## More elementary submodels

May 18, 2011 at 14:09 | Posted in Uncategorized | Leave a commentThis post should be considered a tentative one. What I’m trying to do is isolate a few facts that can be used to prove the various “elementary submodel results” scattered throughout the last chapter of Cardinal Arithmetic. I’ve already gone through several iterations of this, but the following result looks “stable” so I thought I’d put it up here.

Theorem 1Suppose is a singular cardinal, and let be a sufficiently large regular cardinal. Suppose and are elementary submodels of such that

- , and
- .
If , then there is a regular cardinal such that

*Proof:* Assuming , we define

and

Note the following facts:

Note that cannot be a successor ordinal: if it were, then its predecessor would witness the failure of (5). Since , we conclude

and so our proof will be complete if we can show that is in fact a regular cardinal.

Our proof hinges on the fact that is an element of as well. Why is this? By our assumptions, there is a set such that

Since , we conclude

and hence is definable from parameters available inside the model .

Thus, we have . If is not a regular cardinal, then

and therefore

Given our definition of , we see

We can find such that maps onto a cofinal subset of . Since , it follows that there is an such that

But we know , and by (11), and therefore

contradicting (5)

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