Weakly compact cardinals lie at the focal point of a number of diverse
concepts in infinite combinatorics, admitting various characterizations
in terms of these concepts. If $\kappa^{ {<}\kappa} = \kappa$, then
the following are equivalent:
Weak compactness
A cardinal $\kappa$ is weakly compact if and only if it is
<a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a>
and every $\kappa$-satisfiable theory in an
[$\mathcal{L}_{\kappa,\kappa}$](Infinitary%20logic.md "Infinitary logic")
language of size at most $\kappa$ is satisfiable.
Extension property
A cardinal $\kappa$ is weakly compact if and only if for every
$A\subset V_\kappa$, there is a transitive structure $W$ properly
extending $V_\kappa$ and $A^*\subset W$ such that $\langle
V_\kappa,{\in},A\rangle\prec\langle W,{\in},A^*\rangle$.
Tree property
A cardinal $\kappa$ is weakly compact if and only if it is
[inaccessible](Inaccessible.md "Inaccessible")
and has the [](Tree%20property.md).
Filter property
A cardinal $\kappa$ is weakly compact if and only if whenever $M$ is a
set containing at most $\kappa$-many subsets of $\kappa$, then there
is a $\kappa$-[](Filter.md)
$F$ measuring every set in $M$.
Weak embedding property
A cardinal $\kappa$ is weakly compact if and only if for every
$A\subset\kappa$ there is a transitive set $M$ of size $\kappa$ with
$\kappa\in M$ and a transitive set $N$ with an
[embedding](Elementary%20embedding.md "Elementary embedding")
$j:M\to N$ with
<a href="Critical_point" class="mw-redirect" title="Critical point">critical point</a>
$\kappa$.
Embedding characterization
A cardinal $\kappa$ is weakly compact if and only if for every
transitive set $M$ of size $\kappa$ with $\kappa\in M$ there is a
transitive set $N$ and an embedding $j:M\to N$ with critical point
$\kappa$.
Normal embedding characterization
A cardinal $\kappa$ is weakly compact if and only if for every
$\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding
$j:M\to N$ with critical point $\kappa$, such that $N=\{\
j(f)(\kappa)\mid f\in M\ \}$.
Hauser embedding characterization
A cardinal $\kappa$ is weakly compact if and only if for every
$\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding
$j:M\to N$ with critical point $\kappa$ such that $j,M\in N$.
Partition property
A cardinal $\kappa$ is weakly compact if and only if the [](Partition%20property.md)
$\kappa\to(\kappa)^2_2$ holds.
Indescribability property
A cardinal $\kappa$ is weakly compact if and only if it is
$\Pi_1^1$-[indescribable](Indescribable.md "Indescribable").
Skolem Property
A cardinal $\kappa$ is weakly compact if and only if $\kappa$ is
inaccessible and every $\kappa$-unboundedly satisfiable
$\mathcal{L}_{\kappa,\kappa}$-theory $T$ of size at most $\kappa$
has a model of size at least $\kappa$. A theory $T$ is
$\kappa$-unboundedly satisfiable if and only if for any
$\lambda<\kappa$, there exists a model $\mathcal{M}\models T$
with $\lambda\leq\|M\|<\kappa$. For more info see
<a href="https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937#309937" class="external text">here</a>.
Weakly compact cardinals first arose in connection with (and were named
for) the question of whether certain [](Infinitary%20logic.md)
satisfy the compactness theorem of first order logic. Specifically, in a
language with a signature consisting, as in the first order context, of
a set of constant, finitary function and relation symbols, we build up
the language of $\mathcal{L}_{\kappa,\lambda}$ formulas by closing
the collection of formulas under infinitary conjunctions
$\wedge_{\alpha<\delta}\varphi_\alpha$ and disjunctions
$\vee_{\alpha<\delta}\varphi_\alpha$ of any size
$\delta<\kappa$, as well as infinitary quantification
$\exists\vec x$ and $\forall\vec x$ over blocks of variables $\vec
x=\langle x_\alpha\mid\alpha<\delta\rangle$ of size less than
$\kappa$. A theory in such a language is *satisfiable* if it has a
model under the natural semantics. A theory is *$\theta$-satisfiable*
if every subtheory consisting of fewer than $\theta$ many sentences of
it is satisfiable. First order logic is precisely
$L_{\omega,\omega}$, and the classical Compactness theorem asserts
that every $\omega$-satisfiable $\mathcal{L}_{\omega,\omega}$
theory is satisfiable. A uncountable cardinal $\kappa$ is *[](Strongly%20compact.md)*
if every $\kappa$-satisfiable $\mathcal{L}_{\kappa,\kappa}$ theory
is satisfiable. The cardinal $\kappa$ is *weakly compact* if every
$\kappa$-satisfiable $\mathcal{L}_{\kappa,\kappa}$ theory, in a
language having at most $\kappa$ many constant, function and relation
symbols, is satisfiable.
