\DeclareMathOperator{\pp}{pp}

\DeclareMathOperator{\tcf}{tcf}

\def\pcf{\rm{pcf}}

\DeclareMathOperator{\cov}{cov}

\def\cf{\rm{cf}}

\def\REG{\sf {REG}}

\def\restr{\upharpoonright}

\def\bd{\rm{bd}}

\def\subs{\subseteq}

\def\cof{\rm{cof}}

\def\ran{\rm{ran}}

\DeclareMathOperator{\ch}{Ch}

\DeclareMathOperator{\PP}{pp}

\DeclareMathOperator{\Sk}{Sk}

\)

We are going to be looking at Theorem 2.1 of [GiSh:1013] "Applications of pcf for mild large cardinals to elementary embeddings"

Assume $\kappa$ is weakly compact and $\mu>2^\kappa$ is singular of cofinality $\kappa$. If $\lambda$ is a regular cardinal satisfying $\mu<\lambda<\pp^+_{\Gamma(\kappa)}(\mu)$, then there is an increasing sequence $\langle\lambda_i:i<\kappa\rangle$ of regular cardinals converging to $\mu$ such that $\lambda=\tcf(\prod_{i<\kappa}\lambda_i/J^{\bd}_\kappa)$.

In this post, we will simply unpack what the theorem says.

The assumption $\mu<\lambda<\pp^+_{\Gamma(\kappa)}(\mu)$ means the following:

There is a set of regular cardinals $A\subseteq\mu$ and a $\kappa$-complete ideal $J$ on $A$ such that

*Annals of Pure and Applied Logic*164 (2013) 855--865:**Theorem (Gitik and Shelah)**Assume $\kappa$ is weakly compact and $\mu>2^\kappa$ is singular of cofinality $\kappa$. If $\lambda$ is a regular cardinal satisfying $\mu<\lambda<\pp^+_{\Gamma(\kappa)}(\mu)$, then there is an increasing sequence $\langle\lambda_i:i<\kappa\rangle$ of regular cardinals converging to $\mu$ such that $\lambda=\tcf(\prod_{i<\kappa}\lambda_i/J^{\bd}_\kappa)$.

In this post, we will simply unpack what the theorem says.

The assumption $\mu<\lambda<\pp^+_{\Gamma(\kappa)}(\mu)$ means the following:

There is a set of regular cardinals $A\subseteq\mu$ and a $\kappa$-complete ideal $J$ on $A$ such that

- $A$ is unbounded in $\mu$

- $|A|=\kappa$

- $J$ extends the ideal $J^{\bd}[A]$ of bounded subsets of $A$, and

- $\lambda=\tcf(\prod A/ J)$.

Comments:

1. $A$ need not have order-type $\kappa$, and $J$ can potentially be much larger than the ideal of bounded subsets of $A$.

2. Recall a poset has true cofinality $\lambda$ if it has a totally ordered cofinal subset of cardinality $\lambda$.

So our assumptions tell us $\lambda$ can be represented as $\tcf(\prod A/ J)$ for a certain sort of $A$ and $J$.

The conclusion of the theorem is that in fact $\lambda$ has a very nice representation as a true cofinality:

There is a set of regular cardinals $D\subseteq \mu$ (in fact, $D$ will be a subset of $\pcf(A)\cap\mu$) such that

- $D$ is unbounded in $\mu$ of order-type $\kappa$, and

- $\lambda=\tcf(\prod D/ J^{\bd}[D])$

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