure of a Connected Quaternion-JULIA-Set and Symmetries of a Related JULIA-Network

Authors: Udo E. Steinemann
Comments: 20 Pages.

If a variable is replace by its square and subsequently enlarged by a constant during a number of iteration-steps in quaternion-space, a network of (3) sets will be built gradually. As long as for the iteration-constant certain conditions are fulfilled, the network will consist of: an unbounded set (escape-set) with trajectories escaping to infinity during course of the iteration, a bounded set (prisoner-set) with trajectories tending to a sink-point and a further bounded one (JULIA-set) with a fixed-point as repeller having a repulsive effect on all points of both the other sets. The iteration will continue until the attracting sink-point of prisoner-set and the repelling fixed-point on JULIA-set have been found. This situation is reached if predecessor- and successor-state of the iteration became equal. The fixed-point-condition provisionally formulated in general terms of quaternions, can be separated into (3) sub-conditions. When heeding the HAMILTONian-rules for interactions of the imaginary sub-spaces of the quaternion-space, each sub-condition will be appropriate for one imaginary sub-spaces and independently debatable. Knowledge of fixed-points from this fundamental network will one enable to study the structure of a connected JULIA-set. 12The Iteration will start from (1) on real-axis, this is not a restriction on generality because an appropriate scaling on real-axis can always be archived this way. It will become obvious, that the fixed-points in prisoner- and JULIA-set will depend on the iteration-constant only. Thus (16) different constants chosen appropriately will enable to arrange (16) fixed-points of JULIA-sets in the square-points of a hyper-cube and thereby together with the JULIA-sets to built a related JULIA-network. The symmetry-properties of this related JULIA-network can be studied on base of a hyper-cube's symmetry-group extended by some additional considerations.
Category: Set Theory and Logic

The Clique Problem with Collaborative Variables (P=NP)

Authors: John Archie Gillis
Comments: 15 Pages.

The present paper provides a novel approach to solving the clique problem(s). The present methods will work for any clique problem, including those which are determined to be NP-Complete. Determining other cliques, such as cliques of a fixed size (k=3, k=4, etc.) is trivial by comparison but will also be described. The author provides a means for greatly reducing the time that it will take a computer (or human) to solve for: 1. Maximum clique (a clique with the largest possible number of vertices), 2. Listing all maximal cliques (cliques that cannot be enlarged), and 3. Solving the decision problem of testing whether a graph contains a clique larger than a given size. To solve the clique problem, the author feels that we must completely discard previous graphing methods and start from scratch with his new and novel strategy.
Category: Set Theory and Logic

Math: An Invention or a Discovery?

Authors: Joseph Palazzo
Comments: 6 Pages.

Math is a process in which both invention and discovery are essential ingredients in its development. But which one comes first?
Category: Set Theory and Logic

A Premise for the Natural Density of Prime Number Power Multisets: Continuum Hypothesis and Canonical Countability Assertions Refuted

Authors: Derek Tucker
Comments: 2 Pages.

The current canonical treatment of multisets to our knowledge does not discuss their powersets, but such powersets have a natural connection linking prime and natural numbers. Using the natural definition of continuity, and the natural extension of the formula for counting elements in a powerset to count power multisets, we find prime numbers are countably infinite, with an infinitude of infinities with natural density specified by the Riemann zeta function, between the primes and the continuous natural numbers.
Category: Set Theory and Logic

A Propositional Logic Review of Descartes’ Phrase ”Cogito, Ergo Sum”

Authors: Vasilis Valatsos
Comments: 1 Page.

"Cogito, ergo sum" is a Latin phrase used by Rene Descartes as a philosophical proposition, which became a fundamental axiomatic truth within the boundaries of Western Philosophy, and was used to assert the reality of one's own mind. In this paper, we try to review this statement using First Order Propsitional Logic.
Category: Set Theory and Logic

Skolem Revisited - Thoralf's Achilles Heel

Authors: Hannes Hutzelmeyer
Comments: 12 Pages.

The author has developed an approach to logics that comprises, but also goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from geometry, number theory, recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers and a precise analysis of foundation of mathematics in general, including theory of types. A famous paper by Thoralf Skolem of 1934 is usually put at the beginning of publications on non-standard arithmetic. A critical investigation shows that it has serious, if not even insurmountable problems. Firstly one notices that it is based on second-order logic, it has unary and binary function-variables, and binary operator-constants (that map two functions to a function). It seems strange that one makes a fundamental statement about first-order logic systems using second order. In proving Satz 1 on the asymptotic behavior of arithmetic functions Skolem has some inaccuracies and formal errors. These minor problems can be solved by diligent work. But even if one has replaced his metalingual use of his relation symbols Bi by an ontologically correct method there remains secondly the problem of transitivity of the minority relation of functions that is neglected by Skolem. Thirdly, in constructing the strictly ascending function g of Satz 1 use is made of recursion by a dot-dot-dot notation. This is not an admissible procedure in object-language, although there may be a correct way to solve the problem in metalanguage. Therefore one does not only need second-order logic in combination with a precise object-language (in order to avoid ontology problems) but also a precise use of metalanguage (in order to avoid dot-dot-dot) for justifying Skolem's Satz 1 after one has eventually solved the transitivity problem; Skolem's Satz 2 would then be valid. However, as long as Satz 1 is not confirmed it does not pay to treat Satz 3 and Satz 4, leaving open the existence of non-standard models of arithmetic on the basis of Skolem's work
Category: Set Theory and Logic

The Diagonalization Paradox

Authors: Ron Ragusa
Comments: 8 Pages.

In 1891 Georg Cantor published his Diagonal Argument which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers.In this paper we will see how by varying the initial conditions of the demonstration we can use Cantor’s method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the open interval (0,1).
Category: Set Theory and Logic

First Incompleteness Theorem New Way of Proving

Authors: Dmitri Martila
Comments: 2 Pages.

Demonstrated in alternative way, that the First Theorem of Gödel is true, and holds not only for some special mathematical problems, but in general.
Category: Set Theory and Logic

Cantor Diagonalization & Countability of Real Numbers, Famous Paradoxes E.g.richard,russell,skolem,liar , Set Theory,zfc, in Relativistic Mathematical Space-Time

Authors: Pankaj Mani
Comments: 19 Pages.

