Difference between revisions of "Upper attic"

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{{DISPLAYTITLE:The upper attic}}
 
{{DISPLAYTITLE:The upper attic}}
[[File:CapePogueLighthouse_medium.jpg | thumb | Cape Pogue Lighthouse photo by Timothy Valentine]]
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[[File:CapePogueLighthouse_medium.jpg|thumb|Cape Pogue Lighthouse photo by Timothy Valentine]]
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[[Category:Large cardinal axioms]]
  
 
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.  
  
* The [[Kunen inconsistency]]
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* [[Berkeley]] cardinal, [[Berkeley|club Berkeley]], [[Berkeley|limit club Berkeley]] cardinal
* [[Reinhardt]] cardinal
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* [[Reinhardt|weakly Reinhardt]], [[Reinhardt]], [[Reinhardt|super Reinhardt]], [[Reinhardt|totally Reinhardt]] cardinal
* [[L of V_lambda+1 | $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$]]
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* the '''[[Kunen inconsistency]]'''
* [[rank+1 into rank+1]] cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$  
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* '''[[rank into rank]]''' axioms ($I3$=$E_0$, $IE^\omega$, $IE$, $I2$=$E_1$, $E_i$, $I1$=$E_ω$ plus $m$-$C^{(n)}$-$E_i$), [[N-fold_variants#.24.5Comega.24-fold_variants|$\omega$-fold variants]], [[L of V_lambda+1|I0 axiom]] and strengthenings
* [[rank into rank]] cardinal $j:V_\lambda\to V_\lambda$
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* The [[wholeness axioms]], [[I4|axioms $\mathrm{I}_4^n$]]
* The [[wholeness axiom]]
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* [[n-fold variants|$n$-fold variants]] of hugeness (plus $C^{(n)}$ variants), extendibility, supercompactness, strongness, etc...
* [[huge#Super_n-huge | super $n$-huge]] cardinal
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* [[huge|almost huge]], '''[[huge]]''', [[huge|huge*]], [[huge|super almost huge]], [[huge|superhuge]], [[huge|ultrahuge]], [[superstrong|2-superstrong]] cardinal
* [[huge#Superhuge | superhuge]] cardinal
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* [[high-jump]] cardinal, [[high-jump|almost high-jump]] cardinal, [[high-jump|super high-jump]] cardinal, [[high-jump|high-jump with unbounded excess closure]] cardinal
* [[huge]] cardinal
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* [[Woodin#Shelah cardinals|Shelah for supercompactness]]
* [[huge#Almost huge | almost huge]] cardinal
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* [[Vopenka|Vopěnka scheme]], '''[[Vopenka|Vopěnka principle]]''', [[Vopenka#Vopěnka cardinals|Vopěnka-scheme]] cardinal, [[Vopenka#Vopěnka cardinals|Vopěnka]] (=[[Woodin#Shelah cardinals|Woodin for supercompactness]]) cardinal
* [[Vopenka#Formalisations | Vopěnka]] cardinal, [[Vopenka | Vopěnka's principle]]
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* [[extendible|$\alpha$-extendible]], '''[[extendible]]''', [[extendible|$C^{(n)}$-extendible]], [[extendible|$A$-extendible]] cardinals
* [[extendible]] cardinal
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* [[Woodin|Woodin for strong compactness]]
* [[grand reflection]] cardinal
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<!--* [[grand reflection]] cardinal-->
* [[supercompact]] cardinal
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* [[Supercompact#Enhanced supercompact cardinals|enhanced $\lambda$-supercompact]] cardinals, [[Supercompact#Enhanced supercompact cardinals|enhanced supercompact]] cardinal, [[hypercompact|$\lambda$-hypercompact]] cardinals, [[hypercompact]] cardinal
* [[strongly compact]] cardinal
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* [[supercompact|$\lambda$-supercompact]] cardinals, '''[[supercompact]]''' cardinal, [[supercompact|$C^{(n)}$-supercompact]] cardinals
* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact | nearly strongly compact]] cardinals
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* [[strongly compact|$\lambda$-strongly compact]] cardinals, '''[[strongly compact]]''' cardinal
* [[indestructible weakly compact]] cardinal
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* [[nearly supercompact]] and [[nearly supercompact#Nearly strongly compact|nearly strongly compact]] cardinals
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* [[Weakly_compact#Indestructibility of a weakly compact cardinal|indestructible weakly compact]] cardinal
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* The '''[[proper forcing axiom]]''' and [[forcing#Proper forcing|Martin's maximum]]
 
