Gödel’s Theorem

The purpose of this article is to examine aspects of the development of the concept and theory of computability through the theory of recursive functions. Following a brief introduction, Section 2 is devoted to the presuppositions of computability. It focuses on certain concepts, beliefs and theorems necessary for a general property of computability to be formulated and developed into a mathematical theory. The following two sections concern situations in which the presuppositions were realized and the theory of computability was developed. It is suggested in Section 3 that a central item was the problem of generalizing Godel's incompleteness theorem. It is shown that this involved both the characterization of recursiveness and the attempt to clarify and formulate the notion of an effective process as it relates to the syntax of deductive systems. Section 4 concerns the decision problems which grew from the Hilbert program. Section 5 is devoted to the development of an informal' technique in the theory of...

Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory

A generalization of the Second Incompleteness Theorem and some exceptions to it

Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan (improving upon [Kle50] and [Kle52]) showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1is speedable over another system S0on a set of formulaeAiff, for each recursive functionh, there is a formulaαinAsuch that the length of the shortest proof ofαin S0is larger thanhof the shortest proof ofαin S1. (Here we equate the length of a proof with something like the number of characters making it up,notits number of lines.) We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1but unprovable in S0from the collectionA–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below,ϕis an acceptable numbering of the partial recursive functions [Rog87] andΦa (Blum) complexity measure associated withϕ[Blu67], [DW83].

Gentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of these calculi. In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934–35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG. After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus.

Recursive function theory studies functions that can be evaluated by an algorithm based on some computing model such as a Turing machine or computer program. These functions are called partial recursive (p.r.) and are defined on subsets of N, where N = {0, 1,... }, with values in N. (A good introduction to the theory may be found in [2].) This paper explains the Parameter Theorem, illustrates some unusual computer programs, and explains their connection to recursive function theory. Basing our computing model on a modern high level language, we can construct programs whose existence is known through more formal means. Our functions will operate on strings rather than numbers, however. There is no loss in generality since strings can be viewed as members of N, and vice-versa. From a practical aspect, this approach is somewhat more desirable since conventional programming languages often allow making use of long strings, but place more restrictive limits on the size of integers. We first pose a problem: Does there exist a program (written in some conventional programming language) which halts if and only if its input is identical to the program's text (i.e., source code)? For the remainder of this note X is the set of ASCII characters 'a,' 'b, +,' space, line feed, and so forth, needed to write programs in most programming languages. We denote by X * the set of all strings (finite sequences) over X. Let P1 be the set of all one variable partial recursive functions on X *. A function I belongs to P1 if and only if there is a subset of S of X* and some algorithm which computes 't by associating with each element of S a well-defined element of X *. Similarly, 't belongs to P2 if and only if there is a set S c X * X X *, and an algorithm for 't which associates with each element of S a well-defined element of X *. A partial recursive function I in P1 (or in P2) is recursive if the algorithm is defined on all of X * (or X * X X *). This is equivalent to saying that the algorithm which evaluates I always halts. Our computing model will be based on Adat; its high-level string-handling properties simplify later examples. For readability, however, we will express algorithms in a more familiar Pascal-like style. Readers familiar with either programming language should find forthcoming algorithms sufficiently clear. A formal presentation in Ada is given in the Appendix. A string of ASCII characters, such as in FIGURE 1, represents a function in our language which takes strings to strings. Of course, strings of ASCII characters such as The quick brown fox and %*!! represent nothing at all in our programming language. Let A1 be the set of all strings in X * representing functions in our language of one input STRING which outputs a STRING. Similarly let A2 be the set of all strings in

Computation of recursive functionals using minimal initial segments

i. S. A. Berezin, "An algebra of one-place primitive recursive functions with iteration operation of general form," Kibernetika, No. 3, 12-19 (1976). 2. G. P. Gavrilov, "Functional completeness in a countably valued logic," in: Problems of Cybernetics [in Russian], No. 15 (1965), pp. 5-64. 3. D. A. Zakharov, Recursive Functions [in Russian], Novosibirsk (1970). 4. V. V. Koz'minykh, "One-place primitive recursive functions," Algebra Logika, ~, 75-90 (1968). 5. A. I. ~l'tsev, "Iterative algebras and Post varieties," Algebra Logika, ~, 5-24 (1966). 6. A. I. Mal'tsev, "Constructive algebras, I," Usp. Mat. Nauk, 16, No. 3, 3-60 (1961). 7. S. S. Marchenkov, ,"B°unded. . recursions," Math. Balkanlca," 2, 124-142 (1972).~ 8. S S. ~rchenkov, Ellmlnationof recursion schemes in the--Grzegorczyk class ," Mat. Zametki, 5, No. 5, 561-568 (1969). 9. V, L. Mikheev, "A class of algebras of primitive recursive functions," Mat. Zametki, 14, No. 1 143-156 (1973). i0. S. V. Pakhomov, "A simple syntactical definition of all classes of the Grzegorczyk hierarchy," Zap. Nauch. Sem. Leningr. Otd. ~at. Inst. Akad. Nauk SSSR, 40, 127-130 (1974). Ii. H. Rogers, Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill (1967). 12. M. G. Rozinas, "An algebra of many-place primitive recursive functions," Uch. Zap. Ivanovsk. Gos. Pedagog. Inst., i17, 95-111 (1972). 13. A. Cobham, "The intrinsic computational difficulty of functions," in: Logic, Methodology and Philosophy of Science, Amsterdam (1965), pp. 24-30. 14. A. Grzegorczyk, "Some classes of recursive functions," in: Rozprawy Matematiczne, Vol. 4, Warsaw (1953). 15. R. W. Ritchie, "Classes of predictably computable functions," Trans. Am. Math. Soc., 106, 139-173 (1963).

