Definability and Diagonalization

Gödel's Incompleteness Theorems ◽

10.1093/oso/9780195046724.003.0011 ◽

1992 ◽

Author(s):

Raymond M. Smullyan

Keyword(s):

Fixed Points ◽

Recursive Relation ◽

Incompleteness Theorem ◽

Fundamental Fact ◽

Basic Facts ◽

Recursive Set ◽

Recursive Relations

In this chapter we establish some basic facts about Σ1-relations and functions that will be needed for the rest of this study. We also introduce the notion of fixed-points of formulas and prove a fundamental fact about them which is crucial for Gödel’s second incompleteness theorem and related results of the next chapter. A formula F(v1,...,vn) is said to define a relation R(x1,..., xn) in a system S if for all numbers a1,...,an, the two following conditions hold. (1) R(a1,... ,an) ⇒ F(ā1,... , ā n) is provable in S. (2) R̃(a1,...,an) ⇒ F(ā1,... , ā n) is refutable in S. We say that F(v1,...,vn) completely represents R(x1, . . . ,xn ) in S iff F represents R and ~ F represents the complement R of R in S—in other words, if (1) and (2) above hold with “⇒” replaced by “↔”. If F defines R in S and S is consistent, then F completely represents R in S. Proof. Assume hypothesis. We must show that the converses of (1) and (2) above must hold. Suppose F(ā1,... , ān) is provable in S. Then F(ā1,..., ān) is not refutable in S (by the assumption of consistency). Therefore by (2), R̃ (a1,...,an) cannot hold. Hence R(a1,...,an) holds. Similarly, if F(ā1,..., ān) is refutable, then it is not provable. Hence by (1), R(a1,..., an) cannot hold and hence R̃ (a1,...,an). By a recursive set or relation, we mean one such that it and its complement are both Σ1. [There are many different, but equivalent, definitions in the literature of recursive relations. We will consider some others in the sequel to this volume.] It is obvious that a formula F defines a relation R in S iff F separates R from R̃ in S. Suppose now S is a Rosser system and that R is a recursive relation. Then R and R̃ are both Σ1. Hence R is separable from R̃ in S, which means that R is definable in S. And so we have: 1. If S is a Rosser system, then all recursive relations are definable in S. 2. If S is a consistent Rosser system, then all recursive relations are completely representable in S.

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Fixed Points: Some Basic Facts

The Revision Theory of Truth ◽

10.7551/mitpress/5938.003.0004 ◽

1993 ◽

Keyword(s):

Fixed Points ◽

Basic Facts

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A RELATIONAL MODEL OF DEMONIC NONDETERMINISTIC PROGRAMS

International Journal of Foundations of Computer Science ◽

10.1142/s012905419100008x ◽

1991 ◽

Vol 02 (02) ◽

pp. 101-131 ◽

Cited By ~ 15

Author(s):

THANH TUNG NGUYEN

Keyword(s):

Fixed Points ◽

Programming Language ◽

Set Theory ◽

Algebraic Structure ◽

Denotational Semantics ◽

Recursive Relation ◽

Relational Model ◽

Binary Relations ◽

Sequential Programming

The paper gives a self-contained account of a calculus of relations from basic operations through the treatment of recursive relation equations. This calculus serves as an algebraic apparatus for defining the denotational semantics of Dijkstra’s nondeterministic sequential programming language. Nondeterministic programs are modeled by binary relations, objects of an algebraic structure founded upon the operations “union”, “left restriction”, “demonic composition”, “demonic union”, and the ordering “restriction of”. Recursion and iteration are interpreted as fixed points of continuous relationals. Developed in the framework of set theory, this calculus may be regarded as a systematic generalization of the functional style.

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Strongly compact cardinals, elementary embeddings and fixed points

Journal of Symbolic Logic ◽

10.2307/2274133 ◽

1984 ◽

Vol 49 (3) ◽

pp. 808-812

Author(s):

Yoshihiro Abe

Keyword(s):

