weakly compact cardinals and the tree property
Weak compactness implies tree property
Let be a weakly compact cardinal and let be a tree with all levels smaller than . We define a theory in with for each , a constant , and a single unary relation . Then our theory consists of the sentences:
for every incompatible
It should be clear that represents membership in a cofinal branch, since the first class of sentences asserts that no incompatible elements are both in while the second class states that the branch intersects every level.
Clearly , since there are elements in , and hence fewer than sentences in the first group, and of course there are levels and therefore sentences in the second group.
Now consider any with . Fewer than sentences of the second group are included, so the set of for which the corresponding must all appear in for some . But since has branches of arbitrary height, .
Since is weakly compact, it follows that also has a model, and that model obviously has a set of such that whose corresponding elements of intersect every level and are compatible, therefore forming a cofinal branch of , proving that is not Aronszajn.
|Title||weakly compact cardinals and the tree property|
|Date of creation||2013-03-22 12:52:51|
|Last modified on||2013-03-22 12:52:51|
|Last modified by||Henry (455)|