* [[subcompact]] cardinal
 
* [[subcompact]] cardinal
* [[strong#Superstrong cardinal| superstrong]] cardinal
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* [[superstrong]] cardinal, [[superstrong|$C^{(n)}$-superstrong]] hierarchy
* [[Shelah]] cardinal
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* [[Woodin|weakly hyper-Woodin]] cardinal, [[Shelah]] cardinal, [[Woodin|hyper-Woodin]] cardinal
* [[Woodin]] cardinal
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* The '''[[axiom of determinacy]]''' and [[axiom of projective determinacy|its projective counterpart]]
* [[strong]] cardinal and the [[strong | $\theta$-strong]] and [[strong#Hypermeasurable | hypermeasurability]] hierarchy
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* '''[[Woodin]]''' cardinal
* [[tall]] cardinal
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* [[strongly tall]] cardinal
* [[zero dagger| $0^\dagger$]]
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* the [[strong|$\theta$-strong]], [[strong#Hypermeasurable|hypermeasurability]], [[tall|$\theta$-tall]], [[strong|$\theta$-$A$-strong]], [[tall]], '''[[strong]]''', [[strong|$A$-strong]] cardinals
* Nontrivial [[Mitchell rank]], [[Mitchell rank | $o(\kappa)=1$]], [[Mitchell rank | $o(\kappa)=\kappa^{++}$]]  
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* Nontrivial [[Mitchell rank]], [[Mitchell rank|$o(\kappa)=1$]], [[Mitchell rank|$o(\kappa)=\kappa^{++}$]]  
* [[measurable]] cardinal
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*[[zero dagger| $0^\dagger$]] (''zero-dagger'')
* [[weakly measurable]] cardinal
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* [[weakly measurable]] cardinal, '''[[measurable]]''' cardinal
* [[strongly Ramsey]] cardinal
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** singular [[Jonsson|Jónsson]] cardinal
* [[Ramsey]] cardinal
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** $κ^+$-[[filter property]], [[Ramsey|strategic $(\omega+1)$-Ramsey]] cardinal, [[Ramsey|strategic fully Ramsey]] cardinal, [[Ramsey|$ω_1$-very Ramsey]] cardinal, [[Ramsey|$κ$-very Ramsey]] cardinal
* [[virtually Ramsey]] cardinal
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* $κ$-[[filter property]], [[Ramsey|fully Ramsey]] (=[[Ramsey|$κ$-Ramsey]]) cardinal  
* [[Rowbottom]] cardinal
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* [[Ramsey#Strongly Ramsey cardinal|strongly Ramsey]] cardinal, [[Ramsey|strongly Ramsey M-rank]], [[Ramsey#Super Ramsey cardinal|super Ramsey]] cardinal, [[Ramsey|super Ramsey M-rank]]
* [[Jonsson | Jónsson]] cardinal
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* $\alpha$-[[filter property]], [[Ramsey|$\alpha$-Ramsey]] cardinal (for $\omega < \alpha < \kappa$), [[Ramsey|almost fully Ramsey]] (=[[Ramsey|$<κ$-Ramsey]]) cardinal  
* [[Erdos | $\omega_1$-Erdős]] cardinal and [[Erdos | $\gamma$-Erdős]] cardinals for uncountable $\gamma$  
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* [[Ramsey|$\Pi_\alpha$-Romsey]], [[Ramsey|completely Romsey]] (=[[Ramsey|$ω$-very Ramsey]]), [[Ramsey|$\alpha$-hyper completely Romsey]], [[Ramsey|super completely Romsey]] cardinals
* [[zero sharp | $0^\sharp$]]
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* [[Ramsey|$\alpha$-Mahlo–Ramsey]] hierarchy
* [[Erdos | Erdős]] cardinal, and the [[Erdos | $\alpha$-Erdős]] hierarchy for countable $\alpha$
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* [[Ramsey|Ramsey M-rank]]
*[[$\alpha$-iterable| $1$-iterable]] cardinal, and the [[$\alpha$-iterable]] cardinals hierarchy for $1\leq \alpha\leq \omega_1$
