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GÖDEL’S SECOND INCOMPLETENESS THEOREM: HOW IT IS DERIVED AND WHAT IT DELIVERS

Part of: Theory of computing Computability and recursion theory Proof theory and constructive mathematics

Published online by Cambridge University Press:  10 June 2020

SAEED SALEHI
SAEED SALEHI*
Affiliation:
RESEARCH INSTITUTE FOR FUNDAMENTAL SCIENCES UNIVERSITY OF TABRIZ 29 BAHMAN BOULEVARD, P.O. BOX 51666-17766, TABRIZ, IRAN and SCHOOL OF MATHEMATICS INSTITUTE FOR RESEARCH IN FUNDAMENTAL SCIENCES P.O. BOX 19395-5746, TEHRAN, IRAN E-mail: root@saeedsalehi.ir URL: http://saeedsalehi.ir/
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Abstract

The proofs of Gödel (1931), Rosser (1936), Kleene (first 1936 and second 1950), Chaitin (1970), and Boolos (1989) for the first incompleteness theorem are compared with each other, especially from the viewpoint of the second incompleteness theorem. It is shown that Gödel’s (first incompleteness theorem) and Kleene’s first theorems are equivalent with the second incompleteness theorem, Rosser’s and Kleene’s second theorems do deliver the second incompleteness theorem, and Boolos’ theorem is derived from the second incompleteness theorem in the standard way. It is also shown that none of Rosser’s, Kleene’s second, or Boolos’ theorems is equivalent with the second incompleteness theorem, and Chaitin’s incompleteness theorem neither delivers nor is derived from the second incompleteness theorem. We compare (the strength of) these six proofs with one another.

Keywords

first incompleteness theoremsecond incompleteness theoremGödel’s proofRosser’s proofKleene’s proofChaitin’s proofBoolos’ proof

MSC classification

Primary: 03F40: Gödel numberings and issues of incompleteness
Secondary: 03F30: First-order arithmetic and fragments 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.) 03D32: Algorithmic randomness and dimension
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 26 , Issue 3-4 , December 2020 , pp. 241 - 256
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Copyright
© The Association for Symbolic Logic 2020

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