In this paper, we try to revisit some of most foundational issues lying at the foundation of mathematics in space-time relativistic perspective ,rather than conventional absolute space. We are adding a new dimension to the mathematics i.e. Time and view mathematics in Space-Time frame to sort out major issues lying at the foundations in conventional mathematics and making it in line with physical world realities. We will look at the famous Cantor’s Diagonalization approach in Space-time relativistically to show the Countability of Real Numbers and Inifniteness . We aill also look to resolve famous paradoxes e.g. Richard, Russell,Liar, Skolem in Space-time perspective . We will also look at the the foundation of Set theory in Space-time to restore the issues that led to ZFC by elimination. As a consequence, these results will lead to look at Godel Incompleteness theorems for Real Numbers also in Space-Time.
Category: Set Theory and Logic

Rhind Papyrus Problem no 48 Squaring a Circle in 4-D Spacetime

Authors: Manfred U.E. Pohl
Comments: 3 Pages.

The ToE Framework that unites quantum theory with gravitation „Solution to the Problem of Time“ [1] is based on the Solution of the Black-Hole Information Paradox, namely the squaring of a circle (π) in space-time. It is well known how to “square” a circle over an additional dimension, as shown in „Solution to the Problem of Time“ II +III [2]. In addition to the Essay “It takes a Decision to Decide if Decidability is True or False” [3] (concering Gödels incompleteness and impossibility of Hilbert’s Programm) it is shown here the Solution to the Problem No. 48 in the Rhind-Papyrus. Keywords: Pi, God, Unified Principle, Gödel’s Incompleteness, Foundation of Mathematics, TOE.
Category: Set Theory and Logic

Number Theory Beyond Frege on the Necessity of Open Arity

Authors: Hannes Hutzelmeyer
Comments: 13 Pages.

A closer look at mathematical proofs led Gottlob Frege to realize that Aristotle's syllogism logic was not sufficient for many theorems. He developped what today is called first-order predicate logic. It is usually thought that predicate logic is sufficient for the theory of natural numbers. However, this first step of modern logic development again is not sufficient. One needs another step, especially to allow for so-called open arity of arrays. This second step cannot be done in general in object-language based on predicate logic but only by metalanguage. Therefore one needs something like the FUME-method (put forward by the author) which allows for a precise treatment of both language levels. Dot-dot-dot … is not admissible in predicate logics as it needs some kind of recursion. In metalanguage, however, one has to introduce some basic recursion right from the setup (but it is much weaker than primitive recursion). For natural numbers two examples are given, one for a concrete version of Robinson arithmetic and one for recursive arithmetic. Without the second step to metalanguage one cannot express some of the most important so-called theorems of number theory in a direct fashion, leave alone prove them. Actually some are not theorems but metatheorems. The examples comprise Chinese remainders, Gödel's beta-function, little Gauss's summing up of numbers, Euclid's unlimited primes and the canonical representation of a natural number (fundamental theorem of natural arithmetic). After one has included the second step which allows one to talk about open arities in metalanguage one can tackle the problem of talking about number-arrays in object language. One can do this to a certain extent by coding number-arrays by (usually) two numbers. This can be done even in Robinson arithmetic using 'Gödel's beta-function'. But one has to make use of the second step before one can return to object-language. Of course, the introduction of two tiers, i.e. object-language and metalanguage, is necessary for many other areas of mathematics, if not to say, most of them.
Category: Set Theory and Logic

Непригодность общепринятых стереотипных тавтологий классической математики для доказательства теорем при классическом подходе к доказательству теорем

Authors: Viktoria Kondratenko
Comments: 9 Pages.

Оторванность гипотетических теорий от реалий живой материи стала причиной проникновения мистики в научные теории. При мистическом мышлении идея применения аналитического метода решения задач познания в голову не приходит. Диалектическая логика в отличие от мистики утверждает обратное: любые проблемные задачи познания жизнедеятельности процессов и явлений мироздания разрешимы исключительно аналитическим путѐм, при этом единственным методом. Автором создана универсальная и формальная теория решения интеллектуальных (т.е. не имеющих заранее известных алгоритмов решения) задач, связанных с познанием жизнедеятельности естественных и рукотворных процессов в любых явлениях мироздания – метод аксиоматического моделирования Кондратенко, эффективность которого достигается путѐм корректной постановки задачи и еѐ решения чисто формальным методом. Корректность постановки задачи означает, прежде всего, признание несостоятельности всех гипотетических (не подтверждѐнных результатами натурного экспериментирования с предметом познания) теорий. В качестве примера в статье рассматривается парадокс при классическом подходе к доказательству теорем, состоящий в непригодности общепринятых стереотипных тавтологий классической математики для доказательства теорем.
Category: Set Theory and Logic

Формальная модель диалектической логики управления жизнеобеспечивающими функциями психики человека

Authors: Viktoria Kondratenko
Comments: 9 Pages.

Диалектическая логика управления любыми функциональными системами организма человека вытекает из концептуальных знаний о жизнедеятельности этих функциональных систем, добытых исключительно путём натурного экспериментирования с ними. Поэтому концептуальные знания о жизнедеятельности каждой исследуемой функциональной системы человека должны предшествовать описанию сущности логики управления ею. Психика человека является одной из важнейших функциональных систем организма человека. В статье представлены основополагающие концепты жизнедеятельности этой системы, которые позволяют выявить сущность диалектической логики управления её жизнеобеспечивающими функциями, и формальная модель диалектической логики управления функциональной системой психики человека
Category: Set Theory and Logic

A Refutation Of Ilija Barukˇci ́c’s Modus Inversus AndConclusions Following From It

Authors: Nhef Luminati
Comments: 6 Pages.

In this paper the general validity of the methodological approach of frequent vixra.org contributor Ilija Barukčić are examined and refuted.
Category: Set Theory and Logic

Instruction Set Completeness Theorem: Concept, Relevance, Proof, and Example for Processor Architecture

Authors: William F. Gilreath
Comments: 22 Pages. Published in the General Science Journal

The Instruction Set Completeness Theorem is first formally defined and discussed in the seminal work on one-instruction set computing—the book Computer Architecture: A Minimalist Perspective. Yet the original formalism of the Instruction Set Completeness Theorem did not provide a definitive, explicit mathematical proof of completeness, analyze both singular and plural instruction sets that were either complete or incomplete, nor examine the significance of the theorem to computer architecture instruction sets. A mathematical proof of correctness shows the equivalence of the Instruction Set Completeness Theorem to a Turing machine, a hypothetical model of computation, and thereby establishes the mathematical truth of the Instruction Set Completeness Theorem. With a more detailed examination of the Instruction Set Completeness Theorem develops several surprising conclusions for both the instruction set completeness theorem, and the instruction sets for a computer architecture.
Category: Set Theory and Logic

Modus Inversus – if (Premise is False) Then (Conclusion is False)

Authors: Ilija Barukčić
Comments: 9 pages. Copyright © 2018 by Ilija Barukčić, Jever, Germany. Published by:

Objective: When theorems or theories are falsified by a formal prove or by observations et cetera, authors respond many times by different and sometimes inappropriate counter-measures. Even if the pressure by which we are forced to believe in different theories although there are already predictively superior rivals to turn to may be very high, a clear scientific methodology should be able to help us to assure the demarcation between science and pseudoscience. Methods: Karl Popper’s (1902-1994) falsificationist methodology is one of the many approaches to the problem of the demarcation between scientific and non-scientific theories but relies as such too much only on modus tollens and is in fact purely one-eyed. Results: Modus inversus is illustrated in more detail in order to identify non-scientific claims as soon as possible and to help authors not to hide to long behind a lot of self-contradictory and sometimes highly abstract, even mathematical stuff. Conclusions: Modus inversus prevents us from accepting seemingly contradictory theorems or rules in science. Keywords: Science, non-science, modus inversus. E-Mail: Barukcic@t-online.de
Category: Set Theory and Logic

A Flaw in Separation Axioms

Authors: Hannes Hutzelmeyer
Comments: 18 Pages.

