User blog comment:Ynought/Another question/@comment-34422464-20190709152325

\(\textrm{PTO}(L)\geq \omega^{\textrm{CK}}_1\) won't work unless you define a logic stronger than first-order logic it can only do finite proofs, and all of its inference rules are computable. There are other systems such as \(\omega\)-logic which are stronger than FOL and can support non-recursively enumerable theories.