Next, for any cardinal $\kappa$, a *$\kappa$-tree* is a tree of height
$\kappa$, all of whose levels have size less than $\kappa$. More
specifically, $T$ is a *tree* if $T$ is a partial order such that the
predecessors of any node in $T$ are well ordered. The $\alpha^{\rm
th}$ level of a tree $T$, denoted $T_\alpha$, consists of the nodes
whose predecessors have order type exactly $\alpha$, and these nodes
are also said to have *height* $\alpha$. The height of the tree $T$ is
the first $\alpha$ for which $T$ has no nodes of height $\alpha$. A
""$\kappa$-branch"" through a tree $T$ is a maximal linearly ordered
subset of $T$ of order type $\kappa$. Such a branch selects exactly one
node from each level, in a linearly ordered manner. The set of
$\kappa$-branches is denoted $[T\]$. A $\kappa$-tree is an
*Aronszajn* tree if it has no $\kappa$-branches. A cardinal $\kappa$
has the *tree property* if every $\kappa$-tree has a $\kappa$-branch.
A transitive set $M$ is a $\kappa$-model of set theory if
$\|M\|=\kappa$, $M^{\lt\kappa}\subset M$ and $M$ satisfies ZFC$^-$,
the theory ZFC without the power set axiom (and using collection and
separation rather than merely replacement). For any infinite cardinal
$\kappa$ we have that $H_{\kappa^+}$ models ZFC$^-$, and further, if
$M\prec H_{\kappa^+}$ and $\kappa\subset M$, then $M$ is
transitive. Thus, any $A\in H_{\kappa^+}$ can be placed into such an
$M$. If $\kappa^{\lt\kappa}=\kappa$, one can use the downward
Löwenheim-Skolem theorem to find such $M$ with $M^{\lt\kappa}\subset
M$. So in this case there are abundant $\kappa$-models of set theory
(and conversely, if there is a $\kappa$-model of set theory, then
$2^{\lt\kappa}=\kappa$).
The partition property $\kappa\to(\lambda)^n_\gamma$ asserts that
for every function $F:[\kappa\]^n\to\gamma$ there is
$H\subset\kappa$ with $\|H\|=\lambda$ such that
$F\upharpoonright[H\]^n$ is constant. If one thinks of $F$ as coloring
the $n$-tuples, the partition property asserts the existence of a
*monochromatic* set $H$, since all tuples from $H$ get the same color.
The partition property $\kappa\to(\kappa)^2_2$ asserts that every
partition of $\[\kappa\]^2$ into two sets admits a set
$H\subset\kappa$ of size $\kappa$ such that $\[H\]^2$ lies on one
side of the partition. When defining $F:\[\kappa\]^n\to\gamma$, we
define $F(\alpha_1,\ldots,\alpha_n)$ only when
$\alpha_1<\cdots<\alpha_n$.
## Weakly compact cardinals and the constructible universe
Every weakly compact cardinal is weakly compact in
[$L$](Constructible%20universe.md "Constructible universe").
{% cite Jech2003 %}
Nevertheless, the weak compactness property is not generally downward
absolute between transitive models of set theory.
## Weakly compact cardinals and forcing
- Weakly compact cardinals are invariant under small forcing.
<a href="http://www.math.csi.cuny.edu/~fuchs/IndestructibleWeakCompactness.pdf" class="external autonumber">[1]</a>
- Weakly compact cardinals are preserved by the canonical forcing of
the GCH, by fast function forcing and many other forcing notions \[
[](Library.md)
\].
- If $\kappa$ is weakly compact, there is a forcing extension in
which $\kappa$ remains weakly compact and $2^\kappa\gt\kappa$ \[
[](Library.md)
\].
