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Karl-georg Niebergall - One of the best experts on this subject based on the ideXlab platform.

  • Hilbert's programme and gödel's theorems
    Dialectica, 2005
    Co-Authors: Karl-georg Niebergall, Matthias Schirn
    Abstract:

    In this paper, we attempt to show that a weak version of Hilberťs Metamathematics is compatible with Godel's Incompleteness Theorems by employing only what are clearly natural provability predicates. Defining first 4T proves the consistency of a theory S indirectly in one step", we subsequently prove (i) "PA proves its own consistency indirectly in one step" and sketch the proof for (ii) "If S is a recursively enumerable extension of (QF-IA), S proves its own consistency indirectly in one step". The formalizations of the metatheoretical consistency assertions that occur in these theorems are clearly the natural ones. We conclude the paper with reflections on indirect consistency proofs and soundness proofs. 1. Goders Incompleteness Theorems and Consistency Proofs The main goal of Hilberťs foundational project was to vindicate all of classical mathematics by means of a finitist metamathematical consistency "proof'. Hilbert considered classical mathematics to be the paradigm of unassailable truth and believed that finitist means, as conceived by him, were absolutely reliable.1 For decades it has been widely held that Godel's Second Incompleteness Theorem put an end to Hilberťs original proof-theoretic programme. On the face of it, this view seems plausible: if we succeeded in carrying out a consistency proof for all of mathematics in Metamathematics, mathematics would prove its own consistency, given that Metamathematics is only a small fragment of mathematics in its entirety. Yet the very possibility that mathematics proves its own consistency is ruled out by Godel's Second

  • Nonmonotonicity in (the Metamathematics of) Arithmetic
    Erkenntnis, 1999
    Co-Authors: Karl-georg Niebergall
    Abstract:

    This paper is an attempt to bring together two separated areas of research: classical mathematics and Metamathematics on the one side, non-monotonic reasoning on the other. This is done by simulating nonmonotonic “logic” through antitonic theory extensions. In the first half, the specific extension procedure proposed here is motivated informally, partly in comparison with some well-known non-monotonic formalisms. Operators V and, more generally, Uδ are obtained which have some plausibility when viewed as giving nonmonotonic theory extensions. In the second half, these operators are treated from a mathematical and metamathematical point of view. Here an important role is played by Uδ -closed theories and Uδ -fixed points. The last section contains results on V-closed theories which are specific for V.

Petr Hájek - One of the best experts on this subject based on the ideXlab platform.

Matthias Schirn - One of the best experts on this subject based on the ideXlab platform.

  • The Finite and the Infinite: On Hilbert’s Formalist Approach Before and After Gödel’s Incompleteness Theorems
    Logique Et Analyse, 2019
    Co-Authors: Matthias Schirn
    Abstract:

    In this essay, I discuss formalist and finitist aspects of Hilbert’s proof-theoretic programme in the 1920s and 1930s. I begin by pointing out some difficulties arising from his construction of intuitive number theory in his first paper on Metamathematics in the 1920s and by characterizing the nature of his Metamathematics. In what follows, I argue thatHilbert’s paradigm of a transfinite axiom as well as the other “ordinary” transfinite axioms derivable from it fail to supply any formal explication of the term “infinite” or “transfinite”. On the one hand, Hilbert regards both the mathematical and the logical signs as detached from all meaning once the process of formalization of contentual mathematics has been completely carried out. On the other hand, there are several places where he characterizes the finitary or real sentences of the language of formalized arithmetic, in contrast to its transfinite or ideal sentences, expressly as meaningful. I make a proposal to explain why for Hilbert it might still be of interest to be able to rely on meaningful sentences in the language of formalized arithmetic. In section 5, I critically discuss several aspects in Hilbert and Bernays’s extension of the finitist point of view in the second volume of  Foundations of Mathematics (1939). In section 6, I argue that by appreciating the distinctive character of the notion of an “approximative” consistency proof we can make good sense of the nature of the consistency proofs that Hilbert outlines both in his classical papers on proof theory in the 1920s and in the first volume of  Foundations of Mathematics  (1934) . I further argue that a weak version of Hilbert’s programme is compatible with Godel’s second incompleteness theorem by using only what are clearly natural provability predicates.

  • Hilbert's programme and gödel's theorems
    Dialectica, 2005
    Co-Authors: Karl-georg Niebergall, Matthias Schirn
    Abstract:

    In this paper, we attempt to show that a weak version of Hilberťs Metamathematics is compatible with Godel's Incompleteness Theorems by employing only what are clearly natural provability predicates. Defining first 4T proves the consistency of a theory S indirectly in one step", we subsequently prove (i) "PA proves its own consistency indirectly in one step" and sketch the proof for (ii) "If S is a recursively enumerable extension of (QF-IA), S proves its own consistency indirectly in one step". The formalizations of the metatheoretical consistency assertions that occur in these theorems are clearly the natural ones. We conclude the paper with reflections on indirect consistency proofs and soundness proofs. 1. Goders Incompleteness Theorems and Consistency Proofs The main goal of Hilberťs foundational project was to vindicate all of classical mathematics by means of a finitist metamathematical consistency "proof'. Hilbert considered classical mathematics to be the paradigm of unassailable truth and believed that finitist means, as conceived by him, were absolutely reliable.1 For decades it has been widely held that Godel's Second Incompleteness Theorem put an end to Hilberťs original proof-theoretic programme. On the face of it, this view seems plausible: if we succeeded in carrying out a consistency proof for all of mathematics in Metamathematics, mathematics would prove its own consistency, given that Metamathematics is only a small fragment of mathematics in its entirety. Yet the very possibility that mathematics proves its own consistency is ruled out by Godel's Second

Tim Button - One of the best experts on this subject based on the ideXlab platform.

