Peano arithmetic

An introduction to the formal approach that underlies university mathematics. Topics covered are: Sets and Logic, Relations and Functions, Number Systems, Infinity. The book has arisen out of a first year course taught to Mathematics majors at Shiv Nadar University, with the aim of helping students bridge the different requirements of rigour at school and university.

Being and Intelligibility is a philosophical treatise on the meaning of being in its two senses. The book explores the most originary ontological question, namely, " what do we mean when we say that something is? " , and also the most originary existential question, namely, " what is the meaning of human experience? " , in each case, by reference to a fundamental principle of being and intelligibility, which, following the pre-Socratics, is called the " Logos ". The book posits that all rational experience is of objects which are at root a unity (ground) of relations (predicates) which include sequence, magnitude, and proportionality and hence are called " logical objects ". From this simple premise, the book develops its three central theses. The first is that the beingness of beings (called " Being ") and the intelligibility of Being are strictly self-same. The second is that, because nothingness (i.e., absolute not-Being) entails the absence of the rules of its own conception, it is self-contradictory and unintelligible and, therefore, Being is logically necessary. The third is that the fullness of human rational experience cannot be explained in materially reducible terms and requires recognition of the existence of transcendent reality. The book identifies the confused state of modern mainstream metaphysics as resulting from two fundamental errors. The first, which is laid at the feet of René Descartes, is the deconstruction of the unity of the human being into mind and body, which set modern philosophy down the parallel but ill-fated paths of rationalism and empiricism. The former imploded at its Hegelian end of " thought thinking about itself " and the latter expired upon the declaration of A.J. Ayer that metaphysical statements are meaningless because they are not verifiable. The second error, which is attributed initially to Immanuel Kant and is repeated by Martin Heidegger, is the failure to recognize the essential logicality of Being. In the case of Kant, who characterizes logic as a contentless abstraction from empirical categories of understanding, the scope of logic is limited to empirical experience and metaphysics is relegated to faith. In the case of Heidegger, Being holds privileged status over logic, with the results that Heidegger grants coherency to the possibility of nothingness, determines Being to ground all beings but itself to be ungrounded, and denies the necessity of Being. Being and Intelligibility investigates the implications of the essential logicality of Being, which resound throughout the full range of human rational experience. The book shows how logically conceived Being underpins the possibility of objective knowledge of an inherently ordered universe, the homogeneity of logic and mathematics, and the necessary existence of God, as the Supreme Principle of Being and Intelligibility and the ground of all Being including God itself. The book also shows how, from this epistemological and metaphysical context, human Being shows itself to itself from within itself as a substantive, persistent, morally obligated unity among the ordered manifold of its life experiences, whose essential Being is orientation towards God. In connecting Being and logic, Being and Intelligibility restores metaphysics to its proper place at the pinnacle of human understanding, which is precisely where reason, which (as Leibniz tells us) demands that the reasons for all that there is be rendered to it, insists that it must be.

In this thesis, I explore Isaacson's thesis and Wilkie's theorem, providing philosophical and formal results on how they relate to each other. At a first approximation, Isaacson's thesis claims that Peano arithmetic is sound and complete with respect to genuinely arithmetical statements . Using internalist notions familiar from recent work on internal categoricity theorems, I provide a formal notion of genuinely arithmetical statements. As for Wilkie's theorem, it roughly says that, from an external perspective, Peano arithmetic is minimal, in that it is entailed by all categorical axiomatisations of the natural numbers satisfying a certain syntactic restriction. After expositing Wilkie's theorem and the relation of its proof to other known techniques, I discuss its relation to Isaacson's thesis, in particular whether Peano arithmetic is a maximal theory obtained from the categorical characterisation.

The sorites paradox (interpreted as the paradox of small natural numbers) is analyzed using mathematical fuzzy logic. In the first part, we present an extension of BL-fuzzy logic by a new unary connective At of almost true and the crisp Peano arithmetic extended by a fuzzy predicate of feasibility. Then we give examples of possible semantics of At and examples of semantics of feasible numbers. In the second part, we present an analysis of the sorites paradox within fuzzy logic with evaluated syntax and show that under a very natural assumption we obtain a consistent fuzzy theory. Thus, sorites is not paradoxical at all.

I show—contrary to common beliefs tolerated by the “bosses”—that any interpretation of ZF that admits Aristotle's particularisation is not sound; that the standard interpretation of PA is not sound; that PA is consistent but ω-inconsistent; that a sound finitary interpretation of ...

