I’ve been meaning to write up this post for a week or so now, but I haven’t managed to get around to it until now. In this post, I want to briefly go over another cool result from “The Stationary Tower”. We define the Chang Model to be . Such a model was first considered by C. C. Chang, albeit his construction was a bit different. In his paper “Sets Constructible in “, Chang considered the result of going about the construction of in the language for an infinite regular cardinal , where we allow fewer than many instances of quantification and logical connectives. In the case of , the result is simply . In the case where , it turns out the the resulting model, denoted is precisely the model .
Lemma (Chang): satisfies ZF.
Lemma (Chang): satisfies its definition inside itself, and every element of is $\Sigma_2$ definable inside the model from a countable sequence of ordinals.
At this point, I would like to present Kunen’s result that, if there exist uncountably many measurable cardinals, then AC fails in the Chang model. We will follow the proof presented in “The Stationary Tower”. We will need the following technical lemma (which I will be black-boxing for the purpose of this post). Unfortunately, I don’t know who to attribute this to.
Lemma: Given an ordinal , let denote the class of all cardinals such that there exists a countably complete nonprincipal ultrafilter of completeness such that , where is the elementary embedding induced by . Then for all ordinals , is finite.
Theorem (Kunen): Assume that there are uncountably many measurable cardinals, then the Axiom of Choice fails in .
Proof: Let denote the first -many measurable cardinals. For each , let be an ultrafilter witnessing measurability of , and let be the induced elementary embedding. Let , and note that for each . To see this, simply note that each is a fixed point of every other for since it is strongly inaccessible. Thus, is the supremum of fewer than many fixed points of any given , and thus .
Next, note that . We will show that there is no well-ordering of in . Suppose, by way of contradiction, that there is such a well-ordering, in . Then, is definable from a countable set of ordinals in as a parameter. We may fix a regular cardinal such that . By making sufficiently large, and possibly adding to , we may assume that is definable in from . In particular, we may fix a formula such that
Let enumerate . For each , if and only if for some . But, by the above lemma, the collection is finite for each . Thus, there are only countably many such that moves . In particular, there is then some such that, for each , .
We claim that then for every . To see this, by applying , we have that
Furthermore, we know that . Next, since , it follows that . Thus
From this, it follows that .
Now, let , and note that since is a well-ordering of , there is an ordinal such that . Again, we can apply the above lemma to see that there are only finitely many members of with associated elementary embeddings that move . So, fix such that, for each , . Let . We then have the following:
- , so .
But, this gives us a contradiction, as is the coordinate of , and the critical point of , which means that , and therefore that .