- If the existence of weakly compact cardinals is consistent with ZFC,
then there is a model of ZFC in which $\kappa$ is not weakly
compact, but becomes weakly compact in a forcing extension
{% cite Kunen1978 %}.
## Indestructibility of a weakly compact cardinal
*To expand using
<a href="https://arxiv.org/abs/math/9907046" class="external autonumber">[2]</a>*
## Relations with other large cardinals
- Every weakly compact cardinal is
[inaccessible](Inaccessible.md "Inaccessible"),
[Mahlo](Mahlo.md "Mahlo"),
hyper-Mahlo, hyper-hyper-Mahlo and more.
- [Measurable](Measurable.md "Measurable")
cardinals,
[Ramsey](Ramsey.md "Ramsey")
cardinals, and [](Indescribable.md)
cardinals are all weakly compact and a stationary limit of weakly
compact cardinals.
- Assuming the consistency of a
<a href="Strongly_unfoldable" class="mw-redirect" title="Strongly unfoldable">strongly unfoldable</a>
cardinal with ZFC, it is also consistent for the least weakly
compact cardinal to be the least
[unfoldable](Unfoldable.md "Unfoldable")
cardinal.
{% cite Cody2013 %}
- If GCH holds, then the least weakly compact cardinal is not [](Weakly_measurable.md).
However, if there is a
[measurable](Measurable.md "Measurable")
cardinal, then it is consistent for the least weakly compact
cardinal to be weakly measurable.
{% cite Cody2013 %}
- If it is consistent for there to be a [](Nearly%20supercompact.md),
then it is consistent for the least weakly compact cardinal to be
nearly supercompact.
{% cite Cody2013 %}
- For a cardinal $κ=κ^{<κ}$, $κ$ is weakly compact iff it is
0-[Ramsey](Ramsey.md "Ramsey").
{% cite Nielsen2018 %}
## $\Sigma_n$-weakly compact etc.
An inaccessible cardinal $κ$ is $Σ_n$-weakly compact iff it reflects
$Π_1^1$ sentences with $Σ_n$-predicates, i.e. for every $R ⊆ V_κ$
which is definable by a $Σ_n$ formula (with parameters) over $V_κ$ and
every $Π_1^1$ sentence $Φ$, if $\langle V_κ , ∈, R \rangle \models
Φ$ then there is $α < κ$ (equivalently, unboundedly-many $α < κ$)
such that $\langle V_α , ∈, R ∩ V_α \rangle \models Φ$. Analogously
for $Π_n$ and $∆_n$. $κ$ is $Σ_ω$-weakly compact iff it is
$Σ_n$-weakly compact for all $n < ω$.
$κ$ is $Σ_n$-weakly compact $\iff$ $κ$ is $Π_n$-weakly compact
$\iff$ $κ$ is $∆_{n+1}$-weakly compact $\iff$ For every $Π_1^1$
formula $Φ(x_0 , ..., x_k)$ in the language of set theory and every
$a_0 , ..., a_k ∈ V_κ$, if $V κ \models Φ(a_0 , ..., a_k )$, then
there is $λ ∈ I_n := \{λ < κ : λ$ is inaccessible and $V_λ
\preccurlyeq_n V_κ\}$ such that $V_λ \models Φ(a_0 , ..., a_k)$.
In {% cite Bosch2006 %} it is shown
that every $Σ_ω$-w.c. cardinal is
$Σ_ω$-[Mahlo](Mahlo.md "Mahlo")
and the set of $Σ_ω$-Mahlo cardinals below a $Σ_ω$-w.c. cardinal is
$Σ_ω$-stationary, but if κ is $Π_{n+1}$-Mahlo, then the set of
$Σ_n$-w.c. cardinals below $κ$ is $Π_{n+1}$-stationary.
These properties are connected with some forms of absoluteness. For
example, the existence of a $Σ_ω$-w.c. cardinal is equiconsistent with
the
<a href="index.php?title=Generic_absoluteness_axiom&action=edit&redlink=1" class="new" title="Generic absoluteness axiom (page does not exist)">generic absoluteness axiom</a>
$\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ}_ω , Γ)$ where $Γ$ is
the class of projective ccc forcing notions.
This section
from {% cite Leshem2000 Bagaria2006 %}