NDLI: Consistency, Models, and Soundness

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Consistency, Models, and Soundness

Content Provider	Springer Nature Link
Author	Schirn, Matthias
Copyright Year	2010
Abstract	This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. J. Brouwer and D. Hilbert (1899). Part 2 is mainly devoted to Hilbert’s proof theory of the 1920s (1922–1931). I begin with an account of his early attempt to prove directly, and thus not by reduction or by constructing a model, the consistency of (a fragment of) arithmetic. In subsequent sections, I give a kind of overview of Hilbert’s metamathematics of the 1920s and try to shed light on a number of difficulties to which it gives rise. One serious difficulty that I discuss is the fact, widely ignored in the pertinent literature on Hilbert’s programme, that his language of finitist metamathematics fails to supply the conceptual resources for formulating a consistency statement qua unbounded quantification. Along the way, I shall comment on W. W. Tait’s objection to an interpretation of Hilbert’s finitism by Niebergall and Schirn, on G. Gentzen’s allegedly finitist consistency proof for Peano Arithmetic as well as his ideas on the provability and unprovability of initial cases of transfinite induction in pure number theory. Another topic I deal with is what has come to be known as partial realizations of Hilbert’s programme, chiefly advocated by S. G. Simpson. Towards the end of this essay, I take a critical look at Wittgenstein’s views about (in)consistency and consistency proofs in the period 1929–1933. I argue that both his insouciant attitude towards the emergence of a contradiction in a calculus and his outright repudiation of metamathematical consistency proofs are unwarranted. In particular, I argue that Wittgenstein falls short of making a convincing case against Hilbert’s programme. I conclude with some philosophical remarks on consistency proofs and soundness and raise a question concerning the consistency of analysis.
Starting Page	153
Ending Page	207
Page Count	55
File Format	PDF
ISSN	11221151
Journal	Axiomathes
Volume Number	20
Issue Number	2-3
e-ISSN	15728390
Language	English
Publisher	Springer Netherlands
Publisher Date	2010-08-11
Publisher Place	Dordrecht
Access Restriction	Subscribed
Subject Keyword	Compactness theorem Completeness theorem Relative/direct/approximative consistency proof Consistency statement Creation by definition Finitist point of view Game formalism Metamathematics Model $$ Π_{1}^{0} $$ -sentence Soundness Transfinite induction Philosophy Ontology Linguistics (general) Cognitive Psychology Logic
Content Type	Text
Resource Type	Article
Subject	Philosophy Mathematics

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