The author has developed an approach to logics that comprises, but also goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from geometry, number theory, recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers etc. . The conventional treatment of axiomatic set theory (ZFC) is replaced by the abstract calcule sigma so that certain shortcomings can be avoided by the use of Funcish-Mencish language hierarchy: - precise talking about formula strings necessitates a formalized metalanguage - talking about open arities, general tuples, open dimensions of spaces, finite systems of open cardinality and so on necessitates a formalized metalanguage. 'dot dot dot … ' just will not do - the Axiom of Infinity is generalized in order to allow for certain other infinite sets besides the natural number representation according to von Neumann (i.e. general recursion) - the Axioms of Separation is modified as it seems more convenient - there are only enumerably many properties that can be constructed from formula strings, as these are finite strings of characters from a finite alphabet; this should be kept in mind in connection with the Axioms of Replacement - a new look at Cantor's continuum hypothesis in abstract axiomatic set theory leads to the question of so-called basis-incompleteness versus proof-incompleteness - the Axioms of Separation seem to have a flaw; there is a caveat for axiomatic set theory.
Category: Set Theory and Logic

The Problem P=NP

Authors: Thierry DELORT
Comments: 6 Pages.

In this article, we are going to solve the problem P=NP for a particular kind of problems called basic problems of numerical determination. Nonetheless, this solution can be generalized to all problems belonging to class P or NP. We are going to propose 3 fundamental Axioms permitting to solve the problem P=NP, but those Axioms can also be considered as pure logical assertions, intuitively evident and never contradicted, permitting to understand the solution of the problem P=NP. Indeed, we will see that the conclusion of this article solves the considered problem.
Category: Set Theory and Logic

The Existence of Equilibrium States in Dynamic Systems with Attractive Interaction

Authors: Elizabeth Lemeshko
Comments: 4 Pages. Text in Ukrainian. Mohyla Mathematical Journal, Vol 1 (2018) http://mmj.ukma.edu.ua/article/view/152602

Nowadays, science is characterized by needs of the study of various complex processes and phenomena’s. Today’s research of complex and dynamical systems is one of the most advanced ways of research and evolution of the modern world. Models of biology and ecology, physical models, various economic and social models are typical examples of dynamic systems. The concept of an interactive complex system in modern science is a main tool for construction of mathematical models for solving modern civilization problems and development. The dynamical systems approach to conflict is relatively new, but it has beginning in different research fields. Theory of dynamic systems helps us to understand the experiments, build the mathematical model of iterations and examine behavior and relations between opponents, like distribution of resources and territory, population growing etc. This is a challenging problem of finding and achieving a compromise state for opponents on a common territory has different options to define the task and to choose conflict interaction. In 2016, the monograph by V. Koshmanenko where was introduced new approach for dynamic system of conflict that based on interactions of the opponents in the form probability distribution in the disputed area was published. In particular, presented the concept of a complex dynamic system with attractive interaction. The relevance of this research is improving new dynamical system and researching for a new application of abstract models in everyday life. In this paper briefly fundamentals of the theory of dynamical systems described and the theorem on the existence of a equilibrium state in a the new, perspective for research, dynamical system with attractive interaction in terms of probability distributions (measures) and their densities, formulated and proved.
Category: Set Theory and Logic

Visualizing the Distributions and Isosurfaces of Some Standard and Non-Standard Quaternion Fractal Sets

Authors: Shawn Halayka
Comments: 16 Pages.

The length, displacement, and magnitude distributions and isosurfaces related to the trajectories of the points in a quaternion fractal set are visualized.
Category: Set Theory and Logic

The Continued Fractions and the Real Numbers

Authors: Antoine Balan
Comments: 1 page, written in english

We give a new definition of the real numbers by mean of the continued fractions.
Category: Set Theory and Logic

The Proof is in the Pudding: an Outline of New Proof of the Existence of God

Authors: Victor Christianto, Robert Neil Boyd
Comments: 6 Pages. This paper has been submitted to a journal. Comments are welcome

Starting with a review of few known arguments to prove the existence of God, we discuss our argument i.e. Nature's order, Pascal's void and Arrow of Time as Neutrosophic triadic to prove the existence of God. The most convincing one is what we call : the proof is in the pudding, i.e. how direct experience with God is the only way to fill everyone's inner void (cf. Pascal). To write shortly, our spiritual inner void can be filled by direct experience with God. This is what we suggest: the proof is in the pudding.
Category: Set Theory and Logic

Понятие множества и его элементов

Authors: Анатолий Вайчунас
Comments: 2 Pages. in Russian. Только e-meil диалог = onli e-meil dialogue

Понятие множества и его элементов полагается, в теории множеств, интуитивно известным и неопределяемо исходным. Однако, оно является обобщаемым на основе таких понятий как многообразие и разнообразие математических предметов.
Category: Set Theory and Logic

The Logical Content of Inequalities

Authors: Ilija Barukčić
Comments: 28 pages. Copyright © 2019 by Ilija Barukčić, Jever, Germany. All rights reserved. Published by:

Objective. From a theoretical point of view, the demarcation between science and (fantastical) pseudoscience is in necessary for both practical and theoretical reasons. One specific nature of pseudoscience in relation to science and other categories of human reasoning is the resistance to the facts. Methods. Several methods are analyzed which may be of use to prevent that belief is masqueraded genuinely as scientific knowledge. Results. Modus ponens, modus tollens and modus conversus are reanalyzed. Modus sine, logically equivalent to modus ponens is developed. Modus inversus and modus juris are described in detail. Conclusions. In our striving for knowledge, there is still much more scientific work to be done on the demarcation between science and pseudoscience. Keywords: non strict inequality, quantum mechanics, science, pseudoscience
Category: Set Theory and Logic

The Rebuttal of Colin James III's Refutation of Vidamor Cabannas’ Theory of Objectivity

