Diagonalization

These remarks take up the reflexive problematics of Being and Nothingness and related texts from a metalogical perspective. A mutually illuminating translation is posited between, on the one hand, Sartre's theory of pure reflection, the linchpin of the works of Sartre's early period and the site of their greatest difficulties, and, on the other hand, the quasi-formalism of diagonalization, the engine of the classical theorems of Cantor, Gödel, Tarski, Turing, etc. Surprisingly, the dialectic of mathematical logic from its inception through the discovery of the diagonal theorems can be recognized as a particularly clear instance of the drama of reflection according to Sartre, especially in the positing and overcoming of its proper value-ideal, viz. the synthesis of consistency and completeness. Conversely, this translation solves a number of systematic problems about pure reflection's relations to accessory reflection, phenomenological reflection, pre-reflective self-consciousness, conversion, and value. Negative foundations, the metaphysical position emerging from this translation between existential philosophy and metalogic, concurs by different paths with Badiou's Being and Event in rejecting both ontotheological foundationalisms and constructivist anti-foundationalisms.

Jacobi's method is a well-known algorithm in linear algebra to diagonalize symmetric matrices by successive elementary rotations. We report here about the generalization of these elementary rotations towards canonical transformations acting in Hamiltonian phase spaces. This generalization allows to use Jacobi's method in order to compute eigenvalues and eigenvectors of Hamiltonian (and skew-Hamiltonian) matrices with either purely real or purely imaginary eigenvalues by successive elementary "decoupling" transformations. The real importance of Einstein's work was that he introduced Lorentz transformations as something fundamental in physics-P.A.M. Dirac [1]

I develop a new account of Socratic elenchos, starting from the observation that Plato's Socrates asks not one but two logically determinative questions in elenchos: the ti esti, or, “What is —?” question, and the peri tinos, or, “— about what?” question. On this basis, I obtain the following results: The two questions interact systematically to generate a series of strict dilemmas, including those defining the problems of self-knowledge in the Charmides and of the idea of the good in the Republic. These dilemmas anticipate and are explicable by diagonalization, the key to the theorems of Cantor, Gödel, et al. They are also surprisingly convergent with the negative results of the dialectic of in-itself and for-itself in Sartre's Being and Nothingness and related works. The nexus of Platonic and Sartrean reflections on duality, suitably formalized and generalized, is the thesis that there is not, on pain of contradiction, a being which is exactly what it is of and is of exactly what it is, even though such a synthesis of being and being-of can be recognized as the structure of value, or the projection of a perfect being. Metalogical reflection having become the witness of this contradiction, a structural alteration is occasioned in value, enabling the form of the good to distinguish itself from that of the perfect. I call “negative foundations” the metaphilosophical position that results when one puts such dualities, especially as obtained via diagonalization, at the center of logical and ethical reflection. Negative foundations provides a fruitful alternative to familiar foundationalisms and antifoundationalisms.

This article demonstrates the invalidity of Theorem VI of Gödel's monograph of 1931, showing that propositions (15) and (16), derived from definition (8.1), in its proof, are false in PA. This is achieved in two steps. First, the predicate complementary to the well-known Gödel's predicate Bew(x) is ndefined by adding a new relation Wid(x), and new logical connections are accordingly established, Lemma (6). Second, the negations of (15) and (16) are derived by definition (8.1) and Lemma (6). It amounts to saying that (15) and (16) are false and unacceptable for the system. On the account of that, the two well-known cases 1. 17 Gen r is not k-PROVABLE, 2. Neg(17 Gen r) is not k-PROVABLE, can not be drawn, and Theorem VI is therefore invalid.

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

Lecture Notes Contents:
Chapter 2 Matrices and Systems of Linear Equations
2.1 Matrices: definitions and notations
2.2 Matrix algebra
2.3 Terminology and notation for systems of linear equations
2.4 Elementary row operations and row echelon matrices
2.5 Gaussian elimination
2.6 The inverse of a square matrix
Chapter 3 Determinants
3.1 The definition of a determinant
3.2 Properties of determinants
3.3 Cofactor expansions
Chapter 4 Vector Spaces
4.1 Vectors in R^n
4.2 Definition of a vector space
4.3 Subspaces
4.4 Spanning sets
4.5 Linear dependence and linear independence
4.6 Bases and dimension
4.9 The rank-nullity theorem
Chapter 5 Linear Transformations
5.1 Definition of a linear transformation
5.6 The eigenvalue/eigenvector problem
5.7 General results for eigenvalues and eigenvectors
5.8 Diagonalization
and tutorials with different examples.

