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.
- [Berkeley](Berkeley.md "Berkeley")
cardinal, [](Berkeley.md),
[](Berkeley.md)
cardinal
- [](Reinhardt.md),
[Reinhardt](Reinhardt.md "Reinhardt"),
[](Reinhardt.md),
[](Reinhardt.md)
cardinal
- the **[](Kunen_inconsistency.md)**
- **[](Rank_into_rank.md)**
axioms ($I3$=$E_0$, $IE^\omega$, $IE$, $I2$=$E_1$, $E_i$,
$I1$=$E_ω$ plus $m$-$C^{(n)}$-$E_i$), [](N-fold_variants.md#.24.5Comega.24-fold_variants),
[I0
axiom](L_of_V_lambda%2B1 "L of V lambda+1")
and strengthenings
- The [](Wholeness_axioms.md),
<a href="I4" class="mw-redirect" title="I4">axioms $\mathrm{I}_4^n
lt;/a>
- [](N-fold_variants.md)
of hugeness (plus $C^{(n)}$ variants), extendibility,
supercompactness, strongness, etc...
- [](Huge.md),
**[huge](Huge.md "Huge")**,
[huge*](Huge.md "Huge"),
[](Huge.md),
[superhuge](Huge.md "Huge"),
[ultrahuge](Huge.md "Huge"),
[2-superstrong](Superstrong.md "Superstrong")
cardinal
- [high-jump](High-jump.md "High-jump")
cardinal, [](High-jump.md)
cardinal, [](High-jump.md)
cardinal, [](High-jump.md)
cardinal
- [](Woodin.md#Shelah_cardinals)
- [](Vopenka.md),
**[](Vopenka.md)**,
[Vopěnka-scheme](Vopenka.md#Vop.C4.9Bnka_cardinals "Vopenka")
cardinal,
[Vopěnka](Vopenka.md#Vop.C4.9Bnka_cardinals "Vopenka")
(=[](Woodin.md#Shelah_cardinals))
cardinal
- [$\alpha$-extendible](Extendible.md "Extendible")
hierarchy,
**[extendible](Extendible.md "Extendible")**
cardinal,
[$C^{(n)}$-extendible](Extendible.md "Extendible")
hierarchy
- [](Woodin.md)
- [](Supercompact.md#Enhanced_supercompact_cardinals)
cardinals, [](Supercompact.md#Enhanced_supercompact_cardinals)
cardinal,
[$\lambda$-hypercompact](Hypercompact.md "Hypercompact")
cardinals,
[hypercompact](Hypercompact.md "Hypercompact")
cardinal
- [$\lambda$-supercompact](Supercompact.md "Supercompact")
cardinals,
**[supercompact](Supercompact.md "Supercompact")**
cardinal,
[$C^{(n)}$-supercompact](Supercompact.md "Supercompact")
cardinals
- [](Strongly%20compact.md)
cardinals, **[](Strongly%20compact.md)**
cardinal
- [](Nearly%20supercompact.md)
and [](Nearly%20supercompact.md#Nearly_strongly_compact)
cardinals
- [](Weakly%20compact.md#Indestructibility_of_a_weakly_compact_cardinal)
cardinal
- The
**<a href="Proper_forcing_axiom" class="mw-redirect" title="Proper forcing axiom">proper forcing axiom</a>**
and [](Forcing.md#Proper_forcing)
- <a href="Subcompact" class="mw-redirect" title="Subcompact">subcompact</a>
cardinal
- [superstrong](Superstrong.md "Superstrong")
cardinal,
[$C^{(n)}$-superstrong](Superstrong.md "Superstrong")
hierarchy
- [Shelah](Woodin.md#Shelah "Woodin")
cardinal
- The **[](Axiom%20of%20determinacy.md)**
and
<a href="Axiom_of_projective_determinacy" class="mw-redirect" title="Axiom of projective determinacy">its projective counterpart</a>
- **[Woodin](Woodin.md "Woodin")**
cardinal
- <a href="Strongly_tall" class="mw-redirect" title="Strongly tall">strongly tall</a>
cardinal
- the
[$\theta$-strong](Strong.md "Strong"),
[hypermeasurability](Strong.md#Hypermeasurable "Strong")
and
[$\theta$-tall](Tall.md "Tall")
hierarchies,
[tall](Tall.md "Tall") and
**[strong](Strong.md "Strong")**
cardinals
- Nontrivial [](Mitchell%20rank%20and%20Mitchell%20order.md),
[$o(\kappa)=1$](Mitchell%20rank%20and%20Mitchell%20order.md "Mitchell rank"),
[$o(\kappa)=\kappa^{++}$](Mitchell%20rank%20and%20Mitchell%20order.md "Mitchell rank")
- <a href="Zero_dagger" class="mw-redirect" title="Zero dagger">$0^\daggerlt;/a>
(*zero-dagger*)
- [](Weakly_measurable.md)
cardinal,
**[measurable](Measurable.md "Measurable")**
cardinal
