On Godel's treatment of the undecidable in 1931

Abstract

In this article we discuss the proof in the short unpublished paper appeared in the 3rd volume of Godel's Collected Works entitled "On undecidable sentences" (*1931?), which provides an introduction to Godel's 1931 ideas regarding the incompleteness of arithmetic. We analyze the meaning of the negation of the provability predicate, and how it is meant not to lead to vicious circle. We show how in fact in Godel's entire argument there is an omission regarding the cases of non-provability, which, once taken into consideration again, allow a completely different view of Godel's entire argument of incompleteness. Previous results of the author are applied to show that the definition of a contradiction is included in the argument of *1931?. Furthermore, an examination of the application of substitution in the well-known Godel formula as a violation of uniqueness is also briefly presented, questioning its very derivation.

arXiv comment9 pages, This paper will be presented at the Round Table Revisiting Godel's Incompleteness Theorem WCP Rome 2024