## The main proof, part 1.

General results on the approachability ideal ${I[\mu^+]}$ tell us there is a sequence

$\displaystyle \bar{C}=\langle C_\alpha:\alpha<\mu^+\rangle \ \ \ \ \ (1)$

such that

• ${C_\alpha}$ is a closed (not necessarily unbounded!) subset of ${\alpha}$,
• ${C_\alpha}$ has order-type at most ${\kappa^+}$, where ${\kappa={\rm cf}(\mu)}$,
• ${\beta\in{\rm nacc}(C_\alpha)\Longrightarrow C_\beta= C_\alpha\cap\beta}$, and
• the set of ${\delta<\mu^+}$ of cofinality ${\kappa^+}$ with ${\delta=\sup(C_\delta)}$ is stationary.

The above is a result from [Sh:420]; it says that there is a stationary subset of ${S^{\mu^+}_{\kappa^+}}$ in the ideal ${I[\mu^+]}$. The details of this are worked out in my Handbook article — see Theorem 3.18 and Theorem 3.7.

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Without loss of generality, the sequence ${\bar{C}}$ lies in the model ${M_0}$, hence in each ${M_\alpha}$ and ${M^*}$ as well. Let ${\langle \mu_i:i<\kappa\rangle}$ be the ${<_\chi}$-least continuous and increasing sequence of cardinals cofinal in ${\mu}$ satisfying ${\kappa<\mu_0}$; note this means that the sequence is going to be in every elementary submodel of the structure ${\mathfrak{A}}$.

Moving back to the main proof, let us fix ${a\in [\mu^+]^{{\rm cf }\mu}}$; recall that we want to show ${a}$ is a subset of some element ${M^*\cap [\mu]^{<\mu}}$, and the preceding post gives us our strategy.

For each ${\alpha<\mu^+}$ and ${i<\kappa}$, we define

$\displaystyle \mathcal{N}(\alpha,i)={\rm Sk}_{\mathfrak{A}}(\mu_i\cup \{f_\xi:\xi\in C_\alpha\}), \ \ \ \ \ (2)$

and

$\displaystyle \mathcal{N}^+(\alpha, i)={\rm Sk}_{\mathfrak{A}}(\mu_i\cup \{f_\xi:\xi\in C_\alpha\}\cup a). \ \ \ \ \ (3)$

We note the following:

• ${\mathcal{N}(\alpha,i)\subseteq\mathcal{N}^+(\alpha, i)\subseteq M_{\alpha+1}\subseteq M^*}$,
• ${\mathcal{N}(\alpha, i)\in M_{\alpha+1}}$, and
• both ${\mathcal{N}(\alpha,i)}$ and ${\mathcal{N}^+(\alpha,i)}$ have cardinality ${\mu_i}$.

Next, we define

$\displaystyle \mathfrak{a}_{\alpha, i}=\mathcal{N}(\alpha,i)\cap\mu\cap{\sf Reg}\setminus{\mu_i^{++}}, \ \ \ \ \ (4)$

and let ${g_{\alpha, i}}$ be the characteristic function of ${\mathcal{N}^+(\alpha, i)}$ (note the “+”!) in ${\mathfrak{a}_{\alpha, i}}$, i.e., the domain of ${g_{\alpha, i}}$ is ${\mathfrak{a}_{\alpha, i}}$, and for ${\theta\in\mathfrak{a}_{\alpha, i}}$, we have

$\displaystyle g_{\alpha, i}(\theta)=\sup(\mathcal{N}^+(\alpha,i)\cap\theta)<\theta. \ \ \ \ \ (5)$

The function ${g_{\alpha, i}}$ lies in ${\prod\mathfrak{a}_{\alpha, i}}$, but there is no reason to believe it is an element of ${M^*}$. Note, however, that ${\mathfrak{a}_{\alpha, i}}$ IS an element of ${M^*}$. It is also a progressive set of regular cardinals cofinal in ${\mu}$, and therefore by earlier work, we know

$\displaystyle \langle f_\beta\upharpoonright\mathfrak{a}_{\alpha, i}:\beta<\mu^+\rangle \ \ \ \ \ (6)$

is a scale in ${\mathfrak{a}_{\alpha, i}}$ modulo the ideal of bounded subsets of ${\mathfrak{a}_{\alpha, i}}$.

Since ${\prod(\mu\cap{\sf Reg})}$ is ${\mu^+}$-directed modulo the ideal of bounded subsets of ${\mu}$, it follows that for each ${\alpha<\mu^+}$, there exists a ${\gamma(\alpha)<\mu^+}$ such that

$\displaystyle (\forall i<\kappa)\left(g_{\alpha, i}< f_{\gamma(\alpha)}\upharpoonright\mathfrak{a}_{\alpha, i}\mod J^{\rm bd}[\mathfrak{a}_{\alpha, i}]\right) \ \ \ \ \ (7)$

The function mapping ${\alpha}$ to ${\gamma(\alpha)}$ need not be in ${M^*}$. Nevertheless, the set of ${\delta<\mu^+}$ is closed unbounded in ${\mu^+}$, and so we can find a ${\delta}$ of cofinality ${\kappa^+}$ such that

• ${\delta}$ is closed under the map ${\alpha\mapsto\gamma(\alpha)}$, and
• ${C_\delta}$ is closed unbounded in ${\delta}$ (of order-type ${\kappa^+}$).

We’ll end this post on this cliff-hanger…