This paper reveals why Cantor's diagonalization argument fails to prove what it purportedly proves and the logical absurdity of "uncountable sets" that are deemed larger than the set of natural numbers. Cantor's diagonalization proof (CDP) is used to prove various things like Gödel's Incompleteness theorem (GIT) and the uncountability of real numbers (URN). In the case of GIT, it is assumed that you can have a complete list of provable statements. You use CDP to prove a statement that is not on the list, then conclude the statement is not a provable statement; otherwise, it would be part of the list of all provable statements. However, if your complete list is not really complete, then it makes sense that you can prove a new provable statement not on the list. In the case of proving the URN, you use proof by contradiction: assume the real numbers are countable and assume a complete list of them between, say, .000 ... and .111 ... The list is believed to be complete because an infinite number of counting numbers (1 to n) each allegedly have a oneto-one correspondence to each real number on the list. CDP is then used to show there is a real number that is not on the list (a contradiction). Therefore, according to the proof, there is no bijection between the real numbers and the counting numbers. Below, we will work through an example proof using CDP and demonstrate why it comes up short. Let's begin with an nXn matrix of real numbers in base 2. We go along the diagonal and flip each bit from 0 to 1 or 1 to 0 to create a new number not on the list. If we think the number is on the list, say, at row r, we can check it and see that its bit at column r doesn't match. We can repeat this exercise for all rows and see that we truly have a new real number not on the list.

The paper undertakes an analysis of the discussion about the presence and varieties of the de dicto / de re ambiguity for names. I will argue that we are dealing with several phenomena here, which are relatively easy to confuse: first, usual de re; second, a name being normally used as opposed to being used as in a diverging or underdeveloped dialect; third, the reinterpetation of a name as denoting a (temporal, cognitive, etc.) aspect of an individual. The second and the third phenomenon can combine with usual de re.
After surveying the examples, I first filter out those in which deviant dialects are manifest. This issue is resolved via a diagonalisation operation across the special dimension called "dubbing-in-force", which is essentially the part of the naming conventions salient in the discourse situation. Then the cases of semantic reinterpretation are considered, where I stipulate the existence, on a par with the usual domain of individuals, of a dedicated domain of aspect-type entities to which a name can refer in some cases. Th is stipulation does not by itself destroy rigid designation, as names continue to denote rigidly, even if what they denote is aspects of individuals rather than complete individuals. Finally, I turn to customary de re readings for singular terms, where the difference between the aspect/individual contrast and the de re / de dicto ambiguity is emphasised; I provide a Quinean “double vision” scenario which, unlike the original version, involves aspects of individuals. Thus the possibility of superposition is demonstrated for the phenomena studied in the paper.

Jacobi's method is a well-known algorithm in linear algebra to diagonalize symmetric matrices by successive elementary rotations. We report here about the generalization of these elementary rotations towards canonical transformations acting in Hamiltonian phase spaces. This generalization allows to use Jacobi's method in order to compute eigenvalues and eigenvectors of Hamiltonian (and skew-Hamiltonian) matrices with either purely real or purely imaginary eigenvalues by successive elementary ``decoupling'' transformations.

We derive an efficient numerical algorithm for the analysis of certain classes of Hilbert–Schmidt operators that naturally occur in models of wireless radio and sonar communications channels. A common short-time model of these channels writes the channel output as a weighted superposition of time- and frequency shifted copies of the transmitted signal, where the weight function is usually called the spreading function of the channel operator. It is often believed that a good channel model must allow for spreading functions containing Dirac delta distributions. However, we show that many narrowband finite lifelength systems such as wireless radio communications can be well modelled by smooth and compactly supported spreading functions. Further, we exploit this fact to derive a fast algorithm for computing the matrix representation of such operators with respect to well time-frequency localized Gabor bases (such as pulseshaped OFDM bases). Hereby we use a

In this paper I discuss the question of the norm, conceived as a  philosophical problem, rather than an ethical, political, or social one. Firstly, I outline the main declinations of the “critique of normative reason”, which has characterized most of the twentieth century’s philosophy and has pointed out the connection between norm and normalization. Secondly, I focus on the concept of “crisis” and its relationship with the problem of normativity; then I stress the ambiguity of every “state of crisis”, which could lead to claim that crisis is the most vital condition of life. Thirdly, I sketch two moments of Deleuze’s confrontation with the work of Canguilhem, in order to affirm that if we want to see the norm as a philosophical problem, we need to define some fundamental concepts as “becoming” and “relation”—that is, to understand the question of the surd or  diagonal. Finally, my main thesis states that in this perspective we need not to consider one norm’s goodness or badness, but instead the transformative relation between different norms, that presides over the process of norm–shaping.