- singular
[Jónsson](Jonsson.md "Jonsson")
cardinal
- $κ^+$-<a href="Filter_property" class="mw-redirect" title="Filter property">filter property</a>,
[](Ramsey.md)
cardinal, [](Ramsey.md)
cardinal, [](Ramsey.md)
cardinal, [](Ramsey.md)
cardinal
- $κ$-<a href="Filter_property" class="mw-redirect" title="Filter property">filter property</a>,
[](Ramsey.md)
(=[$κ$-Ramsey](Ramsey.md "Ramsey"))
cardinal
- [](Ramsey.md#Strongly_Ramsey_cardinal)
cardinal, [](Ramsey.md),
[](Ramsey.md#Super_Ramsey_cardinal)
cardinal, [](Ramsey.md)
- $\alpha$-<a href="Filter_property" class="mw-redirect" title="Filter property">filter property</a>,
[$\alpha$-Ramsey](Ramsey.md "Ramsey")
cardinal (for $\omega < \alpha < \kappa$), [](Ramsey.md)
(=[lt;κ$-Ramsey](Ramsey.md "Ramsey"))
cardinal
- [$\Pi_\alpha$-Romsey](Ramsey.md "Ramsey"),
[](Ramsey.md)
(=[](Ramsey.md)),
[](Ramsey.md),
[](Ramsey.md)
cardinals
- [$\alpha$-Mahlo–Ramsey](Ramsey.md "Ramsey")
hierarchy
- [](Ramsey.md)
- [](Ramsey.md#Virtually_Ramsey_cardinal)
cardinal,
[Jónsson](Jonsson.md "Jonsson")
cardinal,
[Rowbottom](Rowbottom.md "Rowbottom")
cardinal,
**[Ramsey](Ramsey.md "Ramsey")**
cardinal
- [](Erdos.md)
cardinals, [](Erdos.md)
cardinal
- [](Ramsey.md#Almost_Ramsey_cardinal)
cardinal
- [$\omega_1$-Erdős](Erdos.md "Erdos")
cardinal and
[$\gamma$-Erdős](Erdos.md "Erdos")
cardinals for uncountable $\gamma$, [](Chang's_conjecture.md)
- [$\omega_1$-iterable](Ramsey.md#.24.5Calpha.24-iterable_cardinal "Ramsey")
cardinal, [](Ramsey.md)
cardinal
- **<a href="Zero_sharp" class="mw-redirect" title="Zero sharp">$0^\sharplt;/a>
(*zero-sharp*)**, existence of [](Constructible%20universe.md#Silver_indiscernibles)
- <a href="Silver_cardinal" class="mw-redirect" title="Silver cardinal">Silver cardinal</a>
- the
[$\alpha$-**Erdős**](Erdos.md "Erdos"),
[$\alpha$-iterable](Ramsey.md#.24.5Calpha.24-iterable_cardinal "Ramsey")
and [](Ramsey.md)
hierarchy for countable infinite $\alpha$
- [$\omega$-Erdős](Erdos.md "Erdos")
cardinal, [](Remarkable.md)
cardinal that is not remarkable
- [](Rank_into_rank.md)
cardinal
- the
[$n$-iterable](Ramsey.md#.24.5Calpha.24-iterable_cardinal "Ramsey")
and [](Huge.md)
hierarchy
- [](Woodin.md)
cardinal
- [](Extendible.md)
(=[$2$-remarkable](Remarkable.md "Remarkable")),
[](Extendible.md)
(=[$n+1$-remarkable](Remarkable.md "Remarkable"))
cardinals, [](Remarkable.md)
cardinal, [](Vopenka.md)
- [($1$-)**remarkable**](Remarkable.md "Remarkable")
(=virtually supercompact), [](Measurable.md),
[](Ramsey.md)
cardinals,
<a href="Proper_forcing_axiom" class="mw-redirect" title="Proper forcing axiom">weak Proper Forcing Axiom</a>
- [](Ramsey.md#.24.5Calpha.24-iterable_cardinal)
(=$1$-iterable) cardinal, [](Ramsey.md)
cardinals,
[$\omega$-Ramsey](Ramsey.md "Ramsey")
cardinal
- <a href="Completely_ineffable" class="mw-redirect" title="Completely ineffable">completely ineffable</a>
cardinal (=
$\omega$-<a href="Filter_property" class="mw-redirect" title="Filter property">filter property</a>)
- [](Ineffable.md#Helix)
- [$n$-Ramsey](Ramsey.md "Ramsey"),
[](Ramsey.md),
[](Ramsey.md),
[lt;\omega$-Ramsey](Ramsey.md "Ramsey")
cardinals
- <a href="Weakly_ineffable" class="mw-redirect" title="Weakly ineffable">weakly ineffable</a>
(=almost ineffable=genuine
$0$-[Ramsey](Ramsey.md "Ramsey"))
cardinal,
[ineffable](Ineffable.md "Ineffable")
(=normal
$0$-[Ramsey](Ramsey.md "Ramsey"))
cardinal
- <a href="Subtle" class="mw-redirect" title="Subtle">subtle</a>
cardinal
- [ethereal](Ineffable.md#Ethereal_cardinal "Ineffable")
cardinal
