User blog comment:Scorcher007/The whole googology in one diagram/@comment-11227630-20180323134709/@comment-30754445-20180325072025

This is not true for the computable functions.

All but the weakest axiomatic systems are Turing-complete, which means that you can define any computable function in their language.

Amazingly enough, this appears to be possible even in Peano Arithemtic (see here). You can actually talk about Turing Machines and Busy Beavers in PA. As a result, both sides of the computable/uncomputable border are exactly the same in PA and in all stronger languages.

OTOH it is true that stronger languages can describes stronger uncomputable functions. That part of your diagram is correct.