Difference between revisions of "Weakly compact"
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{{DISPLAYTITLE: Weakly compact cardinal}} | {{DISPLAYTITLE: Weakly compact cardinal}} | ||
− | + | [[Category:Large cardinal axioms]] | |
Weakly compact cardinals lie at the focal point of a number | 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: | 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 $ | + | :; Weak compactness : A cardinal $\kappa$ is weakly compact if and only if it is [[uncountable]] and every $\kappa$-satisfiable theory in an [[Infinitary logic|$\mathcal{L}_{\kappa,\kappa}$]] 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 $\langle V_\kappa,{\in},A\rangle\prec\langle W,{\in},A^*\rangle$. | :; 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. | + | :; 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 $M$. | + | :; 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|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 $\kappa$. | + | :; 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 [[elementary embedding|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 $\kappa$. | :; 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 | + | :; 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 $j,M\in N$. | ;; 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 | + | :; 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]]. | :; 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 [https://mathoverflow.net/questions/309896/a-weakening-of-cardinal-compactness-is-it-equivalent/309937#309937 here]. | ||
+ | :; 2-regular : A cardinal $\kappa$ is weakly compact if and only if every $\kappa$-bounded $F: \kappa_\kappa\rightarrow\kappa_\kappa$ has a witness ($0<\alpha<\kappa$ such that for every $f: \kappa\rightarrow\kappa$ we have $f|\alpha\subseteq\alpha \implies F(f)"\alpha\subseteq\alpha$). ''TODO complete'' <cite>RichterAczel1974:InductiveDefinitions</cite><sup>definition 1.13<!--typo, written as 1.3-->, theorem 1.14</sup> | ||
Weakly compact cardinals first arose | Weakly compact cardinals first arose | ||
− | in connection with | + | in connection with (and were named for) the question of |
− | whether certain infinitary logics satisfy the compactness | + | whether certain [[Infinitary logic|infinitary logics]] satisfy the compactness |
theorem of first order logic. Specifically, in a language | theorem of first order logic. Specifically, in a language | ||
with a signature consisting, as in the first order context, | with a signature consisting, as in the first order context, | ||
of a set of constant, finitary function and relation | of a set of constant, finitary function and relation | ||
− | symbols, we build up the language of $ | + | symbols, we build up the language of $\mathcal{L}_{\kappa,\lambda}$ |
formulas by closing the collection of formulas under | formulas by closing the collection of formulas under | ||
infinitary conjunctions | infinitary conjunctions | ||
Line 34: | Line 36: | ||
satisfiable. First order logic is precisely | satisfiable. First order logic is precisely | ||
$L_{\omega,\omega}$, and the classical Compactness theorem | $L_{\omega,\omega}$, and the classical Compactness theorem | ||
− | asserts that every $\omega$-satisfiable $ | + | asserts that every $\omega$-satisfiable $\mathcal{L}_{\omega,\omega}$ |
theory is satisfiable. A uncountable cardinal $\kappa$ is | theory is satisfiable. A uncountable cardinal $\kappa$ is | ||
''[[strongly compact]]'' if every $\kappa$-satisfiable | ''[[strongly compact]]'' if every $\kappa$-satisfiable | ||
− | $ | + | $\mathcal{L}_{\kappa,\kappa}$ theory is satisfiable. The cardinal |
$\kappa$ is ''weakly compact'' if every | $\kappa$ is ''weakly compact'' if every | ||
− | $\kappa$-satisfiable $ | + | $\kappa$-satisfiable $\mathcal{L}_{\kappa,\kappa}$ theory, in a |
language having at most $\kappa$ many constant, function | language having at most $\kappa$ many constant, function | ||
and relation symbols, is satisfiable. | and relation symbols, is satisfiable. | ||
Line 47: | Line 49: | ||
than $\kappa$. More specifically, $T$ is a ''tree'' if | than $\kappa$. More specifically, $T$ is a ''tree'' if | ||
$T$ is a partial order such that the predecessors of any | $T$ is a partial order such that the predecessors of any | ||
− | node in $T$ are well ordered. The $\alpha^ | + | node in $T$ are well ordered. The $\alpha^\textrm{th}$ level of a |
tree $T$, denoted $T_\alpha$, consists of the nodes whose | tree $T$, denoted $T_\alpha$, consists of the nodes whose | ||
