User blog comment:Vel!/FGH Gripe/@comment-2033667-20150326191109/@comment-5982810-20150327211402

That proposal does not sound contraversal at all. In fact it might be a tad too conservative. I never liked "comparable to" and I never used it myself. It sounds as if your saying that it's only a rough approximation. E^ isn't "roughly comparable to e0 in FGH", it's exactly the same thing in the sense that for every function in FGH below e0, there is a corresponding function in E^ which grows at least as fast, and that f_e0(n) is the first function in FGH that dominates over all functions in E^. So it's not "comparable to", they have exactly the same strength. But how can you reduce that long definition to something simple like "comparable to" without being ambiguous. When people say such and such a system has strength "alpha" they know what is meant by that, but apparently because this phrase has no precedence in professional mathematics you find it to informal. But the problem isn't that the concept is vague so much that it's unproven. Also that we haven't been nearly explicit enough in defining precisely what we mean. But who gives a f--k whether mathematicians use this phrase or not. We can define our own terminology. We just need some agreement about what these things mean.

And all haggling aside, people are going to continue to compare numbers, notations, functions, and systems, because that's what's needed in googology. You have never really acknowledged that this is a necessity, which irks me. So googology as a whole has every motivation to find means of comparison, and the more air tight they are the better. But again, up until now our community is composed of amateurs, and so the best people were able to do was give rough estimates. This wasn't unreasonable given the circumstances. If better methods are available then that's great. If they aren't then we still need to make a best guess on the available information. An the available information so far suggests that stabilization is a natural property of googological functions. If even a small portion of this is true that is good news for googology. It means that we can reliably depend on the order-type of a hierarchy to readily access it's strength against FGH and other systems. I think it might even be possible to prove some fairly broad constraints for this stabilization property, in which case such a theorem could be used all across googology. But it will require a bit of abstraction and careful formulation. Anyway I look forward to the day when googology isn't just leaving all the "real" work to mathematicians