Ridiculously Huge Numbers (Part 40)

[@francoi8]

Beautiful! This is my favorite version of the fast growing hierarchy.

[@KIYOM99]

Thanks for this awesome video! I read that the Loader's number(which I learned to be the fastest growing computible funtion) is so large that it isn't clear what order type it's on the FGH. But if we can define infinite amount of larger and larger ordinals with OCF to plug in to the hierarchies, there must be a place where we reach and exceed it, right? Or is there a limit for how large a countible ordinal can get? I hope maybe you can explain Loader's number about how it's created in future videos because loader.c is just so hard to read, probably as hard as a program can get, but it definitely is a work of art.

[@davidmetzler]

王致雄 I do want to talk about Loader's number soon. It does seem unclear just how clever one would have to be with ordinal notations in order to match Loader. My impression is that no such notation is yet known.

[@FreeAsInFreeBeer]

Let's say we are talking about elements in Ω₀. We have ω₀[n] = n. What's the difference between the tree ordinals α = ω₀ + 1 and α'[n] = n + 1? (What's the difference between the successor of omega_zero and the tree ordinal defined by the fundamental sequence n to n + 1?) From the listed ways of creating elements, it sounds like they should be different. Are they? If so, are β = ω₀² + ω₀ and β'[n] = n² + n different in the same manner?

[@biggiemac42]

I'm a little unsatisfied with the definition of the fundamental sequence of omega_1 in this framework as listing the contents of Omega_1 "in order". Namely the in order part, since you called out some things like the constant function as belonging to Omega_1. I can definitely see how it's still useful, because when you ask for the "whateverth" element of the fundamental sequence, you've provided all the information you need to answer your question. But the concept of ordering feels like a bad match. (And I guess the ordering part is still very important to understanding induction using these ordinals!)

[@robertluo3505]

How come you upload so fast??