I shall argue that a resolution of the PvNP problem requires building an iff bridge between the domain of provability and that of computability. The former concerns how a human intelligence decides the truth of number-theoretic relations, and is formalised by the first-order Peano Arithmetic PA following Dededekind's axiomatisation of Peano's Postulates. The latter concerns how a human intelligence computes the values of number-theoretic functions, and is formalised by the operations of a Turing Machine following Turing's analysis of computable functions. I shall show that such a bridge requires objective definitions of both an `algorithmic' interpretation of PA, and an `instantiational' interpretation of PA. I shall show that both interpretations are implicit in the definition of the subjectively defined `standard' interpretation of PA. However the existence of, and distinction between, the two objectively definable interpretations---and the fact that the former is sound whilst the latter is not---is obscured by the extraneous presumption under the `standard' interpretation of PA that Aristotle's particularisation must hold over the structure N of the natural numbers. I shall argue that recognising the falseness of this belief awaits a paradigm shift in our perception of the application of Tarski's analysis (of the concept of truth in the languages of the deductive sciences) to the `standard' interpretation of PA. I shall then show that an arithmetical formula [F] is PA-provable if, and only if, [F] interprets as true under an algorithmic interpretation of PA. I shall finally show how it then follows from Goedel's construction of a formally `undecidable' arithmetical proposition that there is a Halting-type PA formula which---by Tarski's definitions---is algorithmically verifiable as true, but not algorithmically computable as true, under a sound interpretation of PA.

Transcendent realism is a theistic philosophical system which holds that reality comprises physical and metaphysical entities, including morally conscious souls and God, as the creative and sustaining explanation of the being of beings ("Being") and moral obligation. The three central theses of the philosophy are that (1) Being and the intelligibility of Being are self-same, reducible to logic, and logically necessary, (2) absolute nothingness is incoherent because it entails the absence of the rules of its own conception, and (3) human rational experience is not materially reducible and includes direct and inferential experience of metaphysical Being. Some of the ideas developed in this summary are: (A) that the presuppositions of objective reason include a self-transcendent, knowing soul and the logicality of Being; (B) humankind is a self-concerned, substantive, psychosomatic unity whose Being is being-toward-God, whose soul exists not in, but alongside, the external world, and whose body is the physical manifestation of the soul's relation to the external world; (C) the logicality of Being implies the existence of the one and only God as its Supreme Principle and as Agape (unqualified good will), the definitionally good, self-intending source of the moral obligation of self-transcending souls; and (D) humankind's fundamental obligation, as the agent of God's self-intending good will and the being who brings morality into the world, is to act with agape to all.

Artemov [5] offered the notion of constructive truth and falsity of arithmetical sentences in the spirit of Brouwer-Heyting-Kolmogorov semantics and its formalization, the Logic of Proofs. In this paper, we provide a complete description of constructive truth and falsity for Friedman’s constant fragment of Peano Arithmetic. For this purpose, we generalize the constructive falsity to n-constructive falsity in Peano Arithmetic where n is any positive natural number. Based on this generalization, we also analyze the logical status of well- known Go ̈delean sentences: consistency assertions for extensions of PA, the local reflection principles, the ‘constructive’ liar sentences, Rosser sentences. Finally, we discuss ‘extremely’ independent sentences in the sense that they are classically true but neither constructively true nor n-constructively false for any n.

A Teoria das Funções Recursivas é um modelo matemático da computação. Na formulação usual, as funções recursivas são geradas por dois processos, a Recursão Primitiva e a \mu–Recursão. O Teorema de Equivalência de Lawvere estabelece que a Aritmética de Peano é logicamente equivalente à Recursão Primitiva e, em consequência, os Axiomas de Peano aparentemente servem como fundamento de apenas uma parte da Teoria Matemática da Computação. Neste trabalho apresentamos uma análise da Teoria das Funções Recursivas que parece indicar que a Recursão Primitiva é, de fato, suficiente para formular toda a Teoria de Computação.

Classical theory proves that every primitive recursive function is strongly representable in PA; that formal Peano Arithmetic, PA, and formal primitive recursive arithmetic, PRA, can both be interpreted in Zermelo-Fraenkel Set Theory, ZF; and that if ZF is consistent, then PA+PRA is consistent. We show that PA+PRA is inconsistent; it follows that ZF, too, is inconsistent.