Authors: Vidamor Cabannas
Comments: 11 Pages. More information and comments on Theory of Objectivity and Author Vidamor Cabannas can be found at www.theoryofobjectivity.com

I analyzed the refutation of the Theory of Objectivity performed by Colin James III in vixra.org/abs/1904.0549 and verified that the applied method as well as the values found were correct. However, there is an error in the conclusion, as it refutes the Theory of Objectivity instead of confirming it. The values found in Colin James III's refutation confirm the exact findings of the Theory of Objectivity. Therefore, as the values that were found confirm those presented by the theory, these values are a confirmation and not a refutation of the theory. In other words, the equation N + 1 = n – 1 is tautological, confirming the existence of a geometric entity that occurred before the onset of the universe, which the Theory of Objectivity calls Nothing. However, Nothing is a geometric entity incompatible with the existence of the universe, which forms a contradiction with non-tautological values at the atomic level. This is because in the era of Nothing, there is no space nor any other element other than the geometrical point known as Nothing. There is no reference. This geometric Nothing does not signify absolute zero and has an informative value. The Theory of Objectivity uses a logical and geometric model to demonstrate how the spherical point called Nothing transformed itself into universal space. Thus, it proves that absolute Nothing does not exist.
Category: Set Theory and Logic

Representation of Processive Functions in Robinson Arithmetic ?

Authors: Hannes Hutzelmeyer
Comments: 10 Pages.

In connection with his so-called incompleteness theorem Gödel discovered the beta-function. The beta-function theorem is important for the representation of recursive functions in the concrete calcule ALPHA of Robinson arithmetic. The other features are composition and minimization of primitive recursive functions. Recursive functions are no part of Robinson arithmetic, but they are representable by certain formulae. The author has developed an approach to logics that comprises, but goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers and so on. The concrete calcule LAMBDA of a natural number arithmetic with first-order logic has been defined by the author. It includes straight recursion and composition of functions, it contains a wide range of so-called compinitive functions, with processive functions far beyond primitive recursive functions. They include e.g. Ackermann's function and similar constructions. All recursive functions (that are obtained by minimization too) can be represented in LAMBDA . As long as there is no proof that all processive functions are minimitive recursive (recursive but not primitive recursive) one has the problem of representing them in concrete calcule ALPHA of Robinson arithmetic. As long as the challenge of such a proof is not met there is the conjecture that there are calculative functions that are not representable in Robinson arithmetic. An abstract calcule alphakappa of Robinson-Crusoe arithmetic shows that there exists an even weaker adequate arithmetic than Robinson's.
Category: Set Theory and Logic

Recursive Inaccessibility and the Inductive Hypothesis as to Countable Sets in ZFC

Authors: Amel Mara
Comments: 32 Pages.

The significance of the Inductive Hypothesis is examined with respect to the Principle of Mathematical Induction. A few relevant theorems that involve functions in set theory are specified with respect to the Inductive Hypothesis. The countability of rational numbers is reviewed, as to Cantor’s "intuition" (i.e., the "zig-zag" method of enumerating rational numbers) and constructive formulas that would map the set of natural numbers to a subset of the rational numbers (e.g., a multiplicative inverse function, a divisive function). Von Neumann and Zermelo ordinals are introduced to support the definition of a non-dense, well-ordered set of numbers. It is determined that for a specific transfinite set of ordinals with a maximal element that is a limit ordinal, the set must contain at least one successor ordinal that cannot be recursively accessed in a finite number of steps from a specified base ordinal.
Category: Set Theory and Logic

Higher Categorical Structures in Gödelian Incompleteness: Towards a Topos-Theoretic Model of Metamathematical Limitations

Authors: Paul Chun Kit Lee
Comments: 34 Pages.

This paper presents a novel geometric approach to Gödelian incompleteness phenomena using higher category theory and topos theory. We construct a hierarchy of (∞, 1)-categories that model formal systems as multidimensionalspaces, transforming logical structures into geometric objects. This framework allows us to represent Gödel’s incompleteness results as topological features—singularities or holes—in the fabric of mathematical space. Our use of (∞, 1)-categories is crucial for modeling the higher-order relationships between proofs and metaproofs, providing a natural setting for analyzing self-reference and reflection principles. These logical conceptsare transformed into geometric structures, offering new insights into the nature of incompleteness. We develop a topos-theoretic model that serves as a universal vantage point for surveying the landscape of formal systems. From this perspective, we prove a generalized incompleteness theorem that extends Gödel’sresults to a broader class of formal systems, now interpreted as geometric obstructions in the topos. Leveraging homotopy type theory, we establish a precise correspondence between proof-theoretic strength and homotopical complexity. This connection yields a novel complexity measure for formal systems based onthe geometric properties of their corresponding spaces. Our framework provides new insights into the nature of mathematical truth and the limits of formalization. It suggests a more nuanced view of the hierarchy of mathematical theories, where incompleteness manifests as an intrinsic topological feature of the space of theories.While primarily theoretical, our approach hints at potential applications in theoretical computer science, particularly in complexity theory. We also discuss speculative connections to fundamental questions in physics and cognitive science, presented as avenues for future research. By recasting Gödelian phenomena in geometric terms through higher category theory, we open new avenues for understanding the nature of mathematical reasoning and its inherent limitations. This geometric perspectiveoffers a powerful new language for exploring the foundations of mathematics and the boundaries of formal systems.
Category: Set Theory and Logic

About Countability Sets

Authors: Tomasz Soltysiak
Comments: 7 Pages.

Dispute Cantor's theorem about power sets for infinite sets.Proof of equal number elements of sets of natural and real numbers. Theorem about countability of all sets.
Category: Set Theory and Logic

First Case of Mistake in Cantor’s Proof: Pure Double Contradiction Technique

Authors: Juan Carlos Caso Alonso
Comments: 39 Pages.

The proof of Cantor's Theorem ( |A| < |P(A)| ), changes depending to which mathematician you ask. There are two versions of it. The one I call "Creating an extern element", and the other one I call "Pure double contradiction".This document will try to explain the difference between them, and after that show that the second one should not be never used, for being "not reliable".In case you are interested, you have an extra chapter, with informal data about the first one.
Category: Set Theory and Logic

Any Formal System That Contains Sets Arithmetic and Rational Numbers is Inconsistent

Authors: Jim Rock
Comments: 1 Page.

