## [Sh:430.1-2] Framework for proof

Let us assume that ${\mu}$ is a singular cardinal with ${\rm{pp}(\mu)=\mu^+}$. We will prove that there is a family ${\mathcal{P}\subseteq [\mu]^{<\mu}}$ of cardinality ${\mu^+}$ such that every member of ${[\mu]^{\rm{cf}(\mu)}}$ is a subset of some member of ${\mathcal{P}}$.

Fix a sufficiently large regular cardinal ${\chi}$; we will be working with elementary submodels of the structure ${\langle H(\chi),\in, <_\chi\rangle}$ where ${<_\chi}$ is some appropriate well-ordering of ${H(\chi)}$ used to give us definable Skolem functions.

Let ${\mathfrak{M}=\langle M_\alpha:\alpha<\mu^+\rangle}$ be a ${\mu^+}$-approximating sequence, that is, ${\mathfrak{M}}$ is a ${\in}$-increasing and continuous sequence of elementary submodels of ${H(\chi)}$ such that for each ${\alpha<\mu^+}$,

• ${\mu\in M_0}$,
• ${M_\alpha}$ has cardinality ${\mu}$,
• ${M_\alpha\cap\mu^+}$ is an initial segment of ${\mu^+}$ (so ${\mu+1\subseteq M_\alpha\cap\mu^+}$), and
• ${\langle M_\beta:\beta<\alpha\rangle\in M_{\alpha+1}}$.

Let ${M^*}$ denote ${\bigcup_{\alpha<\mu^+} M_\alpha}$. Then ${M^*}$ is an elementary submodel of ${H(\chi)}$ of cardinality ${\mu^+}$ containing ${\mu^+}$ as both an element and subset. Define

$\displaystyle \mathcal{P}=M^*\cap [\mu]^{<\mu}. \ \ \ \ \ (1)$

Clearly ${\mathcal{P}\subseteq [\mu]^{<\mu}}$ and ${|\mathcal{P}|=\mu^+}$, so we need only verify that for any ${a\in [\mu]^{\rm{cf}(\mu)}}$, there is a ${b\in \mathcal{P}}$ with ${a\subseteq b}$.

In broad terms, this will be done via an “${I[\lambda]}$ argument” with ${\lambda=\mu^+}$, but we’ll fill in the details in further posts.