More elementary submodels

May 18, 2011 at 14:09 | Posted in Uncategorized | Leave a comment

This 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 1 Suppose {\mu} is a singular cardinal, and let {\chi} be a sufficiently large regular cardinal. Suppose {M} and {N} are elementary submodels of {\langle H(\chi), \in, <_\chi\rangle} such that

  1. {\mu\in M\cap N}, and
  2. {(\forall\alpha\in N\cap\mu)(\exists A\in M)(\alpha\in A\wedge A\subseteq N)}.

If {M\cap\mu\nsubseteq N}, then there is a regular cardinal {\kappa\in M\cap N\cap\mu} such that

\displaystyle  \sup(N\cap\kappa)<\sup(M\cap\kappa). \ \ \ \ \ (1)

Proof: Assuming {M\cap\mu\nsubseteq N}, we define

\displaystyle  \gamma(*):=\min(M\cap\mu\setminus N), \ \ \ \ \ (2)

and

\displaystyle  \beta(*):=\min(N\cap{\sf On}\setminus\gamma(*)). \ \ \ \ \ (3)

Note the following facts:

\displaystyle  \gamma(*)<\beta(*)\leq\mu, \ \ \ \ \ (4)

and

\displaystyle   N\cap [\gamma(*),\beta(*))=\emptyset. \ \ \ \ \ (5)

Note that {\beta(*)} cannot be a successor ordinal: if it were, then its predecessor would witness the failure of (5). Since {\gamma(*)+1\in M}, we conclude

\displaystyle  \sup(N\cap\beta(*))\leq\gamma(*)<\gamma(*)+1\leq\sup(M\cap\beta(*)), \ \ \ \ \ (6)

and so our proof will be complete if we can show that {\beta(*)} is in fact a regular cardinal.

Our proof hinges on the fact that {\beta(*)} is an element of {M} as well. Why is this? By our assumptions, there is a set {A\in M} such that

\displaystyle  \beta(*)\in A\text{ and }A\subseteq N. \ \ \ \ \ (7)

Since {N\cap [\gamma(*),\beta(*))=\emptyset}, we conclude

\displaystyle  \beta(*)=\min(A\setminus\gamma(*)), \ \ \ \ \ (8)

and hence {\beta(*)} is definable from parameters available inside the model {M}.

Thus, we have {\beta(*)\in M\cap N}. If {\beta(*)} is not a regular cardinal, then

\displaystyle  {\rm cf}(\beta(*))\in M\cap N\cap\beta(*), \ \ \ \ \ (9)

and therefore

\displaystyle  {\rm cf}(\beta(*))\leq\gamma(*). \ \ \ \ \ (10)

Given our definition of {\gamma(*)}, we see

\displaystyle   M\cap{\rm cf}(\beta(*))\subseteq N. \ \ \ \ \ (11)

We can find {f\in M\cap N} such that {f} maps {{\rm cf}(\beta(*))} onto a cofinal subset of {\beta(*)}. Since {\gamma(*)\in M}, it follows that there is an {\alpha\in M\cap {\rm cf}(\beta(*))} such that

\displaystyle  \gamma(*)<f(\alpha)<\beta(*). \ \ \ \ \ (12)

But we know {f\in N}, and {\alpha\in N} by (11), and therefore

\displaystyle  f(\alpha)\in N\cap [\gamma(*),\beta(*)), \ \ \ \ \ (13)

contradicting (5)

\Box

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