Gӧdel proved that any formal system containing arithmetic is incomplete. We show that any such formal system is inconsistent. We establish a collection of nested sets of rational numbers in a descending hierarchy. The sets higher in the descending hierarchy contain element(s) that are not in the sets below them in the hierarchy. Given such a descending set hierarchy, it is easy to develop two arguments that contradict each other. The conclusion of Argument#2 is false. But, Argument#2 is a valid argument.
Category: Set Theory and Logic

First Principles of Mathematics

Authors: Forrest C. Taylor
Comments: 100 Pages. CC BY-NC-ND 4.0 international license

A dependent type theory is proposed as the foundation of mathematics. The formalism preserves the structure of mathematical thought, making it natural to use. The logical calculus of the type theory is proved to be syntactically complete. Therefore it does not suffer from the limitations imposed by Gödel’s incompleteness theorems. In particular, the concept of mathematical truth can be defined in terms of provability.
Category: Set Theory and Logic

ZFC Inconsistency Explained and Defended

Authors: Jim Rock
Comments: 1 Page.

There is a class of sets that can be constructed within ZFC that both have and do not have a largest element. Two contradictory arguments about these properties are developed and defended.
Category: Set Theory and Logic

An Inconsistent Hierarchy of Sets in [0, 1]

Authors: Jim Rock
Comments: 1 Page. Contains a reply to a common objection made about 2303.0105 v1.

Two contradictory arguments are developed from a hierarchy of sets in [0, 1]. One argument is a proof by contradiction and its conclusion is true. The other argument is an existence argument and while its conclusion is not true, it follows logically from the a valid assumption followed by three true statements that precede the conclusion.
Category: Set Theory and Logic

On Problems and Their Solution Spaces

Authors: Theophilus Agama
Comments: 6 Pages. An important correction has been made in the definition of problem space.

We introduce and develop the logic of existence of solution to problems. We use this theory to answer the question of Florentin Smarandache in logic. We answer this question in the negative.
Category: Set Theory and Logic

Proof of the Lemma About the Absence of Nontrivial Cycles Collatz Sequences

Authors: Savinov Sergey
Comments: 2 Pages.

The article provibes an outline of the proof of the absence of nontrivial cycles in the Collatz sequence.
Category: Set Theory and Logic

A Paradox of “Adjacent” Real Points and Beyond

Authors: Ke Zhang
Comments: 5 pages. Email: alspa@163.com

We reveal adjacent real points in the real set using a concise logical reference. This raises a paradox while the real set is believed as existing and complete. However, we prove each element in a totally ordered set has adjacent element(s); there is no densely ordered set. Furthermore, since the natural numbers can also be densely ordered under certain ordering, the set of natural numbers, which is involved with each infinite set in ZFC set theory, does not exist itself.
Category: Set Theory and Logic

A Basis for Set Theory Without the Use of an Axiom

Authors: Albert Henrik Preiser
Comments: 3 Pages.

A basis for set theory without the use of an axiom.
Category: Set Theory and Logic

A Basis for Set Theory Without the Use of an Axiom.

Authors: Albert Henrik Preiser
Comments: 2 Pages.

Remarks on the existence of sets when they are formed with the all-quantifier.
Category: Set Theory and Logic

Demonstration of a Technique to Construct a One-to-One Correspondence Between N and the Infinite Binary Decimals in (0,1)

Authors: Ron Ragusa
Comments: 8 Pages.

In this paper we will see how by varying the initial conditions of Cantor’s Diagonal Argument we can use the method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the open interval (0, 1). We concede that the initial list of infinite binary decimals is, in fact, incomplete and that the diagonal method does produce a number not contained in the list. Also, we’ll agree that there are an infinite number of binary decimal numbers in the interval that aren’t in the list. We will see how using the same diagonal method we can create infinitely many binary decimal numbers not initially contained in the interval and that each number we so create will correspond with one and only one natural number.
Category: Set Theory and Logic

Set Theory Inc^# _∞^# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.iii): Hyper Inductive Definitions.application in Transcendental Number Theory.generalized Lindemann-Weierstrass Theorem

Authors: Jaykov Foukzon
Comments: 57 Pages.

In this paper intuitionistic set theory INC# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture. Main results are: (i) number e^{e}is transcendental; (ii) the both numbers e+pi and e-pi are irrational.
Category: Set Theory and Logic

Set Theory Inc## Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule.part Iii.hyper Inductive Definitions. Application in Transcendental Number Theory.generalized Lindemann-Weierstrass Theorem.

Authors: Jaykov Foukzon
Comments: 51 Pages.

Set Theory INC## Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule.Part III.Hyper Inductive Definitions. Application in transcendental number theory.Generalized Lindemann-Weierstrass theorem. n this paper intuitionistic set theory INC_{∞^{}}^{} in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture. Main results are:(i) number e^{e} is transcendental; (ii) the both numbers e+π and e-π are irrational.
Category: Set Theory and Logic

Logic and Intuition

Authors: Bertrand Wong
Comments: 4 Pages.

This paper aims to inspire thinking on the capabilities and potential of the human brain. The brain apparently has great potential for development and great untapped capabilities. Practically everyone is keen on improving his mental capacity, especially the capability of logical reasoning, that is, the ability in utilising logic to achieve the desired outcomes. It appears that logic is equated with intelligence and is regarded as the most important aspect of thinking by many (though emotional intelligence is now the new kid in the block which seems to be gaining traction). The author here looks at reasoning or logic, as well as intuition, from a different and perhaps unique perspective.
Category: Set Theory and Logic

Invalidating Cantor’s Continuum Hypothesis and Solving Hilbert’s #1 Problem II

Authors: Stephane H. Maes
Comments: 5 Pages.

In this short paper, we provide a mathematical proof that in set theory, developed in a mathematical universe following the ZFC axioms, Cantor’s continuum hypothesis does not hold: the cardinality of the continuous set of all reals is ��, and not א1, i.e., there are infinity א1 (and maybe more than one) between ��, the cardinality of the continuum, and the cardinality of the infinite set of naturals, א0.The proof is derived from combinatorics, relying on ZFC solely for the model of Cantor and Gödel defining א0. It provides input to the still unresolved first of Hilbert famous 23 math problems of interest.This paper, resolves the first of the 23 Hilbert problems with invalidation of the continuum hypothesis.
Category: Set Theory and Logic

One Fundamental Theorem Concerning Infinite Countable Sets

Authors: Nhat-Anh Phan
Comments: Copyright. All rights reserved. Orthographic error on page 2; added comments on page 8-9.

