User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190718022500/@comment-35470197-20190718072655

1) What does \(\textrm{Lim}_{x \to \infty}\) mean? The variable \(x\) does not occur in \(\Theta(t \# n)/f_{\textrm{PTO}(t \# n)+\textrm{FS}}\). If \(x\) is just the variable of the functions, then both side does not necessarily converge.

I note that we, general googologists, expect that FGH along a well-known system of fundamental sequences should be actually fast growing, but there are few results which ensure such expectation. FGH can even give non-increasing function. Larger ordinals can generate functions which grow slower than those generated by smaller ordinals. We are just intuitively believe that well-known systems of fundamental sequences are suitable for our purpose.

2) Right. I think that \(\textrm{ZFC}\) is the most accepted theory in computable googology.

3) It is a very difficult question for me, and maybe also for other sophisticated googologists. Even if you have two equivalent theories, which have automatically the same PTO, the function \(\Theta\) can perform in distinct ways. Moreover, PTO just scales the proof-theoretic strength for the statements on well-foundedness.

For example, \(\textrm{PA}\) is not so good at defining functions as set theories in the sense it costs formulae of great length. For example, it is quite tiresome to write down the defining formula of \(x^y\) in \(\textrm{PA}\).