User blog comment:P進大好きbot/New Googological Ruler/@comment-31580368-20190718022500/@comment-31580368-20190719024303

This page was my first attempt to deal with Googlology (there I analyzed the functions of the Hypсos and some of my unformalized ones). In fact, it was a draft for my book, which I write for the Russian-speaking community. This book will contain all the definitions, functions and detailed explanations. Now there are 6 chapters in which this material has been reworked with strict definitions (until the KPM level, but every day I continue to work on the book). As for this page (2000 steps), I have already written many times that it is not necessary to refer to it, at the moment it has many errors and unformalized stuff. But people still continue to do this, perhaps I’ll just close access to it.

By the way, returning to the issue of theories with the same PTO. Indeed, the features of the formulation of axioms in a programming language can play a major role in determining the rate of growth. For example, take PTO(PA) = PTO(ACA0) = PTo(KP-ω). PA is 1st order arithmetic, it means in the programming language we can use only normal variables. ACA0 is 2nd order arithmetic, it means in the programming language we can use linear arrays. KP is set theory (despite the fact that rudimentary), it means in the programming language we can use multidimensional arrays. I don't know enough about the turing machine, but in terms of compiling code KP would be clearly preferable. Therefore, it is just a suggestion but it is possible that Θ(PA) < Θ(ACA0) < Θ(KP-ω)

As for this situation PTO(П11-CA0) = PTO(Δ12-CA0). Here the answer may come from the features of the arithmetic hierarchy. Writing cycles for Δ2-formulas can be simpler and more concise than for П1-formulas. Therefore, it is also a suggestion, but it is possible that Θ(П11-CA0) < Θ(Δ12-CA0). However, in this case PTO(KP+∃ω1) = PTO(KP+∃γ admissible on β|(Lβ/Lγ)∩ω1=∅), I have no guesswork.