For A an infinite countable set containing infinitely many distinct natural integersand B an infinite countable set containing infinitely many distinct natural integerssuch that ∀n ∈ A, n ∈ B and ∀m ∈ B, m ∈ A, we demonstrate that it is possiblethat A≠B by exposing infinitely many counter-examples in which, for each counter-example, A and B are respectively two sample spaces of two probability spaces havingdifferent probabilities for similar events. We thus prove that the axiom of extensionality is false for infinite countable sets.
Category: Set Theory and Logic

The Fog Covering Cantor's Paradise: Some Paradoxes on Infinity and Continuum

Authors: Ke Zhang
Comments: 19 pages English + 19 pages Chinese. Mail: alspa@163.com

We challenge Georg Cantor's theory about infinity. By attacking the concept of “countable/uncountable” and diagonal argument, we reveal the uncertainty, which is obscured by the lack of clarity. The problem arises from the basic understandings of infinity and continuum. We perform many thought experiments to refute current standard views. The results support the opinion that no potential infinity leads to an actual infinity, nor is there any continuum composed of indivisibles statically, nor is Cantor's theory consistent in itself.
Category: Set Theory and Logic

Set Theory INC# ∞# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.II) Hyper Inductive Definitions

Authors: Jaykov Foukzon
Comments: Journal of Advances in Mathematics and Computer Science, Page 90-112 DOI: 10.9734/jamcs/2021/v36i430359 Published: 11 June 2021

In this paper intuitionistic set theory INC# ∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture.
Category: Set Theory and Logic

Infinity Put to the Test: Towards a Discrete Revolution in the Mathematics of the XXI Century

Authors: Antonio Leon
Comments: 404 Pages.

From different areas of mathematics, such as set theory, geometry, transfinite arithmetic or supertask theory, in this book more than forty arguments are developed about the inconsistency of the hypothesis of the actual infinity in contemporary mathematics. A hypothesis according to which the uncompletable lists, as the list of the natural numbers, exist as completed lists. The inconsistency of this hypothesis would have an enormous impact on physics, forcing us to change the continuum space-time for a discrete model, with indivisible units (atoms) of space and time. The discrete model would be a great simplification of physical theories, including relativity and quantum mechanics. It would also suppose the solution of the old problem of change, posed by the pre-Socratics philosophers twenty-seven centuries ago.
Category: Set Theory and Logic

Infinity Put to the Test: Towards a Discrete Revolution in the Mathematics of the XXI Century

Authors: Antonio Leon
Comments: 404 Pages.

From different areas of mathematics, such as set theory, geometry, transfinite arithmetic or supertask theory, in this book more than forty arguments are developed about the inconsistency of the hypothesis of the actual infinity in contemporary mathematics. A hypothesis according to which the uncompletable lists, as the list of the natural numbers, exist as completed lists. The inconsistency of this hypothesis would have an enormous impact on physics, forcing us to change the continuum space-time for a discrete model, with indivisible units (atoms) of space and time. The discrete model would be a great simplification of physical theories, including relativity and quantum mechanics. It would also suppose the solution of the old problem of change, posed by the pre-Socratics philosophers twenty-seven centuries ago.
Category: Set Theory and Logic

Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule

Authors: Jaykov Foukzon
Comments: 16 Pages. Journal of Advances in Mathematics and Computer Science, 36(2), 73-88. https://doi.org/10.9734/jamcs/2021/v36i230339

In this article Russell's paradox and Cantors paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.
Category: Set Theory and Logic

Cantor Diagonal Argument

Authors: Antonio Leon
Comments: 9 Pages.

This paper proves a result on the decimal expansion of the rational numbers in the open rational interval (0, 1), which is subsequently used to discuss a reordering of the rows of a table T that is assumed to contain all rational numbers within (0, 1), in such a way that the diagonal of the reordered table T could be a rational number from which different rational antidiagonals (elements of (0, 1) that cannot be in T ) could be defined. If that were the case, and for the same reason as in Cantor’s diagonal argument, the open rational interval (0, 1) would be non-denumerable, and we would have a contradiction in set theory, because Cantor also proved the set of rational numbers is denumerable.
Category: Set Theory and Logic

Cantor Diagonal Argument

Authors: Antonio Leon
Comments: 9 Pages.

This paper proves a result on the decimal expansion of the rational numbers in the open rational interval (0, 1), which is subsequently used to discuss a reordering of the rows of a table T that is assumed to contain all rational numbers within (0, 1), in such a way that the diagonal of the reordered table T could be a rational number from which different rational antidiagonals (elements of (0, 1) that cannot be in T ) could be defined. If that were the case, and for the same reason as in Cantor’s diagonal argument, the open rational interval (0, 1) would be non-denumerable, and we would have a contradiction in set theory, because Cantor also proved the set of rational numbers is denumerable.
Category: Set Theory and Logic

Hilbert Machine

Authors: Antonio Leon
Comments: 7 Pages.

Inspired by the emblematic Hilbert Hotel, Hilbert machine is a conceptual super-machine whose functioning questions the consistency of the actual infinity hypothesis subsumed into the Axiom of Infinity.
Category: Set Theory and Logic

Hilbert Machine

Authors: Antonio Leon
Comments: 7 Pages.

Inspired by the emblematic Hilbert Hotel, Hilbert machine is a conceptual super-machine whose functioning questions the consistency of the actual infinity hypothesis subsumed into the Axiom of Infinity.
Category: Set Theory and Logic

Hilbert Machine

Authors: Antonio Leon
Comments: 7 Pages.

Inspired by the emblematic Hilbert Hotel, Hilbert machine is a conceptual super-machine whose functioning questions the consistency of the actual infinity hypothesis subsumed into the Axiom of Infinity.
Category: Set Theory and Logic

Zeno Dichotomies

Authors: Antonio Leon
Comments: 10 Pages.

This chapter introduces a formalized version of Zeno’s Dichotomy in its two variants (here referred to as Dichotomy I and II) based on the discreteness and separation of ω-order (Dichotomy I) and of ω∗-order (Dichotomy II) defined below in this section. Each of these formalized versions leads to a contradiction pointing to the inconsistency of the hypothesis of the actual infinity.
Category: Set Theory and Logic

Zeno Dichotomies

Authors: Antonio Leon
Comments: 10 Pages.

This chapter introduces a formalized version of Zeno’s Dichotomy in its two variants (here referred to as Dichotomy I and II) based on the discreteness and separation of ω-order (Dichotomy I) and of ω∗-order (Dichotomy II) defined below in this section. Each of these formalized versions leads to a contradiction pointing to the inconsistency of the hypothesis of the actual infinity.
Category: Set Theory and Logic

Thomson Lamp

Authors: Antonio Leon
Comments: 13 Pages.

The argument of Thomson lamp and Benacerraf’s critique are reexamined from the perspective of the w-order legitimated by the hypothesis of the actual infinity subsumed into the Axiom of Infinite. The conclusions point to the inconsistency of that hypothesis.
Category: Set Theory and Logic

Thomson Lamp

Authors: Antonio Leon
Comments: 13 Pages.

The argument of Thomson lamp and Benacerraf’s critique are reexamined from the perspective of the w-order legitimated by the hypothesis of the actual infinity subsumed into the Axiom of Infinite. The conclusions point to the inconsistency of that hypothesis.
Category: Set Theory and Logic

About Structure of a Connected Quaternion-JULIA-Set and Symmetries of a Related JULIA-Network

Authors: Udo E. Steinemann
Comments: 18 Pages.

