Talk:Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic

From ProofWiki
Jump to navigation Jump to search

It is not clear which notion of definability is in play here. Ordinarily in this context, one is concerned with subsets of some model of a theory T -- the set A is definable in the structure M if and only if there is a formula F(x) in the language of T and A is the set of members a of M such that M satisfies F(a). -- 03:32, 30 April 2014‎ Ratboy

I think you hit the nail on the head! The original definition from the source material is (paraphrasing): "If $T$ is a consistent theory in the language of arithmetic, we say a set $S$ is defined in $T$ by $D(x)$ if for all $n$, $n \in S$ if and only if $D(n)$ is a theorem of $T$. $S$ is definable in $T$ if $S$ is defined in $T$ by some such formula $D(x)$." I'd never heard of a notion of definability which is based on provability before checking the source material for this page. Apparently, neither had Burak, hence his (valid) criticisms on the page. I didn't see any such notion when I looked on Wikipedia, either. Going to take a shot at refactoring the Gödel's First Incompleteness Theorem proof so that this page is either unnecessary, or minimally is updated to use modern mathematical language. Also hoping to fill in some of the red links in the proof along the way. JamesMazur2 (talk) 16:34, 9 July 2017 (EDT)
Feel free, nobody's been down this road in some time. If you can approach this in an appropriately rigorous manner (perhaps either constructing a formulation in a similar way as was done with the URM analysis, or even by using that actual formulation), then take it away and rock 'n' roll with it. --prime mover (talk) 18:24, 9 July 2017 (EDT)