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* [[Ramsey#Virtually Ramsey cardinal|virtually Ramsey]] cardinal, [[Jonsson|Jónsson]] cardinal, [[Rowbottom]] cardinal, '''[[Ramsey]]''' cardinal
* [[remarkable]] cardinal
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* [[Erdos|$\alpha$-weakly Erdős]] cardinals, [[Erdos|greatly Erdős]] cardinal
* [[completely ineffable]] cardinal
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* [[Ramsey#Almost Ramsey cardinal|almost Ramsey]] cardinal
* [[ineffable]] cardinal, and the $n$-ineffable cardinals hierarchy
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* [[Erdos|$\omega_1$-Erdős]] cardinal and [[Erdos|$\gamma$-Erdős]] cardinals for uncountable $\gamma$, [[Chang's conjecture]]
* [[weakly ineffable]] cardinal
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* [[Ramsey#.24.5Calpha.24-iterable cardinal|$\omega_1$-iterable]] cardinal, [[Ramsey|$(\omega, \omega_1)$-Ramsey]] cardinal
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* '''[[zero sharp|$0^\sharp$]] (''zero-sharp'')''', existence of [[Constructible universe#Silver indiscernibles|Silver indiscernibles]]
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* [[Silver cardinal]]
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* the [[Erdos|$\alpha$-'''Erdős''']], [[Ramsey#.24.5Calpha.24-iterable cardinal|$\alpha$-iterable]] and [[Ramsey|$(\omega, \alpha)$-Ramsey]] hierarchy for countable infinite $\alpha$
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* [[Erdos|$\omega$-Erdős]] cardinal, [[remarkable|weakly remarkable]] cardinal that is not remarkable
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* [[rank into rank|virtually rank-into-rank]] cardinal
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* the [[Ramsey#.24.5Calpha.24-iterable cardinal|$n$-iterable]] and [[huge|virtually $n$-huge*]] hierarchy
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* [[Woodin|virtually Shelah for supercompactness]] cardinal
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* [[extendible|virtually extendible]] (=[[remarkable|$2$-remarkable]]), [[extendible|virtually $C^{(n)}$-extendible]] (=[[remarkable|$n+1$-remarkable]]) cardinals, [[remarkable|completely remarkable]] cardinal, [[Vopenka|Generic Vopěnka's Principle]]
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* [[remarkable|($1$-)'''remarkable''']] (=virtually supercompact), [[measurable|virtually measurable]], [[Ramsey|strategic $\omega$-Ramsey]] cardinals, [[proper forcing axiom|weak Proper Forcing Axiom]]
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* [[Ramsey#.24.5Calpha.24-iterable cardinal|weakly Ramsey]] (=$1$-iterable) cardinal, [[Ramsey|super weakly Ramsey]] cardinals, [[Ramsey|$\omega$-Ramsey]] cardinal
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* [[completely ineffable]] cardinal (= $\omega$-[[filter property]])
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* [[Basic Theory of Elementary Embeddings]] ([[BTEE|$\mathrm{BTEE}$]])
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* [[ineffable#Helix|the $n$-subtle, $n$-almost ineffable, $n$-ineffable cardinals' hierarchy]]
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* [[Ramsey|$n$-Ramsey]], [[Ramsey|genuine $n$-Ramsey]], [[Ramsey|normal $n$-Ramsey]], [[Ramsey|$<\omega$-Ramsey]] cardinals
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* [[weakly ineffable]] (=almost ineffable=genuine $0$-[[Ramsey]]) cardinal, [[ineffable]] (=normal $0$-[[Ramsey]]) cardinal
 