predecessors have order type exactly $\alpha$, and these | predecessors have order type exactly $\alpha$, and these | ||
Line 54: | Line 56: | ||
through a tree $T$ is a maximal linearly ordered subset of | through a tree $T$ is a maximal linearly ordered subset of | ||
$T$ of order type $\kappa$. Such a branch selects exactly | $T$ of order type $\kappa$. Such a branch selects exactly | ||
− | one node from each level, in a linearly ordered manner. The | + | one node from each level, in a linearly ordered manner. The set of $\kappa$-branches is denoted $[T]$. A $\kappa$-tree |
− | set of $\kappa$-branches is denoted $[T]$. A $\kappa$-tree | + | |
is an ''Aronszajn'' tree if it has no $\kappa$-branches. | is an ''Aronszajn'' tree if it has no $\kappa$-branches. | ||
A cardinal $\kappa$ has the ''tree property'' if every | A cardinal $\kappa$ has the ''tree property'' if every | ||
Line 61: | Line 62: | ||
A transitive set $M$ is a $\kappa$-model of set theory if | A transitive set $M$ is a $\kappa$-model of set theory if | ||
− | $|M|=\kappa$, $M^{\lt\kappa}\subset M$ and $M$ satisfies | + | $|M|=\kappa$, $M^{\lt\kappa}\subset M$ and $M$ satisfies ZFC$^-$, |
− | the theory ZFC without the power set axiom | + | the theory ZFC without the power set axiom (and using collection and separation rather than merely replacement). |
For any | For any | ||
infinite cardinal $\kappa$ we have | infinite cardinal $\kappa$ we have | ||
− | $H_{\kappa^+} | + | that $H_{\kappa^+}$ models ZFC$^-$, and further, if |
$M\prec H_{\kappa^+}$ and $\kappa\subset M$, then $M$ is | $M\prec H_{\kappa^+}$ and $\kappa\subset M$, then $M$ is | ||
transitive. Thus, any $A\in H_{\kappa^+}$ can be placed | transitive. Thus, any $A\in H_{\kappa^+}$ can be placed | ||
Line 71: | Line 72: | ||
the downward Löwenheim-Skolem theorem to find such $M$ | the downward Löwenheim-Skolem theorem to find such $M$ | ||
with $M^{\lt\kappa}\subset M$. So in this case there are abundant | with $M^{\lt\kappa}\subset M$. So in this case there are abundant | ||
− | $\kappa$-models of set theory | + | $\kappa$-models of set theory (and conversely, if there is |
a $\kappa$-model of set theory, then $2^{\lt\kappa}=\kappa$). | a $\kappa$-model of set theory, then $2^{\lt\kappa}=\kappa$). | ||
− | The partition property $\kappa\to | + | The partition property $\kappa\to(\lambda)^n_\gamma$ |
asserts that for every function $F:[\kappa]^n\to\gamma$ | asserts that for every function $F:[\kappa]^n\to\gamma$ | ||
there is $H\subset\kappa$ with $|H|=\lambda$ such that | there is $H\subset\kappa$ with $|H|=\lambda$ such that | ||
Line 81: | Line 82: | ||
existence of a ''monochromatic'' set $H$, since all | existence of a ''monochromatic'' set $H$, since all | ||
tuples from $H$ get the same color. The partition property | tuples from $H$ get the same color. The partition property | ||
− | $\kappa\to | + | $\kappa\to(\kappa)^2_2$ asserts that every partition of |
$[\kappa]^2$ into two sets admits a set $H\subset\kappa$ 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 | size $\kappa$ such that $[H]^2$ lies on one side of the | ||
partition. When defining $F:[\kappa]^n\to\gamma$, we define | partition. When defining $F:[\kappa]^n\to\gamma$, we define | ||
− | $F | + | $F(\alpha_1,\ldots,\alpha_n)$ only when |
$\alpha_1<\cdots<\alpha_n$. | $\alpha_1<\cdots<\alpha_n$. | ||
== Weakly compact cardinals and the constructible universe == | == Weakly compact cardinals and the constructible universe == | ||
− | Every weakly compact cardinal is weakly compact in $L$. | + | Every weakly compact cardinal is weakly compact in [[Constructible universe|$L$]]. <cite>Jech2003:SetTheory</cite> |
Nevertheless, the weak compactness property is not generally downward absolute between transitive models of set theory. | Nevertheless, the weak compactness property is not generally downward absolute between transitive models of set theory. | ||
Line 97: | Line 98: | ||
== Weakly compact cardinals and forcing == | == Weakly compact cardinals and forcing == | ||
− | * Weakly compact cardinals are invariant under small forcing | + | * Weakly compact cardinals are invariant under small forcing. [http://www.math.csi.cuny.edu/~fuchs/IndestructibleWeakCompactness.pdf] |
