The incompleteness theorems after 70 years
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)].
scientific article; zbMATH DE number 2083176
scientific article