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

May 24, 2011 at 10:47 | Posted in Uncategorized | Leave a comment

Here 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 1 Suppose {\mu} is a singular cardinal, let {\mathfrak{A}} be as usual, and let {M} be an elementary submodel such that {\mu\in M}. If there are objects {\eta} and {H} such that

  • {\eta} is a cardinal below {\mu},
  • {\eta=\sup(M\cap\eta)}
  • {H\subseteq M\cap\prod((\eta,\mu)\cap{\sf Reg})}, and
  • for every {\kappa\in M\cap\prod ((\eta,\mu)\cap{\sf Reg})},

    \displaystyle  \sup\{f(\kappa):f\in H\}=\sup(M\cap\kappa), \ \ \ \ \ (1)

    then

    \displaystyle  M\cap\mu\subseteq N:={\rm Sk}_{\mathfrak{A}}(\eta\cup H\cup\{\mu\}). \ \ \ \ \ (2)

Proof:

We see that the assumptions of Proposition 2 from yesterday’s post are satisfied if we define {p} to be {H\cup\{\mu\}}, and therefore the assumptions of Theorem 1 from the May 18, 2011 post are satisfied. If {M\cap\mu} is not a subset of {N}, then there is a regular {\kappa\in M\cap N\cap\mu} such that

\displaystyle  \sup(N\cap\kappa)<\sup(M\cap\kappa). \ \ \ \ \ (3)

But then there is an {f\in H} such that

\displaystyle  \sup(N\cap\kappa)<f(\kappa)<\kappa, \ \ \ \ \ (4)

which is a contradiction as both {f} and {\kappa} are in {N}. \Box

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