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 κ be a weakly compact cardinal and let (T,<T) be a κ tree with all levels smaller than κ. We define a theory in Lκ,κ with for each xT, a constant cx, and a single unary relationPlanetmathPlanetmath B. Then our theory Δ consists of the sentencesMathworldPlanetmath:

  • ¬[B(cx)B(cy)] for every incompatible x,yT

  • xT(α)B(cx) for each α<κ

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 |Δ|=κ, since there are κ elements in T, and hence fewer than κκ=κ sentences in the first group, and of course there are κ levels and therefore κ sentences in the second group.

Now consider any ΣΔ with |Σ|<κ. Fewer than κ sentences of the second group are included, so the set of x for which the corresponding cx must all appear in T(α) for some α<κ. But since T has branches of arbitrary height, T(α)Σ.

Since κ is weakly compact, it follows that Δ also has a model, and that model obviously has a set of cx such that B(cx) 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
Canonical name WeaklyCompactCardinalsAndTheTreeProperty
Date of creation 2013-03-22 12:52:51
Last modified on 2013-03-22 12:52:51
Owner Henry (455)
Last modified by Henry (455)
Numerical id 7
Author Henry (455)
Entry type Result
Classification msc 03E10
Related topic TreeProperty
Related topic Aronszajn