Talk:Huge cardinal

message: In the formula
:κ is almost huge iff there is j : V → M with critical point κ and <j(κ)M ⊂ M.
I am having trouble understanding the notation <j(κ)M ⊂ M.
And in the formula
:κ is n-huge iff there is j : V → M with critical point κ and j^n (κ)M ⊂ M.
should 
:j^n (κ)M ⊂ M 
be 
:jn (κ)M ⊂ M ?
Is this standard notation? Oleg Alexandrov 00:00, 28 May 2005 (UTC)
replies: text: Thanks! Oleg Alexandrov 14:56, 29 May 2005 (UTC)
thread_title: Question

message: Is there such a thing as a totally huge cardinal, meaning one that is nhuge for all n? (I mean, are these considered, and under that name?) The analogy here is with the totally ineffable cardinals.
-- Toby Bartels 08:31, 13 December 2005 (UTC)
replies: text: I think that "totally huge" would be a Reinhardt cardinal which is inconsistent with the axiom of choice.  JRSpriggs 09:44, 3 May 2006 (UTC)

text: If you mean that κ is n-huge with the same elementary embedding j for each n∈ω, then this would be what I would call almost ω-huge (see the section below on that topic). If the j might be different for each n, then it might be weaker than that. JRSpriggs (talk) 08:23, 3 February 2010 (UTC)
thread_title:  Totally huge cardinals 

message: What is a critical point, in this context? The current link seems inappropriate. It surely has nothing to do with  any kind of derivative. The Infidel 15:04, 22 January 2006 (UTC)
replies: text: Critical point (set theory) has been fixed now (I hope).  Please look at it again. JRSpriggs 06:32, 2 May 2006 (UTC)
thread_title:  Critical point 

message: What's the source for this assertion, now vexatiously auto-mirrored all over the web? The term itself is uncommon in the literature I'm familiar with, but usually means nothing more than a cardinal -huge for all , which any rank-into-rank cardinal  satisfies (as noted on this very page). More specifically, this property is strictly weaker than  - any cardinal satisfying  is already -superhuge in this sense, and there's a known ascending chain of consistency strength  < Wholeness Axiom < I3. Hugh Woodin has adopted of late a slightly idiosyncratic definition of -huge:
:there exist ordinals \kappa < \lambda < \gamma such that V_{\kappa} \prec V_{\lambda} \prec V_{\gamma} and
:\kappa is the critical point of a nontrivial automorphism j:V_{\lambda+1} \to V_{\lambda+1}
which is obviously a mild extension of I1, and neither known nor suspected to be inconsistent.
If I were to guess what happened here, I'd venture that this offhand remark by Matt Foreman in Generic Large Cardinals ...using PCF theory one can show that there is no “generic -huge cardinal”, an analogue to a result of Kunen for ordinary large cardinals, has been misinterpreted: 'generic' large cardinals are the critical points of nontrivial elementary embeddings which are defined in some forcing extension . A generic -huge cardinal is not the same thing as an -huge cardinal. But that's only a guess, as this section lacks citations and there is no mention of -huge cardinals in the references given. In any case, barring a relevant mention in the literature, I'd prefer this section vanish along with the mention in Kunen's inconsistency theorem (the latter should definitely go, as the refutation of generic -huge cardinals does not use Kunen's theorem and is analogous only inasmuch as it refutes a plausible, natural extension to axioms believed consistent).
Apologies for barbarous formatting. Ekki, deliquent psychopomp (talk) 23:33, 13 January 2012 (UTC)
replies: text: Ask a question, figure out the answer yourself, heh. There is an allusion to the inconsistency of \omega-huge cardinals in Maddy's Believing the Axioms, though she's referring to closure under sequences of length \omega which, using the usual sort of ordinal indexing, would be \omega+1-hugeness, and Kunen's theorem does preclude that, as it already precludes \omega+1-superstrongness. At least it's a current jargon dispute rather than a factual dispute. Ekki, deliquent psychopomp (talk) 05:46, 14 January 2012 (UTC)
replies: text: Indeed this is probably just a question of terminology. If ω-huge is defined as at Huge cardinal#ω-huge cardinals, then it implies the existence in M of j \upharpoonright \lambda \, and thus j  \lambda \, which is the first thing proven impossible by Kunen's theorem. Since that meaning is contradictory and thus not useful, I am not surprised that someone would choose to give it a different meaning which is not contradictory. JRSpriggs (talk) 08:31, 15 January 2012 (UTC)
thread_title:  {{math|&omega;}}-huge inconsistent?