## The Elementary Submodel argument

February 12, 2011 at 12:07 | Posted in Uncategorized | Leave a commentIn this post, we examine another ingredient we’ll need in the course of the proof: the relationship between elementary submodels of and their “characteristic functions”. There are many variants of this lemma scattered throughout the book and allied literature, but we’ll just worry about what is needed for our current project. Remember that our conventions regarding elementary submodels were laid out in an earlier post.

Lemma 1Let be elementary submodels of , and assume

- is unbounded in , and
- for any we have .

Then .

*Proof:* Suppose not. Define

and

Notice that exists by our assumptions, which also imply that cannot be a regular cardinal, as and

The proof will be complete if we manage to prove to the contrary that MUST be a regular cardinal.

Clearly must be a limit ordinal because is closed under the predecessor function. If is not a regular cardinal, then is a regular cardinal in (and ), and therefore contains a function mapping onto a cofinal subset of .

Since , it follows that and therefore . In the model , there must be an with . But since must be in as well, it follows that

contradicting the definition of .

How is this going to be used? Let us agree for a moment to call models and as above “-twins”. In the context of the main project we have an elementary submodel obtained as the union of a -approximating sequence . The goal is to show that any is a subset of some element of . We do this by producing an elementary submodel satisfying

- ,
- (so ), and
- and are -twins.

The conclusion of the lemma tells us that

which is exactly what we need.

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