Journal ArticleDOI
Hilbert's programme and gödel's theorems
TLDR
In this article, it was shown that a weak version of Hilber's metamathematics is compatible with Godel's Incompleteness Theorems by employing only what are clearly natural provability predicates.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 Secondread more
Citations
More filters
Book ChapterDOI
Deflationism and Instrumentalism
TL;DR: In this article, an instrumentalist reading of deflationism is proposed, which is particularly interesting considering the role a truth predicate can have in facilitating proofs, and provides a new answer to one of the arguments challenging deflationism.
Journal ArticleDOI
Consistency, models, and soundness
TL;DR: In this paper, the authors focus on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction and give a kind of overview of Hilbert's metamathematics of the 1920s.
Journal ArticleDOI
Gödel mathematics versus Hilbert mathematics. I The Gödel incompleteness (1931) statement: axiom or theorem?
TL;DR: In this article , the authors focus on the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics.
Journal ArticleDOI
What finitism could not be
TL;DR: In this paper, the authors show that Tait's claim that finitist functions are primitive recursive functions is disputable and that another, also defended by Tait, is untenable.
References
More filters
Journal ArticleDOI
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I
Book
Metamathematics of First-Order Arithmetic
Petr Hájek,Pavel Pudlák +1 more
TL;DR: This chapter discusses Arithmetic as Number Theory, Set Theory and Logic, Fragments and Combinatorics, and Models of Fragments of Arithmetic.
Book
Grundlagen der Mathematik
David Hilbert,Paul Bernays +1 more
TL;DR: In this article, a vorliegende Buch soll einer eingehenden Orientierung uber den gegenwartigen Stoff der HILBERTschen Beweistheorie dienen.
Book
The logic of provability
TL;DR: In this paper, the completeness and decidability of GL and other modal logics are discussed, including the fixed-point theorem, the fixed point theorem, letterless sentences, and analysis.