Weakly compact cardinal

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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 $L_{\kappa,\kappa}$ 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 it enjoys the partition property $\kappa\to(\kappa)^2_2$.
Indescribability property 
A cardinal $\kappa$ is weakly compact if and only if it is $\Pi_1^1$-indescribable.

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 $L_{\kappa,\lambda}$ 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 $L_{\omega,\omega}$ theory is satisfiable. A uncountable cardinal $\kappa$ is strongly compact if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ theory is satisfiable. The cardinal $\kappa$ is weakly compact if every $\kappa$-satisfiable $L_{\kappa,\kappa}$ 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 $\text{ZFC}^-$, the theory ZFC without the power set axiom (and using collection and separation rather than merely replacement). For any infinite cardinal $\kappa$ we have $H_{\kappa^+}\models\text{ZFC}^-$, 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 $L$.

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 [ 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 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 [1].

Relations with other large cardinals

  • Every weakly compact cardinal is inaccessible, Mahlo, hyper-Mahlo, hyper-hyper-Mahlo and more.
  • Measurable cardinals, Ramsey cardinals, unfoldable cardinals, totally indescribable cardinals are all weakly compact and a stationary limit of weakly compact cardinals.

References

  1. Kunen, Kenneth. Saturated Ideals. J Symbolic Logic 43(1):65--76, 1978. www   bibtex
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