Difference between revisions of "Upper attic"
From Cantor's Attic
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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. | 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. | ||
− | * [[hypercompact|excessively hypercompact cardinal]] | + | * [[hypercompact|excessively hypercompact cardinal]] |
− | * The [[Kunen inconsistency]]: [[Reinhardt]] cardinal, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal | + | * The '''[[Kunen inconsistency]]''': [[Reinhardt]] cardinal, [[Kunen_inconsistency#Super_Reinhardt_cardinal | super Reinhardt]] cardinal, [[Berkeley]] cardinal |
− | * [[Rank into rank]] cardinals $j:V_\lambda\to V_\lambda$, [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$, I0 cardinal [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]] | + | * '''[[Rank into rank]]''' cardinals $j:V_\lambda\to V_\lambda$, [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$, I0 cardinal [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]] |
− | * The [[wholeness axiom]] | + | * The '''[[wholeness axiom]]''' |
− | * [[huge | almost n-huge]] cardinal, [[huge|n-huge]] cardinal, [[superstrong|(n+1)-superstrong cardinal]], [[ | + | * '''[[huge|n-huge]]''' cardinal, [[huge | almost n-huge]] cardinal, [[huge|super-n-huge]] cardinal, [[superstrong|(n+1)-superstrong cardinal]] |
− | * [[Vopenka | Vopěnka's principle]], [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal | + | * [[high-jump]] cardinal, [[high-jump|almost high-jump]] cardinal, [[high-jump|super high-jump]] cardinal, [[high-jump|high-jump with unbounded excess closure]] cardinal |
− | + | * '''[[Vopenka | Vopěnka's principle]]''', [[Vopenka#Vopěnka cardinals | Vopěnka]] cardinal, [[Woodin for supercompactness]] cardinal | |
+ | * '''[[extendible]] cardinal''', [[extendible | $\alpha$-extendible]] cardinal | ||
* [[grand reflection]] cardinal | * [[grand reflection]] cardinal | ||
* [[hypercompact]] cardinal | * [[hypercompact]] cardinal | ||
− | * [[ | + | * '''[[supercompact]] cardinal''', [[supercompact | $\lambda$-supercompact]] cardinal |
* [[PFA]] cardinal | * [[PFA]] cardinal | ||
− | * [[strongly compact]] cardinal | + | * '''[[strongly compact]] cardinal''' |
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals | * [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals | ||
* [[indestructible weakly compact]] cardinal | * [[indestructible weakly compact]] cardinal | ||
* [[subcompact]] cardinal | * [[subcompact]] cardinal | ||
− | * [[superstrong]] cardinal | + | * '''[[superstrong]] cardinal''' |
* [[Shelah]] cardinal | * [[Shelah]] cardinal | ||
− | * [[Woodin]] cardinal, the [[axiom of determinacy]] | + | * '''[[Woodin]] cardinal''', the [[axiom of determinacy]] |
− | * [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy | + | * '''[[strong]] cardinal''' and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy |
* [[tall]] cardinal | * [[tall]] cardinal | ||
* [[zero dagger| $0^\dagger$]], $j:L[U]\to L[U]$ cardinal | * [[zero dagger| $0^\dagger$]], $j:L[U]\to L[U]$ cardinal | ||
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] | * Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]] | ||
− | * [[ | + | * '''[[measurable]] cardinal''', [[weakly measurable]] cardinal |
* [[virtually Ramsey]] cardinal, [[Ramsey]] cardinal, [[strongly Ramsey]] cardinal | * [[virtually Ramsey]] cardinal, [[Ramsey]] cardinal, [[strongly Ramsey]] cardinal | ||
* [[Rowbottom]] cardinal | * [[Rowbottom]] cardinal | ||
* [[Jonsson | Jónsson]] cardinal | * [[Jonsson | Jónsson]] cardinal | ||
− | * [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ | + | * '''[[Erdos | $\omega_1$-Erdős]] cardinal''' and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$ |
− | * [[zero sharp | $0^\sharp$]], $j:L\to L$ cardinal | + | * '''[[zero sharp | $0^\sharp$]]''', $j:L\to L$ cardinal |
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$ | * [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$ | ||
* [[Ramsey | $1$-iterable]] cardinal, and the [[Ramsey | $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$ | * [[Ramsey | $1$-iterable]] cardinal, and the [[Ramsey | $\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$ | ||
* [[remarkable]] cardinal | * [[remarkable]] cardinal | ||
− | * [[ | + | * '''[[ineffable]] cardinal''', [[weakly ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy; [[completely ineffable]] cardinal |
* [[subtle]] cardinal | * [[subtle]] cardinal | ||
* [[ethereal]] cardinal | * [[ethereal]] cardinal | ||
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* [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal | * [[uplifting#weakly superstrong cardinal | weakly superstrong]] cardinal | ||
* [[unfoldable]] cardinal, [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal | * [[unfoldable]] cardinal, [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal | ||
− | * [[indescribable]] cardinal, [[totally indescribable]] cardinal | + | * '''[[indescribable]] cardinal''', [[totally indescribable]] cardinal |
− | * [[weakly compact]] cardinal | + | * '''[[weakly compact]] cardinal''' |
