http://cantorsattic.info/api.php?action=feedcontributions&user=Eaglgenes101&feedformat=atomCantor's Attic - User contributions [en]2022-08-18T23:00:46ZUser contributionsMediaWiki 1.24.4http://cantorsattic.info/index.php?title=Talk:Huge&diff=2903Talk:Huge2019-04-29T15:26:16Z<p>Eaglgenes101: Amateur analysis...</p>
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<div>== $\omega$-almost hugeness underexplored? ==<br />
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The page briefly mentions the $\omega$-almost huge cardinals, but then immediately moves on to $\omega$-huge cardinals without mentioning anything about the properties or possibility/impossibility of $\omega$-almost huge cardinals. Furthermore, none of the other pages in this wiki make any mention of $\omega$-almost hugeness. Is it consistent for them to exist, how strong are they, and is it known how they relate to other large cardinals? [[User:Eaglgenes101|Eaglgenes101]] ([[User talk:Eaglgenes101|talk]]) 14:23, 27 April 2019 (PDT)<br />
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So I thought about it a bit. The first fixed point of the elementary embedding $j$ above $crit(j)$ is the supremum of all of $crit(j)$, $j(crit(j))$, $j^2(crit(j))$, $j^3(crit(j))$, etc..., and said iterates of $j$ form a cofinal set of $j^\omega(crit(j))$. So for any $\alpha < j^\omega(crit(j))$, there is some whole number $n$ such that $j^n(crit(j)) > \alpha$. If I reasoned correctly, this means that $\omega$-almost hugeness is equivalent to $n$-hugeness for all whole number $n$, which is known to be perfectly consistent at least relative to an $I3$ cardinal. [[User:Eaglgenes101|Eaglgenes101]] ([[User talk:Eaglgenes101|talk]]) 08:26, 29 April 2019 (PDT)</div>Eaglgenes101http://cantorsattic.info/index.php?title=Talk:Huge&diff=2876Talk:Huge2019-04-27T21:23:10Z<p>Eaglgenes101: Created page with "== $\omega$-almost hugeness underexplored? == The page briefly mentions the $\omega$-almost huge cardinals, but then immediately moves on to $\omega$-huge cardinals without m..."</p>
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<div>== $\omega$-almost hugeness underexplored? ==<br />
<br />
The page briefly mentions the $\omega$-almost huge cardinals, but then immediately moves on to $\omega$-huge cardinals without mentioning anything about the properties or possibility/impossibility of $\omega$-almost huge cardinals. Furthermore, none of the other pages in this wiki make any mention of $\omega$-almost hugeness. Is it consistent for them to exist, how strong are they, and is it known how they relate to other large cardinals? <br />
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[[User:Eaglgenes101|Eaglgenes101]] ([[User talk:Eaglgenes101|talk]]) 14:23, 27 April 2019 (PDT)</div>Eaglgenes101http://cantorsattic.info/index.php?title=Upper_attic&diff=2704Upper attic2019-02-07T15:49:34Z<p>Eaglgenes101: Undo revision 2703 by Nigga (talk) The joke's not even funny.</p>
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<div>{{DISPLAYTITLE:The upper attic}}<br />
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]<br />
[[Category:Large cardinal axioms]]<br />
<br />
Welcome to the upper attic, the transfinite realm of large cardinals, the higher infinite, carrying us upward from the merely inaccessible and indescribable to the subtle and endlessly extendible concepts beyond, towards the calamity of inconsistency. <br />
<br />
* The '''[[Kunen inconsistency]]''': [[Reinhardt]] cardinal, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal<br />
* '''[[Rank into rank]]''' axioms, [[L of V_lambda+1|I0 axiom]] and strengthenings<br />
* The [[wholeness axioms]]<br />
* [[n-fold variants]] of hugeness, extendibility, supercompactness, strongness, etc...<br />
* '''[[huge]]''' cardinal, [[huge|superhuge]] cardinal, [[huge|ultrahuge]] cardinal, [[superstrong|2-superstrong]] cardinal<br />
* [[high-jump]] cardinal, [[high-jump|almost high-jump]] cardinal, [[high-jump|super high-jump]] cardinal, [[high-jump|high-jump with unbounded excess closure]] cardinal<br />
* [[Woodin#Shelah cardinals|Shelah for supercompactness]]<br />
* [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin#Shelah cardinals|Woodin for supercompactness]] cardinal<br />
* [[Vopenka | Vopěnka's principle]]<br />
* [[extendible]] cardinal, [[extendible | $\alpha$-extendible]] cardinal<br />
