Talk:Huge cardinal

Question
In the formula


 * &kappa; is almost huge iff there is j : V &rarr; M with critical point &kappa; and &lt;j(&kappa;)M &sub; M.

I am having trouble understanding the notation &lt;j(&kappa;)M &sub; M.

And in the formula


 * &kappa; is n-huge iff there is j : V &rarr; M with critical point &kappa; and j^n (&kappa;)M &sub; M.

should
 * j^n (&kappa;)M &sub; M

be
 * j n (&kappa;) M &sub; M ?

Is this standard notation? Oleg Alexandrov 00:00, 28 May 2005 (UTC)

Ben Standeven 04:49, 29 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.


 * 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 &kappa; is n-huge with the same elementary embedding j for each n&isin;&omega;, then this would be what I would call almost &omega;-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 &omega;-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 &omega;-huge is the same as &omega;-huge because &lambda; (the limit of $$j^n(\kappa)$$ as n goes to &omega;) has cofinality &omega; 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 $$\aleph_0$$. 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)

$&omega;$-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 $n&isin;N$, which any rank-into-rank cardinal satisfies (as noted on this very page). More specifically, this property is strictly weaker than $WA_{0}$ - any cardinal satisfying $WA_{0}$ is already $&omega;$-superhuge in this sense, and there's a known ascending chain of consistency strength $WA_{0}$ < Wholeness Axiom < I3. Hugh Woodin has adopted of late a slightly idiosyncratic definition of $&omega;$-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 &ldquo;generic $&omega;$-huge cardinal&rdquo;, 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 $&omega;$-huge cardinal is not the same thing as an $&omega;$-huge cardinal. But that's only a guess, as this section lacks citations and there is no mention of $&omega;$-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 $&omega;$-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 &omega;-huge is defined as at Huge cardinal, 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)