Fixed Points ◽

Elementary Embedding ◽

Inner Model ◽

Power Set ◽

Elementary Embeddings ◽

Basic Facts ◽

The Universe ◽

Normal Ultrafilter ◽

Strongly Compact Cardinals

J. Barbanel [1] characterized the class of cardinals fixed by an elementary embedding induced by a normal ultrafilter on Pκλ assuming that κ is supercompact. In this paper we shall prove the same results from the weaker hypothesis that κ is strongly compact and the ultrafilter is fine.We work in ZFC throughout. Our set-theoretic notation is quite standard. In particular, if X is a set, ∣X∣ denotes the cardinality of X and P(X) denotes the power set of X. Greek letters will denote ordinals. In particular γ, κ, η and γ will denote cardinals. If κ and λ are cardinals, then λ<κ is defined to be supγ<κγγ. Cardinal exponentiation is always associated from the top. Thus, for example, 2λ<κ means 2(λ<κ). V denotes the universe of all sets. If M is an inner model of ZFC, ∣X∣M and P(X)M denote the cardinality of X in M and the power set of X in M respectively.We review the basic facts on fine ultrafilters and the corresponding elementary embeddings. (For detail, see [2].)Definition. Assume κ and λ are cardinals with κ ≤ λ. Then, Pκλ = {X ⊂ λ∣∣X∣ < κ}.It is important to note that ∣Pκλ∣ = λ< κ.

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Automorphisms of supermaximal subspaces

Journal of Symbolic Logic ◽

10.2307/2273783 ◽

1985 ◽

Vol 50 (1) ◽

pp. 1-9 ◽

Cited By ~ 15

Author(s):

R. G. Downey ◽

G. R. Hird

Keyword(s):

Vector Space ◽

Recursive Relation ◽

Effective Procedure ◽

Dimensional Vector ◽

Algebraic Systems ◽

Dimensional Vector Space ◽

Infinite Dimensional ◽

Vector Addition ◽

State Construction ◽

Recursive Set

An infinite-dimensional vector space V∞ over a recursive field F is called fully effective if V∞ is a recursive set identified with ω upon which the operations of vector addition and scalar multiplication are recursive functions, identity is a recursive relation, and V∞ has a dependence algorithm, that is a uniformly effective procedure which when applied to x, a1,…,an, ∈ V∞ determines whether or not x is an element of {a1,…,an}* (the subspace generated by {a1,…,an}). The study of V∞, and of its lattice of r.e. subspaces L(V∞), was introduced in Metakides and Nerode [15]. Since then both V∞ and L(V∞) (and many other effective algebraic systems) have been studied quite intensively. The reader is directed to [5] and [17] for a good bibliography in this area, and to [15] for any unexplained notation and terminology.In [15] Metakides and Nerode observed that a study of L(V∞) may in some ways be modelled upon a study of L(ω), the lattice of r.e. sets. For example, they showed how an e-state construction could be modified to produce an r.e. maximal subspace, where M ∈ L(V∞) is maximal if dim(V∞/M) = ∞ and, for all W ∈ L(V∞), if W ⊃ M then either dim(W/M) < ∞ or dim(V∞/W) < ∞.However, some of the most interesting features of L(V∞) are those which do not have analogues in L(ω). Our concern here, which is probably one of the most striking characteristics of L(V∞), falls into this category. We say M ∈ L(V∞) is supermaximal if dim(V∞/M) = ∞ and for all W ∈ L(V∞), if W ⊃ M then dim(W/M) < ∞ or W = V∞. These subspaces were discovered by Kalantari and Retzlaff [13].

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Infinite Nested Radicals - A Way to Express All Quantities Rational, Irrational Transcendental By a Single Integer Two

Mathematical Journal of Interdisciplinary Sciences ◽

10.15415/mjis.2020.91001 ◽

2020 ◽

Vol 9 (1) ◽

pp. 1-9

Author(s):

Narinder Kumar Wadhawan ◽

Priyanka Wadhawan

Keyword(s):

Binary System ◽

Recursive Relation ◽

Square Root ◽

Fixed Sequence ◽

Recursive Relations

This paper proves that all mathematical quantities including fractions, roots or roots of root, transcendental quantities can be expressed by continued nested radicals using one and only one integer 2. A radical is denoted by a square root sign and nested radicals are progressive roots of radicals. Number of terms in the nested radicals can be finite or infinite. Real mathematical quantity or its reciprocal is first written as cosine of an angle which is expanded using cosine angle doubling identity into nested radicals finite or infinite depending upon the magnitude of quantity. The finite nested radicals has a fixed sequence of positive and negative terms whereas infinite nested radicals also has a sequenceof positive and negative terms but the sequence continues infinitely. How a single integer 2 can express all real quantities, depends upon its recursive relation which is unique for a quantity. Admittedly, there are innumerable mathematical quantities and in the same way, there are innumerable recursive relations distinguished by combination of positive and negative signs under the radicals. This representation of mathematical quantities is not same as representation by binary system where integer two has powers 0, 1, 2, 3…so on but in nested radicals, powers are roots of roots.