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular ifPis some property, then the statement “for eachmthere isnsuch thatP(m, n)” means that we can construct a (computable) functionθsuch thatP(m, θ(m))for allm. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.

Gödel’s Incompleteness Theorem was a major discovery in modern logic that has consistently attracted the attention of scientific and philosophical circles. However, since Gödel’s Incompleteness Theorem was put forward, the scientific and philosophical significance of its proof has been questioned, especially Wittgenstein, who strongly criticized it. Wittgenstein’s main argument against this theorem was that he believed Gödel sentence was a logical contradiction, that there was no such thing as a true and unprovable proposition at all, that it was a self-referential and self-contradictory concept, and that we should abandon the natural language explanation of this proposition. The author points out that Gödel sentence is a logically invalid formula constructed outside the domain of the principle of representability and lacks recursion; this contradiction stems from the abuse of formalization in language. At the same time, the author discovered a paradox in the proof of Gödel’s Incompleteness Theorem, that is, Gödel sentence itself is a kind of Russell’s Paradox. The author believes that the philosophical root of Russell’s Paradox and Gödel sentence lies in adhering to the idea of actual infinity, while the Infinite Exchange Paradox overturns the Cantor’s diagonal argument, which shakes the foundation of the idea of actual infinity. The above findings strongly support Wittgenstein’s view. Therefore, the conclusion of this paper is that Wittgenstein’s questioning of Gödel sentence is completely correct, the proof of Gödel’s Incompleteness Theorem is based on a logically invalid formula, and the contradiction of the proof originates from Gödel sentence itself, thus it cannot be attributed to the incompleteness of the system, so the proof is wrong.

In this paper we proved that the function class CFRF and its proper subclass CFPRF are respectively the partial recursive functions and primitive recursive functions of context free languages (CFLs). Also we discussed the relation between them and recursive functions defined on other domains. It is indicated that the functions of natural numbers and/or symbol strings (words) are functions of CFLs. Several frequently used primitive recursive functions on words were given, including logical connectives, conditional expressions. Also the powerful operators (bounded maximization and minimization operators) for constructing primitive recursive functions were defined. Two important nontrivial algorithms, the characteristic function of arbitrary CFL and the parse function of CFL sentences were constructed. Based on them, the method for extending or restricting function domain was described.

Minimization and NP multifunctions

Tarski's theory of definability: common themes in descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic

In spite of the many efforts made to clarify von Neumann's methodology of science, one crucial point seems to have been disregarded in recent literature: his closeness to Hilbert's spirit. In this paper I shall claim that the scientific methodology adopted by von Neumann in his later foundational reflections originates in the attempt to revaluate Hilbert's axiomatics in the light of Gödel's incompleteness theorems. Indeed, axiomatics continues to be pursued by the Hungarian mathematician in the spirit of Hilbert's school. I shall argue this point by examining four basic ideas embraced by von Neumann in his foundational considerations: a) the conservative attitude to assume in mathematics; b) the role that mathematics and the axiomatic approach have to play in all that is science; c) the notion of success as an alternative methodological criterion to follow in scientific research; d) the empirical and, at the same time, abstract nature of mathematical thought. Once these four basic ideas have been accepted, Hilbert's spirit in von Neumann's methodology of science will become clear.

A model for computation over an arbitrary (ordered) ring R is presented. In this general setting, universal machines, partial recursive functions, and NP-complete problems are obtained. While the theory reflects of classical over Z (e.g. the computable functions are the recursive functions), it also reflects the special mathematical character of the underlying ring R (e.g. complements of Julia sets provide natural examples of recursively enumerable undecidable sets over the reals) and provides a natural setting for studying foundational issues concerning algorithms in numerical analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>