Can we get rid of all the statements like this on the wiki?

- Its rate of growth is comparable to \(f_{\psi_0(\varepsilon_{\Omega_\omega + 1})}(x)\).

- This notation has strength at \(\theta(\Omega^\omega)\).

My question is **what on Earth do they mean?** They are *fractally wrong* and represent pretty much every mathematical sin ever. What does "comparable" mean when comparing functions? What is the fundamental sequence system used for FGH? Where do these statements come from and who proved them?

*Fundamental sequence systems don't drastically change the growth rates of functions, so why bother?*

Yes they do! Joel David Hamkins showed that with a few minor changes to the Wainer hierarchy, bumping down only the *beginnings* of a few fundamental sequences, it's possible to make \(f_{\varepsilon_0}(n) = f_\omega(n)\).

*But that's just an artificial case!*

Maybe. But it shows that FGH is pretty damn delicate. If you omit any part of the definition, you might find yourself on the wrong side of a phase transition. So **define your FS systems!**

*But Chris Bird used these kinds of statements!*

Chris Bird is wrong.

*Here's a definition of comparable, and I'm using this fundamental sequence system. Are you happy now?*

No, because you need to prove it.

*But that's hard!*

So don't prove it, and don't spout unproven claims.

*Okay, smartass, what can we use instead?*

This is what I want to get at. Instead of using these statements, I suggest the following:

- "Buchholz showed that his function eventually dominates every function provably total recursive in Pi11 - CA0 + BI."
- "Rado showed that the Rado sigma function eventually dominates every computable function."
- "The xi function eventually dominates every function computable in SKIO calculus."
- "The function \(f(n) = n \uparrow^n n\) eventually dominates every primitive recursive function."

These are actual, verifiable statements, and statements like these have been proven for a number of functions, including a large portion of Friedman's work.

*But compared FGH, these statements give us a very sparse sense of scale!*

I'd rather have a sparse sense of scale than a scale that doesn't work at all.

*They're also REALLY hard to prove!*

Sure, but it's not like the comparability ones are any easier.

*Nobody understands what these theories are!*

You should if you're discussing set theory.

*Aren't there some functions that don't have such proofs?*

Yes, array notation and ExE are examples. I would rather have *no* claims of growth rate than unproven and ill-defined claims of growth rate.

*Can we still use FGH to bound things?*

Yes, if the statements are well-formed and proven.

*Who cares about a little informality?*

I do, because these statements are spreading throughout the wiki. Numerous people are swearing by them and believing them to be some sort of backbone to googology as a field, when there is literally **zero formal basis** for them. They can't be verified or refuted because they are meaningless.

I like having encyclopedic content that's actually true. What about you?

*Eh, Googology Wiki is just for fun, who cares about professional standards? You're just being pedantic.*

That's no excuse for having statements littered throughout wiki that are wrong in every way possible.

I've talked about this issue a lot. It's tempting to just go ahead and purge all these statements from the wiki, but that would be pretty brash and an abuse of privileges. I'd like for us to reach a consensus on this once and for all. Please discuss.