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- Weakly compact cardinals and the constructible universe
- Weakly compact cardinals and forcing
- Indestructibility of a weakly compact cardinal
- Relations with other large cardinals
- $\Sigma_n$-weakly compact etc.

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
uncountable
and every $\kappa$-satisfiable theory in an
$\mathcal{L}_{\kappa,\kappa}$
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
and has the tree
property.

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 $\kappa$-complete nonprincipal
filter
$F$ 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
$j:M\to N$ with
critical point
$\kappa$.

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
property
$\kappa\to(\kappa)^2_2$ holds.

Indescribability property

A cardinal $\kappa$ is weakly compact if and only if it is
$\Pi_1^1$-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
here.

Weakly compact cardinals first arose in connection with (and were named
for) the question of whether certain infinitary
logics
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
compact*
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
$\alpha_1<\cdots<\alpha_n$.

Every weakly compact cardinal is weakly compact in $L$. (Jech, 2003)

Nevertheless, the weak compactness property is not generally downward absolute between transitive models of set theory.

- Weakly compact cardinals are invariant under small forcing. [1]
- Weakly compact cardinals are preserved by the canonical forcing of
the GCH, by fast function forcing and many other forcing notions [
*citation needed*]. - If $\kappa$ is weakly compact, there is a forcing extension in
which $\kappa$ remains weakly compact and $2^\kappa\gt\kappa$ [
*citation needed*]. - 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 (Kunen, 1978).

*To expand using
[2]*

- Every weakly compact cardinal is inaccessible, Mahlo, hyper-Mahlo, hyper-hyper-Mahlo and more.
- Measurable cardinals, Ramsey cardinals, and totally indescribable cardinals are all weakly compact and a stationary limit of weakly compact cardinals.
- Assuming the consistency of a strongly unfoldable cardinal with ZFC, it is also consistent for the least weakly compact cardinal to be the least unfoldable cardinal. (Cody et al., 2013)
- If GCH holds, then the least weakly compact cardinal is not weakly measurable. However, if there is a measurable cardinal, then it is consistent for the least weakly compact cardinal to be weakly measurable. (Cody et al., 2013)
- If it is consistent for there to be a nearly supercompact, then it is consistent for the least weakly compact cardinal to be nearly supercompact. (Cody et al., 2013)
- For a cardinal $κ=κ^{<κ}$, $κ$ is weakly compact iff it is 0-Ramsey. (Nielsen & Welch, 2018)

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 (Bosch, 2006) it is shown that every $Σ_ω$-w.c. cardinal is $Σ_ω$-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 generic absoluteness axiom $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ}_ω , Γ)$ where $Γ$ is the class of projective ccc forcing notions.

This section from (Leshem, 2000; Bagaria, 2006)

- Jech, T. J. (2003).
*Set Theory*(Third). Springer-Verlag. https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf - Kunen, K. (1978). Saturated Ideals.
*J. Symbolic Logic*,*43*(1), 65–76. http://www.jstor.org/stable/2271949 - Cody, B., Gitik, M., Hamkins, J. D., & Schanker, J. (2013).
*The Least Weakly Compact Cardinal Can Be Unfoldable, Weakly Measurable and Nearly θ-Supercompact*. - Nielsen, D. S., & Welch, P. (2018).
*Games and Ramsey-like cardinals*. - Bosch, R. (2006). Small Definably-large Cardinals.
*Set Theory. Trends in Mathematics*, 55–82. https://doi.org/10.1007/3-7643-7692-9_3 - Leshem, A. (2000). On the consistency of the definable tree property on \aleph_1.
*J. Symbolic Logic*,*65*(3), 1204–1214. https://doi.org/10.2307/2586696 - Bagaria, J. (2006). Axioms of generic absoluteness.
*Logic Colloquium 2002*. https://doi.org/10.1201/9781439865903