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

Published online by Cambridge University Press:  10 June 2020

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/

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.

Type
Articles
Copyright
© The Association for Symbolic Logic 2020

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