In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, bi-interpretability and mutual interpretability, take the form of isomorphism. We study the usual notions like monomorphism and product in the various theories. We provide some examples to separate notions across categories. In contrast, we show that, in some cases, notions in different categories do coincide. E.g., we can, under such-and-such conditions, infer synonymity of two theories from their being equivalent in the sense of a coarser equivalence relation. We illustrate that the categories offer an appropriate framework for conceptual analysis of notions. For example, we provide a ‘coordinate free’ explication of the notion of axiom scheme. Also we give a closer analysis of the object-language/ meta-language distinction. Our basic category can be enriched with a form of 2-structure. We use this 2-structure to characterize a salient subclass of...

I show--contrary to common beliefs tolerated by the 'bosses'--that any interpretation of ZF that admits Aristotle's particularisation is not sound; that the standard interpretation of PA is not sound; that PA is consistent but omega-inconsistent; that a sound finitary interpretation of PA is definable in terms of Turing-computability; and that PA cannot be consistently extended to ZF.

We show unexpected connection of Set Theoretical Forcing with Quantum Mechanical lattice of projections over some separable Hilbert space. The basic ingredient of the construction is the rule of indistinguishability of Standard and some Nonstandard models of Peano Arithmetic. The ingeneric reals introduced by M. Ozawa will correspond to simultaneous measurement of incompatible observables. We also discuss some results concerning model theoretical analysis of Small Exotic Smooth Structures on topological 4-space. Forcing appears rather naturally in this context and the rule of indistinguishability is crucial again. As an unexpected application we are able to approach Maldacena Conjecture on $AdS/CFT$ correspondence in the case of AdS_5xS^5 and Super YM Conformal Field Theory in 4 dimensions. We conjecture that there is possibility of breaking Supersymetry via sources of gravity generated in 4 dimensions by exotic smooth structures on R^4 emerging in this context.

We show that Tarski's inductive definitions admit evidence-based interpretations of the first-order Peano Arithmetic PA that allow us to define the satisfaction and truth of the quantified formulas of PA constructively over the domain of the natural numbers in two essentially different ways: (a) in terms of algorithmic verifiabilty; and (b) in terms of algorithmic computability. We argue that the algorithmically computable PA-formulas provide a finitary interpretation of PA from which we may conclude that PA is consistent in an intuitionistically unobjectionable manner.

I show--contrary to common beliefs tolerated by the 'bosses'--that any interpretation of ZF that admits Aristotle's particularisation is not sound; that the standard interpretation of PA is not sound; that PA is consistent but omega-inconsistent; that a sound finitary interpretation of PA is definable in terms of Turing-computability; and that PA cannot be consistently extended to ZF.

Classical interpretations of Goedel's formal reasoning imply that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is essentially unverifiable. However, a language of general, scientific, discourse cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic is verifiably complete. We show how some paradoxical concepts of Quantum mechanics can be expressed, and interpreted, naturally under a constructive definition of mathematical truth.

I shall argue that a resolution of the PvNP problem requires building an iff bridge between the domain of provability and that of computability. The former concerns how a human intelligence decides the truth of number-theoretic relations, and is formalised by the first-order Peano Arithmetic PA following Dededekind's axiomatisation of Peano's Postulates. The latter concerns how a human intelligence computes the values of number-theoretic functions, and is formalised by the operations of a Turing Machine following Turing's analysis of computable functions. I shall show that such a bridge requires objective definitions of both an `algorithmic' interpretation of PA, and an `instantiational' interpretation of PA. I shall show that both interpretations are implicit in the definition of the subjectively defined `standard' interpretation of PA. However the existence of, and distinction between, the two objectively definable interpretations---and the fact that the fo...

We conclude from Goedel's Theorem VII of his seminal 1931 paper that every recursive function f(x_1, x_2) is representable in the first-order Peano Arithmetic PA by a formula [F(x_1, x_2, x_3)] which is algorithmically verifiable, but not algorithmically computable, if we assume that the negation of a universally quantified formula of the first-order predicate calculus is always indicative of the existence of a counter-example under the standard interpretation of PA. We conclude that the standard postulation of the Church-Turing Thesis does not hold if we define a number-theoretic formula as effectively computable if, and only if, it is algorithmically verifiable; and needs to be replaced by a weaker postulation of the Thesis as an equivalence.

