weakly compact cardinals and the tree property

A cardinal is weakly compact if and only if it is inaccessible and has the tree property.

Weak compactness implies tree property

Let $\kappa$ be a weakly compact cardinal and let $(T,<_{T})$ be a $\kappa$ tree with all levels smaller than $\kappa$. We define a theory in $L_{\kappa,\kappa}$ with for each $x\in T$, a constant $c_{x}$, and a single unary relation $B$. Then our theory $\Delta$ consists of the sentences:

• $\neg\left[B(c_{x})\wedge B(c_{y})\right]$ for every incompatible $x,y\in T$

• $\bigvee_{x\in T(\alpha)}B(c_{x})$ for each $\alpha<\kappa$

It should be clear that $B$ represents membership in a cofinal branch, since the first class of sentences asserts that no incompatible elements are both in $B$ while the second class states that the branch intersects every level.

Clearly $|\Delta|=\kappa$, since there are $\kappa$ elements in $T$, and hence fewer than $\kappa\cdot\kappa=\kappa$ sentences in the first group, and of course there are $\kappa$ levels and therefore $\kappa$ sentences in the second group.

Now consider any $\Sigma\subseteq\Delta$ with $|\Sigma|<\kappa$. Fewer than $\kappa$ sentences of the second group are included, so the set of $x$ for which the corresponding $c_{x}$ must all appear in $T(\alpha)$ for some $\alpha<\kappa$. But since $T$ has branches of arbitrary height, $T(\alpha)\models\Sigma$.

Since $\kappa$ is weakly compact, it follows that $\Delta$ also has a model, and that model obviously has a set of $c_{x}$ such that $B(c_{x})$ whose corresponding elements of $T$ intersect every level and are compatible, therefore forming a cofinal branch of $T$, proving that $T$ is not Aronszajn.

Title weakly compact cardinals and the tree property WeaklyCompactCardinalsAndTheTreeProperty 2013-03-22 12:52:51 2013-03-22 12:52:51 Henry (455) Henry (455) 7 Henry (455) Result msc 03E10 TreeProperty Aronszajn