What's the supremum of a sequence that stops increasing after a finite number of steps? I'm currently looking at Γ_0+1, Γ_0, Γ_0, Γ_0, Γ_0, Γ_0, Γ_0, Γ_0, etc..., which is the sequence of the functions corresponding to the fundamental hierarchy of Γ_0, applied to Γ_0

Γ_0 is the first fixed point of B -> f_B(ω) under my notation, and the way up is visible, but rather foggy from what I currently have

Okay maybe that specific example is bunk, since it actually does increase rather than sticking to Γ_0, but still, the general question remains

@eaglgenes101 sup = max when max exists.

So the supremum of the sequence I listed is Γ_0+1?

Looks a bit weird, but ordinals are a lot weird already, so I'll take it

@eaglgenes101 normal variants of basic ordinal collapsing functions is probably what you want.

Depending on the supremum definition chosen, f_(Γ_0+1)(ω) could have been Γ_0, Γ_0+1, Γ_1, or even Γ_ω

youtube.com/user/davidmetzler/playlists

Also some related vids ^

"Depending on the supremum definition chosen"

supremum of a sequence/set/whatev is almost always the least upper bound.

If this is true, then f_(Γ_0)(Γ_0) ends up as a grand total of... Γ_0+1

Not much, but f_(Γ_0)(Γ_0+1) then evaluates to Γ_1, which is getting somewhere

lel

Turns out that using the fast-growing hierarchy on ordinals is turning into an exercise to see how far I can get with only a small amount of impredication

¯\_(ツ)_/¯

I've already replicated the result that the feferman schutte ordinal is the smallest ordinal that requires impredication to notate

To paraphrase the wikipedia article on OCFs, the power of an OCF is mostly from how many uncountable ordinals it can define and throw at fixed points to dislodge the ordinal notation from them

more or less yeah

ZFC without powerset can use replacement to contort itself into all sorts of countable sequences, but without powerset or another axiom asserting or implying the existence of uncountable sets, it can't get anywhere beyond countables

I just spotted a bit higher in the large cardinals and now I'm looking at elementary embeddings

Can zfc define "small" embeddings, such as one of aleph zero into itself?