w^(w^n*2)=(w^w^n)^w^n. Assuming n=e_x, (w^w^e_x)^w^e_x=e_x^e_x>e_x. If n≠e_x for any x, then n<w^n<w^n*2<w^(w^n*2) so Ξ can be neither epsilon nor nonepsilon, and therefore is illdefined.Moosey wrote:Came across this by defining a different Ord-type function:
if I define Ξ (Greek letter Xi) as the smallest (?) ordinal such that n= w^((w^n)2)
Just "uncountable κ that is the κth regular cardinal" works.Moosey wrote:Can someone explain inaccessible cardinals? Apparently they are κs such that κ= an uncountable κ such that it is the ב_κ th regular cardinal—
A cardinal such that any sequence of lower cardinals that exceeds any cardinal less than κ at some point must have length at least κ.Moosey wrote:What’s a regular cardinal?
I know that ב_n+1 = 2^(ב_n) so I sorta understand that.
You can't prove it won't happen (unless you have an axiom stating that inaccessibles do not exist or something like that).Moosey wrote:The other thing I understand is that κ is 1-inaccessible if it is the κth inaccessible cardinal, and I guess it’s probably n+1 inaccessible if it’s the κth n-inaccessible cardinal.
The other other thing I get is that κ is hyper-inaccessible if it’s κ-inaccessible.
The other other other thing I think I get is that κ is 1-hyper-inaccessible if it’s the κth hyper-inaccessible number, and
it’s n+1-hyper-inaccessible if it’s the κth n-hyper-inaccessible number.
I love the jargon at the end of this page:
http://cantorsattic.info/Inaccessible#D ... essibilityI suppose the logical continuation is that κ is actually-hyper-hyper-inaccessible if it’s hyper^κ inaccessible, whether that’s even possible or not.cantor’s attic wrote:κ is hyper^a-inaccessible if it is hyper-inaccessible and for every b<a it is κ-hyper^b-inaccessible where κ is a-hyper^b-inaccessible if it is hyper^b-inaccessible...
That works too, but it's much larger than just nesting aleph_κ w times on 0.Moosey wrote:EDITLATERTHANTHERESTOFPOST:
Oh, I get it— inaccessible numbers are to aleph_n what epsilon numbers are to w^n:
All inaccessible numbers κ = aleph_κ.
Cardinals such that you can't describe them without also describing a bunch of lower cardinals (for a certain definition of "describe")Moosey wrote: </late edit>
Also, can somebody describe (cue laughtrack) indescribable cardinals? (I don’t expect you to speak of the ineffables, of course.)
https://googology.wikia.org/wiki/User_b ... ion_to_OCFsMoosey wrote: Also reflection and regular cardinals and pretty much everything else building up to indescribables, in a comprehensible way?
Weaker than Mahlos. It's not very easy to create an interesting and original large cardinal property (otherwise the Cantor's Attic list would be overflowing with large cardinal axioms)Moosey wrote: Speaking of comprehensibility, someone should define incomprehensible cardinals. I’ll take a crack at it maybe...
Attempt I:
An incomprehensible cardinal is a cardinal κ such that κ is the κth compressed-κ-inaccessible cardinal
A compressed-n-inaccessible cardinal is defined as follows:
A compressed-0-inaccessible cardinal is a hyper^κ-inaccessible cardinal.
A compressed-n+1-inaccessible cardinal κ is the κth compressed-n-inaccessible cardinal
Inc_0 (the 1st incomprehensible cardinal) is the Inc_0th compressed-Inc_0-inaccessible cardinal.