User blog comment:Primussupremus/What are the limits of Googology?/@comment-27513631-20170531200824

Depends on in what way, tbh. I'll restrict myself to the natural numbers because there is - dispute - as to whether the creation/study of large ordinals is googology or not.


 * Provably in a given theory \(T\): Depends on \(T\). If the consistency strength of any finite fragment of \(T\) is less than that of the entire theory and \(T\) is recursively enumerable, then yes, we have reached the limit: we merely need to diagonalize over functions not provably recursive/total in the first \(n\) axioms. Otherwise, I don't know what methods are known, besides converting between theories that have the same \(\Delta^0_1\)/parametered-\(\Pi^0_2\)*-provable sentences. (*might have got that second part wrong)


 * Recursive: Yes. Take a recursively enumerable theory \(T\), take a Turing machine that sums the Turing machine halt times of the \(n\)-state Turing machines which can be proven to halt in \(n\) steps, and you have a recursive function of \(n\) that outgrows all \(T\)-provably recursive functions.


 * Recursively-constructed: No, and possibly never. This is the one that most googologists are interested in. For a formal definition, I'd propose that a collection of functions \(C\) is \(\alpha\)-recursively constructed if there is a recursive function \(F:C \to \alpha\)* with cofinal range such that \(f\in C\) is provably recursive in \(T+TI_{\alpha}\), some theory \(T\) (such as PA or PRA, choice of T affects valid choices of \(\alpha\)) with the additional scheme of transfinite induction on an arbitrary recursive wellorder of order type \(\alpha\). Hopefully didn't screw that up. Anyway, that would probably be intricately tied to ordinal analyses of certain theories, to the point that we could effectively only create an \(\alpha\)-recursively constructed collection of functions if we had an ordinal analysis of a theory of proof-theoretic strength of, above, or slightly below \(\alpha\). *(recursive ordinals being interpreted as recursive wellorders, the higher-order function being interpreted as recursive by assuming the existence of a partial recursive function \(a:s \to C\), s a recursive language over a finite alphabet, and both \(F\) and \(a\) being implicitly curried so as to make their domains first-order)


 * Non-recursive definable: yes. We can effectively restrict the construction of Rayo's number to \(\Pi_n\) predicates, fixed \(n\), to get a cofinal sequence of \(\omega\) first-order definable functions (ordered by growth rate). I am not aware of any intermediate well-defined categories, unfortunately, besides (say) restrictions to \(\Pi_2\)-definable functions, for instance.


 * Meta-definable functions: no, possibly never. This one very much depends on the theory we work with, as for instance, the answer will be very different for new foundations or positive set theory or intuitionistic type theory compared to a well-founded set theory, like ZFC. This is where Rayo's number etc. live, due to not being first-order definable in the theory's language, but being first-order definable in a recursive extension of the theory's language and axioms, which may (as in Rayo's number and the Bigeddons) or may not (as in BIG FOOT) be valid over an arbitrary base universe, although it should be unique if valid. There has been almost no work on what is meta-definable over a universe, so it's incredibly hard to know if there is a theoretical bound to this or not.