* [[subtle]] cardinal
 
* [[subtle]] cardinal
* [[ethereal]] cardinal
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* [[ineffable#Ethereal cardinal|ethereal]] cardinal
* [[unfoldable]] cardinal, [[unfoldable#Strongly unfoldable | strongly unfoldable]] cardinal
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* [[uplifting#Strongly Uplifting|strongly uplifting]] (=[[unfoldable#Superstrongly Unfoldable|superstrongly unfoldable]]) cardinal
* [[Totally indescribable]] cardinal
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* [[weakly superstrong]] cardinal
* [[indescribable]] cardinal
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* [[unfoldable]] cardinal, [[unfoldable#Strongly Unfoldable|strongly unfoldable]] cardinal
* [[weakly compact]] cardinal
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* [[shrewd|$η$-shrewd]], [[shrewd]], [[shrewd|$\mathcal{A}$-$η$-$\mathcal{F}$-shrewd]], [[shrewd|$\mathcal{A}$-$η$-shrewd]], [[shrewd|$\mathcal{A}$-shrewd]] cardinals
* [[Mahlo#Hyper-Mahlo | $1$-Mahlo]], the [[Mahlo#Hyper-Mahlo | $\alpha$-Mahlo]] hierarchy and [[Mahlo#Hyper-Mahlo | hyper-Mahlo]] cardinals  
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* $\Sigma^m_n$- and '''$\Pi^m_n$-[[indescribable]]''', [[totally indescribable]], [[indescribable|$η$-indescribable]] cardinals
* [[Mahlo]] cardinal
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* [[weakly compact|$\Sigma_n$-weakly compact]] cardinals, [[weakly compact|$\Sigma_\omega$-weakly compact]] cardinal, '''[[weakly compact]]''' (=$\Pi_1^1$-[[indescribable]]=$0$-[[Ramsey]]) cardinal
* [[ORD is Mahlo]]
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* The [[Positive set theory|positive set theory]] $\text{GPK}^+_\infty$
* [[reflecting#Inaccessible reflecting cardinal | inaccessible $\Sigma_2$-reflecting]], [[reflecting#Inaccessible reflecting cardinal | inaccessible $\Sigma_n$-reflecting]] and [[reflecting#Inaccessible reflecting cardinal| inaccessible reflecting]] cardinals
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* [[Mahlo|$\Sigma_n$-Mahlo]] cardinals, [[Mahlo|$\Sigma_\omega$-Mahlo]] cardinal, [[Mahlo|weakly Mahlo]] cardinal, (strongly) '''[[Mahlo]]''' cardinal, [[Mahlo#Hyper-Mahlo|$1$-Mahlo]], the [[Mahlo#Hyper-Mahlo|$\alpha$-Mahlo]] hierarchy, [[Mahlo#Hyper-Mahlo|hyper-Mahlo]] cardinals, [[Mahlo|$Ω^α$-Mahlo]] cardinals
* [[inaccessible#Degrees of inaccessibility | $1$-inaccessible]], the [[inaccessible#Degrees of inaccessibility | $\alpha$-inaccessible]] hierarchy and [[inaccessible#Hyper-inaccessible | hyper-inaccessible]] cardinals
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* [[uplifting#pseudo uplifting cardinal|pseudo uplifting]] cardinal, [[uplifting]] cardinal
* [[inaccessible#Universes | Grothendieck universe axiom]], equivalent to the existence of a proper class of [[inaccessible]] cardinals
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* [[ORD is Mahlo|$\text{Ord}$ is Mahlo]]
* [[inaccessible]] cardinal, also known as strongly inaccessible
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* [[reflecting#Sigma_2 correct cardinals|$\Sigma_2$-reflecting]], [[reflecting|$\Sigma_n$-reflecting]] and [[reflecting]] cardinals
* [[inaccessible#Weakly inaccessible cardinal| weakly inaccessible]] cardinal
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* [[Jäger's collapsing functions and ρ-inaccessible ordinals]]
* [[worldly]] cardinal and the [[worldly#Degrees of worldliness | $\alpha$-wordly]] hierarchy, [[worldly#Degrees of worldliness | hyper-worldly]] cardinal
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* [[inaccessible#Degrees of inaccessibility|$1$-inaccessible]], the [[inaccessible#Degrees of inaccessibility|$\alpha$-inaccessible]] hierarchy, [[inaccessible#Hyper-inaccessible|hyper-inaccessible]] cardinals, [[inaccessible|$Ω^α$-inaccessible]] cardinals
* the [[Transitive ZFC model#Transitive model universe axiom | transitive model universe axiom]]  
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* [[inaccessible#Universes|Grothendieck universe axiom]] (the existence of a proper class of [[inaccessible]] cardinals)
* [[Transitive ZFC model]]
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* [[inaccessible#Weakly inaccessible cardinal|weakly inaccessible]] cardinal, (strongly) '''[[inaccessible]]''' cardinal
* the [[Transitive ZFC model#Minimal transitive model of ZFC | minimal transitive model]]
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* [[Morse-Kelley set theory|Morse-Kelley]] set theory
* [[Con ZFC | Con(ZFC)]] and [[Con ZFC#Consistency hierarchy | $\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy | iterated consistency hierarchy]]  
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* '''[[worldly]]''' cardinal and the [[worldly#Degrees of worldliness|$\alpha$-wordly]] hierarchy, [[worldly#Degrees of worldliness|hyper-worldly]] cardinal
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* the [[Transitive ZFC model#Transitive model universe axiom|transitive model universe axiom]]  
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* [[transitive ZFC model|transitive model of $\text{ZFC}$]]
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* the [[Transitive ZFC model#Minimal transitive model of ZFC|minimal transitive model]]
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* '''[[Con ZFC|$\text{Con(ZFC)}$]]''' and [[Con ZFC#Consistency hierarchy|$\text{Con}^\alpha(\text{ZFC})$]], the [[Con ZFC#Consistency hierarchy|iterated consistency hierarchy]]
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* '''[[ZFC|Zermelo-Fraenkel]]''' set theory
  
 
* down to [[the middle attic]]
 
* down to [[the middle attic]]

Latest revision as of 05:59, 17 November 2019

Cape Pogue Lighthouse photo by Timothy Valentine

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.