<!--* [[grand reflection]] cardinal--><br />
* [[hypercompact]] cardinal<br />
* '''[[supercompact]]''' cardinal, [[supercompact | $\lambda$-supercompact]] cardinal<br />
* '''[[strongly compact]]''' cardinal [[strongly compact | $\lambda$-strongly compact]] cardinal<br />
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals<br />
* [[Weakly_compact#Indestructibility of a weakly compact cardinal|indestructible weakly compact]] cardinal<br />
* The '''[[proper forcing axiom]]''' and [[forcing#Proper forcing|Martin's maximum]]<br />
* [[subcompact]] cardinal<br />
* [[superstrong]] cardinal<br />
* [[Woodin#Shelah|Shelah]] cardinal<br />
* The '''[[axiom of determinacy]]''' and [[axiom of projective determinacy|its projective counterpart]]<br />
* '''[[Woodin]]''' cardinal<br />
* [[strongly tall]] cardinal<br />
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchies, [[tall]] cardinal, [[tall|$\theta$-tall]] hierarchy<br />
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] <br />
*[[zero dagger| $0^\dagger$]] (''zero-dagger'')<br />
* '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal<br />
* [[Ramsey#Super Ramsey cardinal|super Ramsey]] cardinal<br />
* [[Ramsey#Strongly Ramsey cardinal|strongly Ramsey]] cardinal<br />
* '''[[Ramsey]]''' cardinal, [[Jonsson | Jónsson]] cardinal, [[Rowbottom]] cardinal, [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] cardinal<br />
* [[Ramsey#Almost Ramsey cardinal|almost Ramsey]] cardinal<br />
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$, [[Chang's conjecture]]<br />
* [[Ramsey#.24.5Calpha.24-iterable cardinal|$\omega_1$-iterable]] cardinal<br />
* '''[[zero sharp | $0^\sharp$]]''' (''zero-sharp''), existence of [[Constructible universe#Silver indiscernibles|Silver indiscernibles]]<br />
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$<br />
* the [[Ramsey#.24.5Calpha.24-iterable cardinal| $\alpha$-iterable]] cardinals hierarchy for $1\leq\alpha<\omega_1$<br />
* [[remarkable]] cardinal<br />
* [[Ramsey#.24.5Calpha.24-iterable cardinal|weakly Ramsey]] cardinal<br />
* [[ineffable]] cardinal, [[weakly ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy; [[completely ineffable]] cardinal<br />
* [[subtle]] cardinal<br />
* [[ineffable#Ethereal cardinal|ethereal]] cardinal<br />
* [[unfoldable#Superstrongly Unfoldable | superstrongly unfoldable]] cardinal, [[uplifting#strongly uplifting | strongly uplifting]] cardinal <br />
* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal<br />
* [[unfoldable]] cardinal, [[unfoldable#Strongly Unfoldable | strongly unfoldable]] cardinal<br />
* [[indescribable]] hierarchy, [[totally indescribable]] cardinal<br />
* '''[[weakly compact]]''' cardinal<br />
* The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$ <br />
* '''[[Mahlo]]''' cardinal, [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals<br />
* [[uplifting]] cardinal, [[uplifting#pseudo uplifting cardinal | pseudo uplifting]] cardinal<br />
* [[ORD is Mahlo|$\text{Ord}$ is Mahlo]]<br />
* [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$-reflecting]], [[reflecting | $\Sigma_n$-reflecting]] and [[reflecting]] cardinals<br />
* [[Jäger's collapsing functions and ρ-inaccessible ordinals]] <br />
* [[inaccessible#Degrees of inaccessibility | $1$-inaccessible]], the [[inaccessible#Degrees of inaccessibility | $\alpha$-inaccessible]] hierarchy and [[inaccessible#Hyper-inaccessible | hyper-inaccessible]] cardinals<br />
* [[inaccessible#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals<br />
* '''[[inaccessible]]''' cardinal, '''[[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]]''' cardinal<br />
* [[Morse-Kelley set theory|Morse-Kelley]] set theory<br />
* '''[[worldly]]''' cardinal and the [[worldly#Degrees of worldliness | $\alpha$-wordly]] hierarchy, [[worldly#Degrees of worldliness | hyper-worldly]] cardinal<br />
* the [[Transitive ZFC model#Transitive model universe axiom | transitive model universe axiom]] <br />
* [[transitive ZFC model|transitive model of $\text{ZFC}$]]<br />
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]<br />
* '''[[Con ZFC | $\text{Con(ZFC)}$]]''' and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]]<br />
* '''[[ZFC|Zermelo-Fraenkel]]''' set theory<br />
<br />
* down to [[the middle attic]]</div>Eaglgenes101