User blog comment:Vel!/FGH Gripe/@comment-5982810-20150325004329/@comment-24.103.234.74-20150326005602

I'm just going to address a bunch of points you brought up, even though they weren't addressed at me.

'  I, for one, don't buy the claim "FS's don't matter that much." ' - Vel!

I have a feeling that you doubt it on purely theoretical grounds because you can't prove it for every and all cases. But your not taking into account that googologists are practioners as well as theoreticians. We aren't so much interesting in the general theoretical question of FS's, but rather whether it makes much of a difference from my system to your system. This is a partly competitive endevour and what we are mainly interested is in comparing actually worked out systems by different googologists. There is good reasons to suspect that almost all the googological systems are stabilized against each other and FGH, with a minor hiccup here and there. The reason being because the sheer growth rate makes tiny perturbations vanish. It seems to flaunt common sense, but stabilization has already been observed and proved in a few instances. So all that's being claimed Vel is that the various systems are essentially interchangable. This is something that is taken for granted for example when you order the googological numbers by *order of size*. If you refuse to accept these as more than conjectural, then guess what: you now have no way to order the numbers on that page!

' Admittedly I'm unable to refute it, but that's because 1) I can't argue formally against an informal claim ' - vel!

Again I think you're confusing this with some kind of absolute claim. No one is making such an absolute claim that no change in the fundamental sequences makes a difference. But there is an uncountable number of fundamental sequences for any countable limit ordinal, some extremely absurd; do you think we can actually describe them all? Of the ones we could reasonably describe, how many do you think we could define in a reasonable amount of space (in our universe)? Of those how much effort do you think people are going to make, when even modestly simple FS's like those seen in the Weiner Hierarchy get the job done and produce numbers far far far beyond anything else we know?! Wouldn't the efficient thing to do be to stick with "simple" definitions for FS's? Of those "simple" choices that people are likely to make what do you think the largest gap in relative growth rate is? The largest I know of is only a factor of 2. This is between BEAF and my array-variant. I never for a minute thought it made any difference and that my arrays would somehow be stronger than the equivalent BEAF arrays because of it. Why are we suddenly so number dumb. Don't we know what kind of scales we're talking about? You really think "2" is going to making an f---ing difference? Find me a larger relative growth rate used in practice.

' 2) any example of a degenerate FS system I give you will be passed off as "pathological" or "smartass" or whatever. ' - Vel!

Don't worry, I am addressing that prong as well. First off, we can discard a case as pathological if (1) the hierarchy is not strictly monotonic. So that rules out examples like the venetian blind trick on e0 that has every function f_a(n) the same for ainfinity a[n] = a , and most importantly that we define the sequence as a[0] = b, and a[n+1] = phi(a[n]) for some ordinal function phi(a). phi(a) may be as simple as a+1. This appears to prevent sequences which do not diagonalize properly over the previous system. Of course I'm not absolutely sure this is strigent enough either, but here's the main point. All these "smartass" w/e's you've been presenting, while interesting theoretical counter examples that warn us to not make too many broad assumptions aren't getting us any actual results. What are actual results: being able to compare functions and numbers! Instead we keep being moved further and further away from such results and put in greater and greater doubt because of all the "what ifs". We don't need any more "what ifs". What we need is some actual "we know such and such under these sets of conditions".

" But I get the sense that you really don't care about formality. If that's the case -- and ultimately we're arguing about whether informality is acceptable in our community -- then God help us because I have no idea how we'll reach an agreement. ' - Vel!

It's not about being against formality. But you keep forgetting that we are a community of '''amateurs. '''Googology has been playing catch up with professional mathematics ever since Milton uttered the first "googol". Informality is all we have had up until now, and only more recently has more serious work gone into googology. So yes Vel, informality WAS/IS acceptable in our community. We don't need to pretend that's not the case. Informality is acceptable in the ABSENSE of an alternative. What are we going to do, ask a professional mathematician about our every petty grievance: "mister professional, can you figure out for me whether golapulus is actually bigger than TREE(3)?". No. We are going to have to do this work ourselves if we want to see it done. No one is going to do it for us, because WE are the ones that are interested in such questions. In the interim, while we don't have the tools, it is not a sin to say "we guess that golapulus is bigger because of such and such reasons, but a formal proof has not been formulated". At least that's honest without having to completely say "we have no claim to make, not even a good guess".

Personally I think we should start building up our own mathematical standards and methods. This means using sound logic applied to an axiomatic system, developing techniques from the ground up applied specifically to our needs. We can borrow from whatever professional mathematics we can get our hands on, but we shouldn't treat it like a f--king crutch. If we don't know something it doesn't mean we can't figure it out for ourselves. If we come up with ideas that don't perfectly fit into the mold of professional mathematics (FSEs), we shouldn't let that stop us.