- [](Uplifting.md#Strongly_Uplifting)
(=[](Unfoldable.md#Superstrongly_Unfoldable))
cardinal
- <a href="Weakly_superstrong" class="mw-redirect" title="Weakly superstrong">weakly superstrong</a>
cardinal
- [unfoldable](Unfoldable.md "Unfoldable")
cardinal, [](Unfoldable.md#Strongly_Unfoldable)
cardinal
- [$η$-shrewd](Shrewd.md "Shrewd"),
[shrewd](Shrewd.md "Shrewd"),
[$\mathcal{A}$-$η$-shrewd](Shrewd.md "Shrewd"),
[$\mathcal{A}$-shrewd](Shrewd.md "Shrewd")
cardinals
- $\Sigma^m_n$- and
**$\Pi^m_n$-[indescribable](Indescribable.md "Indescribable")**,
<a href="Totally_indescribable" class="mw-redirect" title="Totally indescribable">totally indescribable</a>,
[$η$-indescribable](Indescribable.md "Indescribable")
cardinals
- [[weakly compact]]
cardinals, [[weakly compact]]cardinal, **[[weakly compact]]**
(=$\Pi_1^1$-[[weakly compact]]e.md "Indescribable")=$0$-[Ramsey](/web/20191104221438/http://cantorsattic.info/Ramsey "Ramsey"))
cardinal
- The [](Positive_set_theory.md)
$\text{GPK}^+_\infty$
- [$\Sigma_n$-Mahlo](Mahlo.md "Mahlo")
cardinals,
[$\Sigma_\omega$-Mahlo](Mahlo.md "Mahlo")
cardinal, [](Mahlo.md)
cardinal, (strongly)
**[Mahlo](Mahlo.md "Mahlo")**
cardinal,
[$1$-Mahlo](Mahlo.md#Hyper-Mahlo "Mahlo"),
the
[$\alpha$-Mahlo](Mahlo.md#Hyper-Mahlo "Mahlo")
hierarchy,
[hyper-Mahlo](Mahlo.md#Hyper-Mahlo "Mahlo")
cardinals,
[$Ω^α$-Mahlo](Mahlo.md "Mahlo")
cardinals
- [](Uplifting.md#pseudo_uplifting_cardinal)
cardinal,
[uplifting](Uplifting.md "Uplifting")
cardinal
- [](ORD_is_Mahlo.md)
- [$\Sigma_2$-reflecting](Reflecting.md#Sigma_2_correct_cardinals "Reflecting"),
[$\Sigma_n$-reflecting](Reflecting.md "Reflecting")
and
[reflecting](Reflecting.md "Reflecting")
cardinals
- [](Jäger's_collapsing_functions_and_ρ-inaccessible_ordinals.md)
- [$1$-inaccessible](Inaccessible.md#Degrees_of_inaccessibility "Inaccessible"),
the
[$\alpha$-inaccessible](Inaccessible.md#Degrees_of_inaccessibility "Inaccessible")
hierarchy,
[hyper-inaccessible](Inaccessible.md#Hyper-inaccessible "Inaccessible")
cardinals,
[$Ω^α$-inaccessible](Inaccessible.md "Inaccessible")
cardinals
- [](Inaccessible.md#Universes)
(the existence of a proper class of
[inaccessible](Inaccessible.md "Inaccessible")
cardinals)
- [](Inaccessible.md#Weakly_inaccessible_cardinal)
cardinal, (strongly)
**[inaccessible](Inaccessible.md "Inaccessible")**
cardinal
- <a href="Morse-Kelley_set_theory" class="mw-redirect" title="Morse-Kelley set theory">Morse-Kelley</a>
set theory
- **[worldly](Worldly.md "Worldly")**
cardinal and the
[$\alpha$-wordly](Worldly.md#Degrees_of_worldliness "Worldly")
hierarchy,
[hyper-worldly](Worldly.md#Degrees_of_worldliness "Worldly")
cardinal
- the
<a href="Transitive_ZFC_model#Transitive_model_universe_axiom" class="mw-redirect" title="Transitive ZFC model">transitive model universe axiom</a>
- <a href="Transitive_ZFC_model" class="mw-redirect" title="Transitive ZFC model">transitive model of $\text{ZFC}lt;/a>
- the
<a href="Transitive_ZFC_model#Minimal_transitive_model_of_ZFC" class="mw-redirect" title="Transitive ZFC model">minimal transitive model</a>
- **<a href="Con_ZFC" class="mw-redirect" title="Con ZFC">$\text{Con(ZFC)}lt;/a>**
and
<a href="Con_ZFC#Consistency_hierarchy" class="mw-redirect" title="Con ZFC">$\text{Con}^\alpha(\text{ZFC})lt;/a>,
the
<a href="Con_ZFC#Consistency_hierarchy" class="mw-redirect" title="Con ZFC">iterated consistency hierarchy</a>
- **[Zermelo-Fraenkel](ZFC.md "ZFC")**
set theory
- down to
<a href="The_middle_attic" class="mw-redirect" title="The middle attic">the middle attic</a>