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In the formula
I am having trouble understanding the notation <j(κ)M ⊂ M.
And in the formula
should
be
Is this standard notation? Oleg Alexandrov 00:00, 28 May 2005 (UTC)
Ben Standeven 04:49, 29 May 2005 (UTC)
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)
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)
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)
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:
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)
![]() | This article is rated Start-class on Wikipedia's
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In the formula
I am having trouble understanding the notation <j(κ)M ⊂ M.
And in the formula
should
be
Is this standard notation? Oleg Alexandrov 00:00, 28 May 2005 (UTC)
Ben Standeven 04:49, 29 May 2005 (UTC)
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)
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)
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)
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:
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)