What does gödel's second theorem say

Philosophia Mathematica 9 (1):37-71 (2001)

Authors
Michael Detlefsen
University of Notre Dame
Abstract
We consider a seemingly popular justification (we call it the Re-flexivity Defense) for the third derivability condition of the Hilbert-Bernays-Löb generalization of Godel's Second Incompleteness Theorem (G2). We argue that (i) in certain settings (rouglily, those where the representing theory of an arithmetization is allowed to be a proper subtheory of the represented theory), use of the Reflexivity Defense to justify the tliird condition induces a fourth condition, and that (ii) the justification of this fourth condition faces serious obstacles. We conclude that, in the types of settings mentioned, the Reflexivity Defense does not justify the usual ‘reading’ of G2—namely, that the consistency of the represented theory is not provable in the representing theory.
Keywords Gödel  Gödel's incompleteness theorems  Gödel's second incompleteness theorem  Hilbert-Bernays derivability conditions  Löb's derivability conditions  representing metamathematical notions
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DOI 10.1093/philmat/9.1.37
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Hilbert's Program Then and Now.Richard Zach - 2007 - In Dale Jacquette (ed.), Philosophy of Logic. Amsterdam: North Holland. pp. 411–447.
2004 Summer Meeting of the Association for Symbolic Logic.Wolfram Pohlers - 2005 - Bulletin of Symbolic Logic 11 (2):249-312.

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