Forum:Measure the strength of a theory

I've read about different ways to measure the computable strength of a theory (number theory or set theory). e.g. Then, what's the relationships among PTO(T), the growth rate of BBT, and \(\kappa_T\)? &#123;hyp/^,cos&#125; (talk) 08:34, February 14, 2018 (UTC)
 * 1) The proof-theoretic ordinal of theory T. That's an ordinal.
 * 2) Provable "busy beavers". For positive integer n, take every proof in T with less than n symbols, which shows a Turing machine with 2 colors, less than n states halts. Then BBT(n) the maximum numbers of 1's that can be written by that set of Turing machines. Use function BBT to indicate the strength of T.
 * 3) Growth rate limit. Let \(\kappa_T\) be the least ordinal \(\alpha\) such that \(f_\alpha\) eventually dominates all functions provably total in T. That's an ordinal.