If a variable is replace by its square and subsequently enlarged by a constant during a number of iteration-steps in quaternion-space, a network of (3) sets will be built gradually. As long as for the iteration-constant certain conditions are fulfilled, the network will consist of: an unbounded set (escape-set) with trajectories escaping to infinity during course of the iteration, a bounded set (prisoner-set) with trajectories tending to a sink-point and a further bounded one (JULIA-set) with a fixed-point as repeller having a repulsive effect on all points of both the other sets. The iteration will continue until the attracting sink-point of prisoner-set and the repelling fixed-point on JULIA-set have been found. This situation is reached if predecessor- and successor-state of the iteration became equal. The fixed-point-condition provisionally formulated in general terms of quaternions, can be separated into (3) sub-conditions. When heeding the HAMILTONian-rules for interactions of the imaginary sub-spaces of the quaternion-space, each sub-condition will be appropriate for one imaginary sub-spaces and independently debatable. Knowledge of fixed-points from this fundamental network will one enable to study the structure of a connected JULIA-set. The Iteration will start from (1) on real-axis, this is not a restriction on generality because an appropriate scaling on real-axis can always be archived this way. It will become obvious, that the fixed-points in prisoner- and JULIA-set will depend on the iteration-constant only. Thus (16) different constants chosen appropriately will enable to arrange (16) fixed-points of JULIA-sets in the square-points of a hyper-cube and thereby together with the JULIA-sets to built a related JULIA-network. The symmetry-properties of this related JULIA-network can be studied on base of a hyper-cube's symmetry-group extended by some additional considerations.
Category: Set Theory and Logic

Comment Number 9 to Theory of Objectivity

Authors: Vidamor Cabannas
Comments: 102 Pages. More information and comments on Theory of Objectivity and Author Vidamor Cabannas can be found at www.theoryofobjectivity.com

The Theory of Objectivity is also called "The Third Way" or "The Third Theory", as a reference to be a third alternative explanation of the emergence of the universe, different from the Big Bang Theory and Creationism. This Comment Number 9 aims to demonstrate the number of sides that make up the spherical point that occurs before the appearance of the Universe and confirm that it is not possible to build a minimal, perfect and logical sphere without it being composed in its maximum circumference by less than sixty-four straight sides, as presented in Objectivity Theory. In sequence, the consequences of the logical construction of the geometric elements derived from the perfect sphere are demonstrated, assessing a spatial reality that surpasses the three conventional dimensions of human mathematics, proving the existence of a fourth spatial dimension and a fifth logical dimension for every geometric element.
Category: Set Theory and Logic

A Propositional Logic Review of Descartes’ Phrase “Cogito, Ergo Sum”

Authors: Vasilis Valatsos
Comments: 1 Page.

“Cogito, ergo sum” is a Latin phrase used by René Descartes as a philosophical proposition, which became a fundamental axiomatic truth within the boundaries of Western Philosophy, and was used to assert the reality of one’s own mind. In this paper, we try to review this statement using Propsitional Logic.
Category: Set Theory and Logic

The Diagonalization Paradox

Authors: Ron Ragusa
Comments: 11 Pages.

In 1891 Georg Cantor published his Diagonal Method which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers. In this paper we will see how by varying the initial conditions of Cantor’s proof we can use the diagonal method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the interval (0, 1). In the appendix we demonstrate that using the diagonal method recursively will, at the limit of the process, fully account for all the infinite binary decimals in (0, 1). The proof will cement the one-to-one correspondence between the natural numbers and the infinite binary decimals in (0, 1).
Category: Set Theory and Logic

The Diagonalization Paradox

Authors: Ron Ragusa
Comments: 11 Pages.

In 1891 Georg Cantor published his Diagonal Method which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers. In this paper we will see how by varying the initial conditions of Cantor’s proof we can use the diagonal method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the interval (0, 1). In the appendix we demonstrate that using the diagonal method recursively will, at the limit of the process, fully account for all the infinite binary decimals in (0, 1). The proof will cement the one-to-one correspondence between the natural numbers and the infinite binary decimals in (0, 1).
Category: Set Theory and Logic

Time & Relativity in Mathematics

Authors: Pankaj Mani
Comments: 24 Pages.

In this paper, we try to revisit some of the most fundamental issues lying at the foundation of mathematics in space-time relativistic perspective ,rather than conventional absolute space. We are adding a new dimension "Time" to the mathematics and review it in Space-Time relativistic framework to resolve the major foundational issues and making it in line with physical world realities. We shall look at the famous Cantor’s Diagonalization approach in to show the Countability of Real Numbers and explain Infiniteness in that perspective. We shall also look to resolve the famous paradoxes e.g. Richard, Russell,Liar, Skolem. We shall also look at the foundation of Set theory historically in Space-time to restore the issues that had led to ZFC by elimination and restrictions. As a consequence, we shall also revisit Godel Incompleteness theorems for Real Numbers and also otherwise explain the "inconsistency" in the new relativistic framework where they might be extended to all relativistically. This could possibly lead to entirely new way of looking at conventional mathematics in broad sense where reference frame plays key role in mathematics at higher level and in fact mathematics is relative! In fact Simpson’s Paradox in Statistics and Data Analysis is also related where Statistical Data are seen in absolute sense but in reality they are relativistic in different frames of references ! The absence of relativistic reference frames in statistical analysis often could lead to paradoxes e.g. Simpson’s paradoxes having wide relevance in real-world applications.
Category: Set Theory and Logic

Time & Relativity in Mathematics - Revisiting the Foundation: Set Theory, ZFC, Godel Incompleteness Results, Cantor Diagonalization & Countability of Real Numbers, Famous Paradoxes (Skolem, Richard etc.) in Relativistic Mathematical Space-Time

Authors: Pankaj Mani
Comments: 19 Pages.

Time & Relativity in Mathematics. In this paper, we try to revisit some of the most fundamental issues lying at the foundation of mathematics in space-time relativistic perspective ,rather than conventional absolute space. We are adding a new dimension "Time" to the mathematics and review it in Space-Time relativistic framework to resolve the major foundational issues and making it in line with physical world realities. We shall look at the famous Cantor’s Diagonalization approach in to show the Countability of Real Numbers and explain Infiniteness in that perspective. We shall also look to resolve the famous paradoxes e.g. Richard, Russell,Liar, Skolem. We shall also look at the the foundation of Set theory historically in Space-time to restore the issues that had led to ZFC by elimination and restrictions. As a consequence, we shall also revisit Godel Incompleteness theorems for Real Numbers and also otherwise explain the "inconsistency" in the new framework. This could possibly lead to entirely new way of looking at conventional mathematics in broad sense.
Category: Set Theory and Logic

Godel Incompleteness Results, Cantor Diagonalization & Countability of Real Numbers, Famous Paradoxes E.g. Richard, Russell, Skolem, Liar , Set Theory, ZFC, in Relativistic Mathematical Space-Time

Authors: Pankaj Mani
Comments: 19 Pages.

