## Claim 3.3A, or, More Elementary Submodels (iii)

May 24, 2011 at 10:47 | Posted in Uncategorized | Leave a commentHere we conclude a trio of posts dealing with elementary submodels and Skolem hulls. My goal in this post is to prove a slight strengthening of Observation 3.3A on page 379 of Cardinal Arithmetic, a result which will fill a critical role in the proof of the main theorem. Notice that what we are doing is just putting the two previous posts together into a single result.

Lemma 1Suppose is a singular cardinal, let be as usual, and let be an elementary submodel such that . If there are objects and such that

- is a cardinal below ,
- , and
- for every ,
then

*Proof:*

We see that the assumptions of Proposition 2 from yesterday’s post are satisfied if we define to be , and therefore the assumptions of Theorem 1 from the May 18, 2011 post are satisfied. If is not a subset of , then there is a regular such that

But then there is an such that

which is a contradiction as both and are in .

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