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

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)

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