[FOM] 18 Word Proof of the Godel, Rosser and Smullyan Incompleteness Theorems

[Charlie V]

Godel’s 1931 First Incompleteness Theorem is equivalent to the assertion that truth and provability do not coincide.  Rosser’s 1936 extension is equivalent to the assertion that provability and unrefutability do not coincide.  Smullyan’s 1961 Dual Form Theorem is equivalent to the assertion that truth and unrefutability do not coincide.

We can prove that these three sets, the true, provable and unrefutable sentences, are three different sets by referring to the property of being recursively enumerable, as stated in theorems of the Theory of Computation:

The sets of true, provable and unrefutable sentences pairwise differ w.r.t. whether they or their complement is r.e.

Charlie Volkstorf
Cambridge, MA