Talk:Huge cardinal

Question

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)

• j^n refers to the nth iterate of the function j. It should be superscripted again, but I don't know if that can be done in HTML. The <j(kappa) bit means that M need only be closed under sequences with length less than kappa, rather than sequences of length kappa.

Ben Standeven 04:49, 29 May 2005 (UTC)

Thanks! Oleg Alexandrov 14:56, 29 May 2005 (UTC)

Totally huge cardinals

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)

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)

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)

Critical point

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)

Critical point (set theory) has been fixed now (I hope). Please look at it again. JRSpriggs 06:32, 2 May 2006 (UTC)

Almost ω-huge

Even though ω-huge is inconsistent, almost ω-huge might be consistent. In fact, I suspect that it is the same as the rank-into-rank axiom I2. Do you know? JRSpriggs (talk) 03:07, 3 February 2010 (UTC)

I now see that almost ω-huge is the same as ω-huge because λ (the limit of $j^n(\kappa )$ as n goes to ω) has cofinality ω instead of being regular. Thus it is forbidden by Kunen's inconsistency theorem. Presumably that means that it is not the same as I2. JRSpriggs (talk) 06:08, 15 February 2010 (UTC)

It's not any kind of problem that $\lambda$ has cofinality ${\mathrm{\aleph }}_0$. This is precisely how the $\lambda$ in all of the rank-into-rank axioms is defined (as the ${\omega }^{th}$ functional iterate of $j$). The cofinal $\omega$-sequence is obvious, as is $\lambda$'s status as a strong limit.

An $I2$ cardinal is however equivalent to an $\omega$-superstrong cardinal (this result is due to Kanamori). Perhaps that'd be worth adding to the superstrong cardinals article. Ekki, deliquent psychopomp (talk) 04:11, 14 January 2012 (UTC)

ω-huge inconsistent?

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 n-huge for all n∈N, which any rank-into-rank cardinal satisfies (as noted on this very page). More specifically, this property is strictly weaker than WA₀ - any cardinal satisfying WA₀ is already ω-superhuge in this sense, and there's a known ascending chain of consistency strength WA₀ < Wholeness Axiom < I3. Hugh Woodin has adopted of late a slightly idiosyncratic definition of ω-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 ω-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 V[G]. A generic ω-huge cardinal is not the same thing as an ω-huge cardinal. But that's only a guess, as this section lacks citations and there is no mention of ω-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 ω-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)

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)

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)