In this paper, we try to revisit some of the most fundamental issues lying at the foundation of mathematics in space-time relativistic perspective ,rather than conventional absolute space. We are adding a new dimension “Time” to the mathematics and review it in Space-Time relativistic framework to resolve the major foundational issues and making it in line with physical world realities. We shall look at the famous Cantor’s Diagonalization approach in to show the Countability of Real Numbers and explain Infiniteness in that perspective. We shall also look to resolve the famous paradoxes e.g. Richard, Russell,Liar, Skolem . We shall also look at the the foundation of Set theory historically in Space-time to restore the issues that had led to ZFC by elimination and restrictions. As a consequence, we shall also revisit Godel Incompleteness theorems for Real Numbers and also otherwise explain the “inconsistency” in the new framework. This could possibly lead to entirely new way of looking at conventional mathematics in broader sense.
Category: Set Theory and Logic

Rhind Papyrus Problem no 48 Squaring a Circle in 4-D Spacetime

Authors: Manfred U.E. Pohl
Comments: 4 Pages.

The ToE Framework that unites quantum theory with gravitation „Solution to the Problem of Time“ [1] is based on the Solution of the Black-Hole Information Paradox, namely the squaring of a circle (π) in space-time. It is well known how to “square” a circle over an additional dimension, as shown in „Solution to the Problem of Time“ II +III [2]. In addition to the Essay “It takes a Decision to Decide if Decidability is True or False” [3] (concering Gödels incompleteness and impossibility of Hilbert’s Programm) it is shown here the Solution to the Problem No. 48 in the Rhind-Papyrus. Keywords: Pi, God, Unified Principle, Gödel’s Incompleteness, Foundation of Mathematics, TOE.
Category: Set Theory and Logic

Modus Inversus – if (Premise is False) Then (Conclusion is False)

Authors: Ilija Barukčić
Comments: 9 Pages.

Objective: When theorems or theories are falsified by a formal prove or by observations et cetera, authors respond many times by different and sometimes inappropriate counter-measures. Even if the pressure by which we are forced to believe in different theories although there are already predictively superior rivals to turn to may be very high, a clear scientific methodology should be able to help us to assure the demarcation between science and pseudoscience. Methods: Karl Popper’s (1902-1994) falsificationist methodology is one of the many approaches to the problem of the demarcation between scientific and non-scientific theories but relies as such too much only on modus tollens and is in fact purely one-eyed. Results: Modus inversus is illustrated in more detail in order to identify non-scientific claims as soon as possible and to help authors not to hide to long behind a lot of self-contradictory and sometimes highly abstract, even mathematical stuff. Conclusions: Modus inversus prevents us from accepting seemingly contradictory theorems or rules in science. Keywords: Science, non-science, modus inversus. E-Mail: Barukcic@t-online.de
Category: Set Theory and Logic

The Mathematical Expressions of Quranic Exegeses and the Mathematical Definition of the Quranic Correctness

Authors: Tahara Hiroki
Comments: 4 Pages.

I succeeded to give mathematical expressions to any correct Quranic Exegeses and define the Quranic correctness as the unique existence of Tahara I function. In a precise mathematical sense, the expressions and the definition are ill-defined however they might have meanings to prove the Quranic correctness.
Category: Set Theory and Logic

The Mathematical Expressions of Quranic Exegeses and the Mathematical Definition of the Quranic Correctness

Authors: Hiroki Tahara
Comments: 4 Pages.

I succeeded to give mathematical expressions to any correct Quranic Exegeses and define the Quranic correctness as the unique existence of Tahara I function. In a precise mathematical sense, the expressions and the definition are ill-defined however they might have meanings to prove the Quranic correctness.
Category: Set Theory and Logic

The Mathematical Expressions of Quranic Exegeses and the Mathematical Definition of the Quranic Correctness

Authors: Hiroki Tahara
Comments: 4 Pages.

I succeeded to give mathematical expressions to any correct Quranic exegeses and define the Quranic correctness as the unique existence of Tahara I function. In a precise mathematical sense, the expressions and the definition are ill-defined however they might have meanings to prove the Quranic correctness.
Category: Set Theory and Logic

The Problem P=NP

Authors: Thierry Delort
Comments: 30 Pages.

In this article, we are going to solve the problem P=NP for a particular kind of problems called basic problems of numerical determination. We are going to propose 3 fundamental Axioms permitting to solve the problem P=NP for basic problems of numerical determination, those Axioms can also be considered as pure logical assertions, intuitively evident and never contradicted, permitting to understand the solution of the problem P=NP for basic problems of numerical determination. We will see that those Axioms imply that the problem P=NP in undecidable for basic problems of numerical determination. Nonetheless we will see that it is possible to give a theoretical justification (which is not a classical proof) of the proposition "P≠NP". We will then study a 2nd problem, named "PN=DPN problem" analogous to the problem P=NP but which is fundamental in mathematics.
Category: Set Theory and Logic

Visualizing the Distributions and Isosurfaces of Some Traditional and Non-Traditional Quaternion Fractal Sets

Authors: Shawn Halayka
Comments: 18 Pages.

After a concise introduction, the length, displacement, and magnitude distributions and isosurfaces related to some quaternion fractal sets are visualized.
Category: Set Theory and Logic

The Logical Content of Inequalities

Authors: Ilija Barukčić
Comments: 20 pages. Copyright © 2019 by Ilija Baruk�?ić, Jever, Germany. All rights reserved. Published by: Ilija Baruk�?ić "The Interior Logic of Inequalities" IJMTT, 65 (7) (2019): 146-155. doi: 10.14445/22315373/IJMTT-V65I7P524

Objective. From a theoretical point of view, the demarcation between science and (fantastical) pseudoscience is in necessary for both practical and theoretical reasons. One specific nature of pseudoscience in relation to science and other categories of human reasoning is the resistance to the facts. Methods. Several methods are analyzed which may be of use to prevent that belief is masqueraded genuinely as scientific knowledge. Results. Modus ponens, modus tollens and modus conversus are reanalyzed. Modus sine, logically equivalent to modus ponens is developed. Modus inversus and modus juris are described in detail. Conclusions. In our striving for knowledge, there is still much more scientific work to be done on the demarcation between science and pseudoscience. Keywords: non strict inequality, quantum mechanics, science, pseudoscience
Category: Set Theory and Logic