We discuss simple subtheories of Peano arithmetic over languages which include the monus function. The system $\mathrm{ZDL}$ corresponds with $\mathrm{PA}^-$. The choice of language permits our theories to have special universal axiomatizations; their classes of models have corresponding model theoretic properties.

Contents
Part I  Arithmetic
Chapter 1  Sets and Orders
1.1 Sets
1.2 Relations
1.3 Ordinals
1.4 Frames
Chapter 2  Formal Arithmetic
2.1 Arithmetic Language with the Truth Predicate
2.2 Classical Valuation
2.3 Truth Arithmetic and Peano Arithmetic
2.4 Non-standard Models of Truth Arithmetic
Chapter 3  Primitive Recursive Functions and Recursive Functions
3.1 Primitive Recursive Functions
3.2 Some Primitive Recursive Functions
3.3 Primitive Recursive Relations
3.4 Recursive Functions and Relations
Chapter 4  Gödel’s Numbers and Related Functions
4.1 Gödel’s Numbers
4.2 Sequence Numbers
4.3 Course-of-value Recursion
4.4 Some Syntactical Functions and Relations
Chapter 5  Arithmetical Definability of Recursive Functions
5.1 Arithmetical Definability
5.2 Composition and Minimization
5.3 Gödel’s Beta Function
5.4 Primitive Recursive Operation
Chapter 6  Undefinability of Truth
6.1 Diagonal Lemma
6.2 Tarski’s Theorem
6.3 Montague’s Theorem and McGee’s Theorem
6.4 Maximal Consistent sets of Instances of T-scheme

Part II  Truth
Chapter 7  Hierarchy Theories
7.1 Definability Problem of Truth
7.2 Language Hierarchies

7.3 Axiomatization Problem of Truth

7.4 Axiomatic Theories of Ascending and Descending Hierarchies
Chapter 8  Kleene’s strong Three-valued Scheme and Kripke’s Fixed-Point Theorem

8.1 Kleene’s strong Three-valued Scheme
8.2 Fixed Points and Truth Predicate
8.3 Fixed-Point Theorem
8.4 A Classification of Sentences
Chapter 9  Super-valuation and and Minimal Fixed Points
9.1 Precisification of Vague Predicates
9.2 Super-valuational Scheme
9.3 A Comparison of Minimal Fixed Points
9.4 Fixed Points and Paradoxes
Chapter 10  Periodicity of Revision Sequences
10.1 Revision Sequences
10.2 Revision Periods
10.3 Stability
10.4 Revision Sequence and Paradoxes
Chapter 11  Categoricity and Fixed Points
11.1 Categoricity of Sentences
11.2 Set-theoretical Description of Categoricity
11.3 Nearly-Categoricity
11.4 Categorical Sentences and Grounded Sentences
Chapter 12  Semantic Dependence Relation
12.1 Dependence Relation
12.2 Dependence and Reference
12.3 Fixed points of Dependence Operator
12.4 Dependence Fixed Points and other Fixed Points

Part III  Paradoxes
Chapter 13  T-scheme and the Liar
13.1 T-scheme in the Construction of Truth Predicate
13.2 A Characterization of the Liar
13.3 Virtuous and Vicious Cycles
13.4 Elimination or Description of Paradoxes
Chapter 14  Paradoxicality Degrees of Paradoxes
14.1 Degrees of Paradoxicality
14.2 A Comparison between the Liar and Jourdain’s Card
14.3 Jump Liars
14.4 A Characterization of Jump Liars
Chapter 15  Finite Card Paradoxes
15.1 Sets of Finite Card Sentences
15.2 A Generalization of Sets of Finite Card Sentences
15.3 Walks and Their Depth
15.4 A Characterization of Finite Card Paradoxes
Chapter 16  Boolean Paradoxes
16.1 Boolean Paradoxes and their Revision Periods
16.2 A Construction of Boolean Paradoxes
16.3 A Characterization of Boolean Paradoxes
16.4 Structure of Paradoxicality Degrees of Boolean Paradoxes
Chapter 17  Paradoxes and Self-reference
17.1 Yabloesque Paradoxes
17.2 Finite Card Paradoxes and Yabloesque Paradoxes
17.3 Non-Self-reference
 of Yabloesque Paradoxes
17.4 Self-reference of Finite Reference Systems
Chapter 18  Paradoxes and Circularity
18.1 Transfinite Card Paradoxes
18.2 McGee’s Paradox
18.3 A Characterization of Transfinite Card Paradoxes and McGee’s Paradox
18.4 Circularity of Finite Reference Systems
Bibliography
Index