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Dominical categories: recursion theory without elements

Journal of Symbolic Logic ◽

10.1017/s0022481200029649 ◽

1987 ◽

Vol 52 (3) ◽

pp. 594-635 ◽

Cited By ~ 15

Author(s):

Robert A. di Paola ◽

Alex Heller

Keyword(s):

Recursion Theory ◽

Incompleteness Theorem ◽

Partial Functions ◽

Polyadic Algebras ◽

Recursive Functions ◽

Principal Reason ◽

New Formulation ◽

Abstract Formulation ◽

Recursive Set

Dominical categories are categories in which the notions of partial morphisms and their domains become explicit, with the latter being endomorphisms rather than subobjects of their sources. These categories form the basis for a novel abstract formulation of recursion theory, to which the present paper is devoted. The abstractness has of course its usual concomitant advantage of generality: it is interesting to see that many of the fundamental results of recursion theory remain valid in contexts far removed from their classic manifestations. A principal reason for introducing this new formulation is to achieve an algebraization of the generalized incompleteness theorem, by providing a category-theoretic development of the concepts and tools of elementary recursion theory that are inherent in demonstrating the theorem.Dominical recursion theory avoids the commitment to sets and partial functions which is characteristic of other formulations, and thus allows for an intrinsic recursion theory within such structures as polyadic algebras. It is worthy of notice that much of elementary recursion theory can be developedwithout referencetoelements.By Gödel's generalized incompleteness theorem for consistent arithmetical systemTwe mean any statement of the following sort:(1) if every recursive set is definable inT, thenTis essentially undecidable [41]; or(2) if all recursive functions are definable inT, thenTis essentially undecidable [41]; or(3) if every recursive set is definable inT, thenT0andR0(the sets of Gödel numbers of the theorems and refutables ofT) are recursively inseparable [39]; or(4) if all re sets are representable inT, thenT0is creative [28], [39]; or(5) ifTis a Rosser theory (i.e., all disjoint re sets are strongly separable inT), thenT0andR0are effectively inseparable [39].

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On axiomatizability within a system

Journal of Symbolic Logic ◽

10.2307/2266324 ◽

1953 ◽

Vol 18 (1) ◽

pp. 30-32 ◽

Cited By ~ 114

Author(s):

William Craig

Keyword(s):

Recursive Function ◽

Recursive Relation ◽

Recursively Enumerable Set ◽

Formal Systems ◽

Primitive Recursive Function ◽

Recursively Enumerable ◽

Primitive Recursive ◽

Recursive Set

Let C be the closure of a recursively enumerable set B under some relation R. Suppose there is a primitive recursive relation Q, such that Q is a symmetric subrelation of R (i.e. if Q(m, n), then Q(n, m) and R(m, n)), and such that, for each m ϵ B, Q(m, n) for infinitely many n. Then there exists a primitive recursive set A, such that C is the closure under R of A. For proof, note that , where f is a primitive recursive function which enumerates B, has the required properties. For each m ϵ B, there is an n ϵ A, such that Q(m, n) and hence Q(n, m); therefore the closure of A under Q, and hence that under R, includes B. Conversely, since Q is a subrelation of R, A is included in C. Finally, that A is primitive recursive follows from [2] p. 180.This observation can be applied to many formal systems S, by letting R correspond to the relation of deducibility in S, so that R(m, n) if and only if m is the Gödel number of a formula of S, or of a sequence of formulas, from which, together with axioms of S, a formula with the Gödel number n can be obtained by applications of rules of inference of S.

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Note on a conjecture of Skolem

Journal of Symbolic Logic ◽

10.2307/2266735 ◽

1946 ◽

Vol 11 (3) ◽

pp. 73-74 ◽

Cited By ~ 1

Author(s):

Emil L. Post

Keyword(s):

Recursive Function ◽

Recursive Relation ◽

Natural Numbers ◽

Excellent Review ◽

Recursive Functions ◽

Diagonal Argument ◽

Primitive Recursive ◽

Recursive Relations

In his excellent review of four notes of Skolem on recursive functions of natural numbers Bernays states: “The question whether every relation y = f(x1,…, xn) with a recursive function ƒ is primitive recursive remains undecided.” Actually, the question is easily answered in the negative by a form of the familiar diagonal argument.We start with the ternary recursive relation R, referred to in the review, such that R(x, y, 0), R(x, y, 1), … is an enumeration of all binary primitive recursive relations.

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