− | * | + | * '''[[Mahlo]] cardinal''', [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals |
− | + | ||
* [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal, [[uplifting]] cardinal | * [[uplifting#psuedo uplifting cardinal | psuedo uplifting]] cardinal, [[uplifting]] cardinal | ||
* [[ORD is Mahlo]] | * [[ORD is Mahlo]] | ||
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* [[inaccessible#Degrees of inaccessibility | $1$-inaccessible]], the [[inaccessible#Degrees of inaccessibility | $\alpha$-inaccessible]] hierarchy and [[inaccessible#Hyper-inaccessible | hyper-inaccessible]] cardinals | * [[inaccessible#Degrees of inaccessibility | $1$-inaccessible]], the [[inaccessible#Degrees of inaccessibility | $\alpha$-inaccessible]] hierarchy and [[inaccessible#Hyper-inaccessible | hyper-inaccessible]] cardinals | ||
* [[inaccessible#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals | * [[inaccessible#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals | ||
− | * [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal, (strongly) [[inaccessible]] cardinal, | + | * '''[[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal''', (strongly) '''[[inaccessible]] cardinal''', |
* [[Kelly-Morse]] set theory | * [[Kelly-Morse]] set theory | ||
− | * [[worldly]] cardinal and the [[worldly#Degrees of worldliness | $\alpha$-wordly]] hierarchy, [[worldly#Degrees of worldliness | hyper-worldly]] cardinal | + | * '''[[worldly]] cardinal'''' and the [[worldly#Degrees of worldliness | $\alpha$-wordly]] hierarchy, [[worldly#Degrees of worldliness | hyper-worldly]] cardinal |
* the [[Transitive ZFC model#Transitive model universe axiom | transitive model universe axiom]] | * the [[Transitive ZFC model#Transitive model universe axiom | transitive model universe axiom]] | ||
* [[Transitive ZFC model]] | * [[Transitive ZFC model]] | ||
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]] | * the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]] | ||
− | * [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] | + | * '''[[Con ZFC | Con(ZFC)]]''' and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]] |
* down to [[the middle attic]] | * down to [[the middle attic]] |
Latest revision as of 07:15, 13 September 2017
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.
- excessively hypercompact cardinal
- The Kunen inconsistency: Reinhardt cardinal, super Reinhardt cardinal, Berkeley cardinal
- Rank into rank cardinals $j:V_\lambda\to V_\lambda$, rank+1 into rank+1 cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$, I0 cardinal $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$
- The wholeness axiom
- n-huge cardinal, almost n-huge cardinal, super-n-huge cardinal, (n+1)-superstrong cardinal
- high-jump cardinal, almost high-jump cardinal, super high-jump cardinal, high-jump with unbounded excess closure cardinal
- Vopěnka's principle, Vopěnka cardinal, Woodin for supercompactness cardinal
- extendible cardinal, $\alpha$-extendible cardinal
- grand reflection cardinal
- hypercompact cardinal
- supercompact cardinal, $\lambda$-supercompact cardinal
- PFA cardinal
- strongly compact cardinal
- nearly supercompact and nearly strongly compact cardinals
- indestructible weakly compact cardinal
- subcompact cardinal
- superstrong cardinal
- Shelah cardinal
- Woodin cardinal, the axiom of determinacy
- strong cardinal and the $\theta$-strong and hypermeasurability hierarchy
- tall cardinal
- $0^\dagger$, $j:L[U]\to L[U]$ cardinal
- Nontrivial Mitchell rank, $o(\kappa)=1$, $o(\kappa)=\kappa^{++}$
- measurable cardinal, weakly measurable cardinal
- virtually Ramsey cardinal, Ramsey cardinal, strongly Ramsey cardinal
- Rowbottom cardinal
- Jónsson cardinal
- $\omega_1$-Erdős cardinal and $\gamma$-Erdős cardinals for uncountable $\gamma$
- $0^\sharp$, $j:L\to L$ cardinal
- Erdős cardinal, and the $\alpha$-Erdős hierarchy for countable $\alpha$
- $1$-iterable cardinal, and the $\alpha$-iterable cardinals hierarchy for $1\leq \alpha\leq \omega_1$
- remarkable cardinal
- ineffable cardinal, weakly ineffable cardinal, and the $n$-ineffable cardinals hierarchy; completely ineffable cardinal
- subtle cardinal
- ethereal cardinal
- superstrongly unfoldable cardinal, strongly uplifting cardinal
- weakly superstrong cardinal
- unfoldable cardinal, strongly unfoldable cardinal
- indescribable cardinal, totally indescribable cardinal
- weakly compact cardinal
- Mahlo cardinal, $1$-Mahlo, the $\alpha$-Mahlo hierarchy, hyper-Mahlo cardinals
- psuedo uplifting cardinal, uplifting cardinal
- ORD is Mahlo
- $\Sigma_2$-reflecting, $\Sigma_n$-reflecting and reflecting cardinals
- $1$-inaccessible, the $\alpha$-inaccessible hierarchy and hyper-inaccessible cardinals
- Grothendieck universe axiom, equivalent to the existence of a proper class of inaccessible cardinals
- weakly inaccessible cardinal, (strongly) inaccessible cardinal,
- Kelly-Morse set theory
- worldly cardinal' and the $\alpha$-wordly hierarchy, hyper-worldly cardinal
- the transitive model universe axiom
- Transitive ZFC model
- the minimal transitive model
- Con(ZFC) and $\text{Con}^\alpha(\text{ZFC})$, the iterated consistency hierarchy
- down to the middle attic