## The Elementary Submodel argument

In this post, we examine another ingredient we’ll need in the course of the proof: the relationship between elementary submodels of ${\mathfrak{A}}$ 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 1 Let ${N\subseteq M}$ be elementary submodels of ${\mathfrak{A}}$, and assume

• ${N\cap\mu}$ is unbounded in ${\mu}$, and
• for any ${\theta\in N\cap\mu\cap{\sf Reg}}$ we have ${\sup(M\cap\theta)=\sup(N\cap\theta)}$.

Then ${M\cap\mu= N\cap\mu}$.

Proof: Suppose not. Define

$\displaystyle \alpha=\min(M\cap\mu\setminus N), \ \ \ \ \ (1)$

and

$\displaystyle \beta=\min(N\cap\mu\setminus\alpha). \ \ \ \ \ (2)$

Notice that ${\beta}$ exists by our assumptions, which also imply that ${\beta}$ cannot be a regular cardinal, as ${\beta\in N\cap\mu}$ and

$\displaystyle \sup(N\cap\beta)\leq\alpha<\alpha+1\leq\sup(M\cap\beta). \ \ \ \ \ (3)$

The proof will be complete if we manage to prove to the contrary that ${\beta}$ MUST be a regular cardinal.

Clearly ${\beta}$ must be a limit ordinal because ${N}$ is closed under the predecessor function. If ${\beta}$ is not a regular cardinal, then ${\gamma={\rm cf}(\beta)}$ is a regular cardinal in ${N}$ (and ${M}$), and therefore ${N}$ contains a function ${f}$ mapping ${\gamma}$ onto a cofinal subset of ${\beta}$.

Since ${\gamma\in N}$, it follows that ${\gamma<\alpha}$ and therefore ${M\cap\gamma=N\cap\gamma}$. In the model ${M}$, there must be an ${\epsilon<\gamma}$ with ${\alpha. But since ${\epsilon}$ must be in ${N}$ as well, it follows that

$\displaystyle \alpha

contradicting the definition of ${\beta}$. $\Box$

How is this going to be used? Let us agree for a moment to call models ${M}$ and ${N}$ as above “${\mu}$-twins”. In the context of the main project we have an elementary submodel ${M^*}$ obtained as the union of a ${\mu^+}$-approximating sequence ${\langle M_\alpha:\alpha<\mu^+\rangle}$. The goal is to show that any ${a\in [\mu]^{{\rm cf}(\mu)}}$ is a subset of some element of ${M^*\cap [\mu]^{<\mu}}$. We do this by producing an elementary submodel ${N}$ satisfying

• ${|N|<\mu}$,
• ${N\in M^*}$ (so ${N\cap\mu\in M^*\cap[\mu]^{<\mu}}$), and
• ${N}$ and ${M:={\rm Sk}_\mathfrak{A}(N\cup a)}$ are ${\mu}$-twins.

The conclusion of the lemma tells us that

$\displaystyle a\subseteq M\cap\mu = N\cap \mu\in M^*\cap [\mu]^{<\mu} \ \ \ \ \ (5)$

which is exactly what we need.