Difference between revisions of "Weakly compact"

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* (weak compactness property) 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.
 
* (weak compactness property) 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 $A^*\of W$ 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 $\kappa$-complete nonprincipal filter $F$ measuring every set in $M$.
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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 $ZFC^-$,
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$|M|=\kappa$, $M^{\lt\kappa}\subset M$ and $M$ satisfies $\text{ZFC}^-$,
 
the theory ZFC without the power set axiom (and using collection and separation rather than merely replacement).  
 
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
$H_{\kappa^+}\satisfies\ZFC^-$, and further, if
+
$H_{\kappa^+}\models\text{ZFC}^-$, 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
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The partition property $\kappa\to(\lambda)^n_\gamma$
 
The partition property $\kappa\to(\lambda)^n_\gamma$
 
asserts that for every function $F:[\lambda]^n\to\gamma$
 
asserts that for every function $F:[\lambda]^n\to\gamma$
there is $H\of\lambda$ with $|H|=\kappa$ such that
+
there is $H\subset\lambda$ with $|H|=\kappa$ such that
$F\restrict [H]^n$ is constant. If one thinks of $F$ as
+
$F\upharpoonright[H]^n$ is constant. If one thinks of $F$ as
 
coloring the $n$-tuples, the partition property asserts the
 
coloring the $n$-tuples, the partition property asserts the
 
existence of a ""monochromatic"" set $H$, since all
 
existence of a ""monochromatic"" set $H$, since all

Revision as of 08:53, 27 December 2011

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.

  • (weak compactness property) 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 property) 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 property) 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 property) 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 $\kappa\to(\kappa)^2_2$.

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 $\vec x=\langle x_\alpha\st\alpha<\delta\rangle$ 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 $F:[\lambda]^n\to\gamma$ there is $H\subset\lambda$ with $|H|=\kappa$ 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 $L$.

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.
  • Weakly compact cardinals are preserved by the canonical forcing of the GCH, by fast function forcing and many other forcing notions.
  • If $\kappa$ is weakly compact, there is a forcing extension in which $\kappa$ remains weakly compact and $2^\kappa\gt\kappa$.
  • 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. (Kunen citation needed)


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.