User blog comment:Alemagno12/First Order Theory Theory/@comment-27513631-20170809194629

A few comments:

What's stopping this FOTT from being a FOTT-theory, and failing for the same reason as Oblivion?

A (recursively enumerable, which almost all are) theory can have predicates and booleans encoded into \(\mathbb N\), and the set of provably true statements is therefore \(\Sigma^0_1\) (i.e. of the form \(\exists m \cdots \exists n(\ldots)\) where \(\ldots\) is a combination of \(0\), \(1\), \(+\), \(\times\), \(\geq\), \(\lnot\) and \(\wedge\)). I don't think it's useful for your version much, but it might be nice to know in general.

What about \(\varphi(S|T) \leftrightarrow \lnot\varphi(S|T)\) and other similar statements that define theories? Do theories have to be consistent, and use the law of excluded middle etc.?

For your second point in the final paragraph, T(1) < T(2), so statements etc. in T(2) are also in T(1); I'm pretty sure this is backwards? Also, why do T(n) > T(m) instead of n > m?

What is the syntax of some T(n)? It includes X, but why?

What theories does T(n) contain? T(m) for m < n??

Think that's it.