* Weakly compact cardinals are preserved by the canonical forcing of the GCH, by fast function forcing and many other forcing notions {{Citation needed}}. | * 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 $\kappa$ 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, but becomes weakly compact in a forcing extension <CITE>Kunen1978:SaturatedIdeals</CITE>. | + | * 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:SaturatedIdeals</CITE>. |
+ | |||
+ | == Indestructibility of a weakly compact cardinal == | ||
+ | ''To expand using [https://arxiv.org/abs/math/9907046]'' | ||
== Relations with other large cardinals == | == Relations with other large cardinals == | ||
− | * Every weakly compact cardinal is inaccessible, Mahlo, hyper-Mahlo, hyper-hyper-Mahlo and more. | + | * Every weakly compact cardinal is [[inaccessible]], [[Mahlo]], hyper-Mahlo, hyper-hyper-Mahlo and more. |
− | * Measurable cardinals, Ramsey cardinals, | + | * [[Measurable]] cardinals, [[Ramsey]] cardinals, and [[indescribable|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. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite> | ||
+ | *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. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite> | ||
+ | *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. <cite>CodyGitikHamkinsSchanker2013:TheLeastWeaklyCompactCardinal</cite> | ||
+ | * For a cardinal $κ=κ^{<κ}$, $κ$ is weakly compact iff it is 0-[[Ramsey]]. <cite>NielsenWelch2018:GamesRamseylike</cite> | ||
+ | |||
+ | ==$\Sigma_n$-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:SmallDefinablyLargeCardinals</cite> 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 [[axioms of generic absoluteness]]. For example: | ||
+ | * The existence of a $Σ_n$-weakly compact cardinal is equiconsistent with $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, Γ_n)$ where $Γ_n$ is the class of ccc posets that are $Σ_n$ or $Π_n$ definable in $H(ω_1)$ with parameters. | ||
+ | * The existence of a $Σ_ω$-weakly compact cardinal is equiconsistent with $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of projective ccc forcing notions. | ||
+ | |||
+ | This section from<cite>Leshem2000:OCDefinableTreeProperty</cite><cite>Bagaria2002:AxiomsOfGenericAbsoluteness</cite> | ||
+ | |||
+ | ==Recursive analogue== | ||
+ | ''$Π_3$-[[reflecting ordinal|reflecting]]'' or ''2-[[admissible]]'' ordinals are analogous to weakly compact ($Π_1^1$-indescribable) cardinals and can be called ''recursively weakly compact''<cite>Madore2017:OrdinalZoo</cite><cite>RichterAczel1974:InductiveDefinitions</cite><sup>after definition 1.12</sup> | ||
{{References}} | {{References}} |
Latest revision as of 06:38, 15 May 2022
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.
- 2-regular
- A cardinal $\kappa$ is weakly compact if and only if every $\kappa$-bounded $F: \kappa_\kappa\rightarrow\kappa_\kappa$ has a witness ($0<\alpha<\kappa$ such that for every $f: \kappa\rightarrow\kappa$ we have $f|\alpha\subseteq\alpha \implies F(f)"\alpha\subseteq\alpha$). TODO complete [1]^{definition 1.13, theorem 1.14}
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 $\alpha^\textrm{th}$ 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 $\alpha_1<\cdots<\alpha_n$.
Contents
Weakly compact cardinals and the constructible universe
Every weakly compact cardinal is weakly compact in $L$. [2]
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. [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 [3].
Indestructibility of a weakly compact cardinal
To expand using [2]
Relations with other large cardinals
- 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. [4]
- 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. [4]
- 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. [4]
- For a cardinal $κ=κ^{<κ}$, $κ$ is weakly compact iff it is 0-Ramsey. [5]
$\Sigma_n$-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 [6] 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 axioms of generic absoluteness. For example:
- The existence of a $Σ_n$-weakly compact cardinal is equiconsistent with $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, Γ_n)$ where $Γ_n$ is the class of ccc posets that are $Σ_n$ or $Π_n$ definable in $H(ω_1)$ with parameters.
- The existence of a $Σ_ω$-weakly compact cardinal is equiconsistent with $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ_ω}, Γ)$ where $Γ$ is the class of projective ccc forcing notions.
Recursive analogue
$Π_3$-reflecting or 2-admissible ordinals are analogous to weakly compact ($Π_1^1$-indescribable) cardinals and can be called recursively weakly compact[9][1]^{after definition 1.12}
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