  • The Metamathematics of Putnam's model-theoretic arguments
    2017
    Co-Authors: Tim Button
    Abstract:

    Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the Metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.Philip Scowcroft has written a very useful review of this paper, on MathSciNet, MR2785345 (2012e:03005).Published in Erkenntnis 74.3: 321–49.

  • The Metamathematics of Putnam’s Model-Theoretic Arguments
    Erkenntnis, 2011
    Co-Authors: Tim Button
    Abstract:

    Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the Metamathematics of all three model-theoretic arguments, and I argue against Bays ( 2001 , 2007 ) that Putnam is largely immune to metamathematical challenges.

  • The Metamathematics of Putnam's Model-Theoretic Arguments
    Erkenntnis, 2011
    Co-Authors: Tim Button
    Abstract:

    Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the Metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.

Raymond M. Smullyan - One of the best experts on this subject based on the ideXlab platform.

  • Recursion Theory for Metamathematics
    1993
    Co-Authors: Raymond M. Smullyan
    Abstract:

    This work is a sequel to the author's Gödel's Incompleteness Theorems, though it can be read independently by anyone familiar with Gödel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the Metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field.

  • Recursion Theory for Metamathematics - Generative Sets and Creative Systems
    Recursion Theory for Metamathematics, 1993
    Co-Authors: Raymond M. Smullyan
    Abstract:

    we now have the background to study the beautiful subject of creative and productive sets inaugurated by Emil Post [1944]. This plays a key rôle in the metamathematical study of incompleteness and undecidability. §1. Productive and Creative Sets. To say that a set A is not r.e. is equivalent to saying that for any r.e. subset ωi of A, there is a number in A not in ωi . A is called productive if there is a recursive function f(x)—called a productive function for A—such that for any number i, if ωi Í A, then f(i) Î A - ωi . [Informally, this means that it is not only true that no r.e. subset of A is A, but given any such r.e. subset, we can effectively find a number which is in A but not in the subset.] A set A is called creative (after Post) if it is r.e., and its complement is productive. Let us note that a productive function for the complement of a set A is a recursive function f(x) such that for any number i such that ωi is disjoint from A, the number f(i) lies outside both ωi and A. A system S is called productive if the set P of Gödel numbers of the provable formulas of S is a productive set; S is called creative if the set P is creative. As we will see, the complete theory N is not only not axiomatizable but is productive, and the system P.A. is not only undecidable but creative. Post’s Sets C and K. A simple example of a creative set is Post’s set C —the set of all numbers x such that x Î ωx. Thus for any number i, i Î C ↔ i Î ωi. If ωi is disjoint from C, then i is outside both ωi and C and, therefore, the identity function I(x) is a productive function for C͂. Therefore, C͂ is productive, and since C is obviously r.e., C is creative.

  • Recursion Theory for Metamathematics - Universal and Doubly Universal Systems
    Recursion Theory for Metamathematics, 1993
    Co-Authors: Raymond M. Smullyan
    Abstract:

    We now turn to two theorems (Theorems A and B below) that will play a major role in this study. We will give three different proofs of them in the course of this volume, since each proof reveals certain interesting features of its own. . . . Theorem A. If( A1 , A2 ) is semi-D.U. and A1 and A2 are both r.e., then A1 and A2 are both universal sets. Theorem B. I f ( A1 , A2 ) is semi-D.U. and A1 and A2 are both r.e., then (A1, A2) is D.U. . . Of course, Theorem A is a trivial corollary of Theorem B, but our proofs of Theorem A reveal facts not revealed by our proofs of Theorem B. We give our first proofs in this chapter. After each proof, we establish a metamathematical corollary: Theorem A yields the result of Ehrenfeucht-Feferman [1960] that for any consistent axiomatizable Rosser system S for sets in which all recursive functions of one argument are strongly definable, all r.e. sets are representable in S. Theorem B yields the stronger result of Putnam-Smullyan [1960]— that any such system S is an exact Rosser system for sets. This result is apparently incomparable in strength with Shepherdson’s result that any consistent axiomatizable Rosser system for binary relations is an exact Rosser system for sets. Both results, of course, yield different proofs that every consistent axiomatizable extension of (R) is an exact Rosser system for sets. §1. Generativity and Universality. We have shown that every universal set is generative. Our first proof of Theorem A will be based on the converse. Theorem 1. Every generative set is universal. We will, in fact, prove something considerably stronger which will have other applications as well. Consider a collection C of r.e. sets.

  • recursion theory for Metamathematics
    1993
    Co-Authors: Raymond M. Smullyan
    Abstract:

    1. Recursive Enumerability and Recursivity 2. Undecidability and Recursive Inseparability 3. Indexing 4. Generative Sets and Creative Systems 5. Double Generativity and Complete Effective Inseparability 6. Universal and Doubly Universal Systems 7. Shepherdson Revisited 8. Recursion Theorems 9. Symmetric and Double Recursion Theorems 10. Productivity and Double Productivity 11. Three Special Topics 12. Uniform Godelization