We show that the classical interpretations of Tarski's inductive definitions actually allow us to define the satisfaction and truth of the quantified formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two essentially different ways: (a) in terms of algorithmic verifiabilty; and (b) in terms of algorithmic computability. We show that the classical Standard interpretation I_PA(N, Standard) of PA essentially defines the satisfaction and truth of the formulas of the first-order Peano Arithmetic PA in terms of algorithmic verifiability. It is accepted that this classical interpretation---in terms of algorithmic verifiabilty---cannot lay claim to be finitary; it does not lead to a finitary justification of the Axiom Schema of Finite Induction of PA from which we may conclude---in an intuitionistically unobjectionable manner---that PA is consistent. We now show that the PA-axioms---including the Axiom Schema of Finite Induction---are, however, a...

Historical and philosophical background providing justification for an alternative understanding of nothingness that can serve as the basis for an alternative to the empty set and the number 0. Nonexistence arises from using the existential quantifier. A different quantifier could lead to a different notion of nothing such as absence - "Here is [some numbers]" and its negation "here is not [some numbers]". Zero defined as the absence of the Natural numbers in the Peano Axioms for example.

In a forthcoming book, professional computer scientist and physicist Paul Budnik presents an exposition of classical mathematical theory as the backdrop to an elegant thesis: we can interpret any model of a formal system of Peano Arithmetic in an appropriate, digital, computational language. In this paper we attempt - without addressing the question of whether or not Budnik succeeds in establishing his thesis convincingly - to identify dogmas of standard interpretations of classical mathematical theory that appear to be implicit in Budnik's exposition, and to correspond to them dogmas of a constructive interpretation of classical theory.

Standard interpretations of Goedel's "undecidable" proposition, [(Ax)R(x)], argue that, although [~(Ax)R(x)] is PA-provable if [(Ax)R(x)] is PA-provable, we may not conclude from this that [~(Ax)R(x)] is PA-provable. We show that such interpretations are inconsistent with a standard Deduction Theorem of first order theories.

In this paper we establish that the well-known Arithmetic System is consistent in the traditional sense. The proof is done within this Arithmetic System. Key words: consistency in the traditional sense, consistency in the absolute sense, Arithmetic System, Peano's Arithmetic System

We present a new syntactical proof that first-order Peano Arithmetic with Skolem axioms is conservative over Peano Arithmetic alone for arithmetical formulas. This result - which shows that the Excluded Middle principle can be used to eliminate Skolem functions - has been previously proved by other techniques, among them the epsilon substitution method and forcing. In this paper, we employ Interactive Realizability, a computational semantics for Peano Arithmetic which extends Kreisel's modified realizability to the classical case.

We conclude from Godel's Theorem VII of his seminal 1931 paper that every recursive function f(x1,x2) is representable in the first-order Peano Arithmetic PA by a formula (F(x1,x2,x3)) which is algorithmically verifi- able, but not algorithmically computable, if we assume that the negation of a universally quantified formula of the first-order predicate calculus is always indicative of the existence of a counter-example under the stan- dard interpretation of PA. We conclude that the standard postulation of the Church-Turing Thesis does not hold if we define a number-theoretic formula as effectively computable if, and only if, it is algorithmically ver- ifiable; and needs to be replaced by a weaker postulation of the Thesis as an equivalence. Keywords Algorithmic computability, algorithmic verifiability, Aristotle's particu- larisation, Church-Turing Thesis, effective computability, first-order, Godel �-function, Peano Arithmetic PA, standard interpretation, Tarski, uniform m...

Otter-lambda is Otter modified by adding code to implement an algo- rithm for lambda unification. Otter is a resolution-based, clause-language first-order prover that accumulates deduced clauses and uses strategies to control the deduction and retention of clauses. This is the first time that such a first-order prover has been combined in one program with a unification algorithm capable of instantiating variables to lambda terms to assist in the deductions. The resulting prover has all the advantages of the proof-search algorithm of Otter (speed, variety of inference rules, excellent handling of equality) and also the power of lambda unification. We illustrate how these capabilities work well together by using Otter- lambda to find proofs by mathematical induction. Lambda unification instantiates the induction schema to find a useful instance of induction, and then Otter's first-order reasoning can be used to carry out the base case and induction step. If necessary, induction c...