The incompleteness theorems after 70 years
type: Item
instance of: scholarly article
title: The incompleteness theorems after 70 years
zbMATH Open document ID: 1053.03033
author: Henryk Kotlarski
published in: Annals of Pure and Applied Logic
publication date: 2004-08-06T00:00:00Z
review text: In the recent years there has been much work done on alternative proofs of Gödel's incompleteness theorems and related results. This paper summarizes some of that work and presents some reasonably self-contained proofs. The main tools are the Arithmetized Completeness Theorem in the form adapted by \textit{H. Kotlarski} [J. Symb. Log. 59, 1414--1419 (1994; Zbl 0816.03025)] and partial inductive satisfaction classes. Kotlarski gives a proof of Tarski's Theorem on Undefinability of Truth in the form: Let \(T\) be a consistent theory in the language of arithmetic extending PA. Then, there is no formula \(\eta\) such that, for every sentence \(\xi\) (identified with its Gödel number), \(T\vdash \xi\equiv \eta(\xi)\). Then, he presents a proof of the second incompleteness theorem based on Berry's paradox, due to \textit{M. Kikuchi} [Math. Log. Q. 40, 528--532 (1994; Zbl 0805.03052)] and his own proof based on the ``busy beaver problem''. There is also a comment on Adamowicz's proof of the second incompleteness theorem using \(\exists_1\)-closed models from \textit{Z. Adamowicz} and \textit{T. Bigorajska} [J. Symb. Log. 66, 349--356 (2001; Zbl 0981.03043)].
reviewed by: Roman Kossak
Mathematics Subject Classification ID: 03F30
Mathematics Subject Classification ID: 03F03
Mathematics Subject Classification ID: 03C62
zbMATH DE Number: 2083176
zbMATH Keywords: Gödel's incompleteness theorems
zbMATH Keywords: arithmetized completeness theorem
zbMATH Keywords: satisfaction classes
MaRDI profile type: Publication
full work available at URL: j.apal.2003.10.012
OpenAlex ID: W1992712548
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DOI: 10.1016/J.APAL.2003.10.012
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description: scientific article; zbMATH DE number 2083176