Friday, July 01, 2016

On a theorem of Gitik and Shelah I

\(
\DeclareMathOperator{\pp}{pp}
\DeclareMathOperator{\tcf}{tcf}
\def\pcf{\rm{pcf}}
\DeclareMathOperator{\cov}{cov}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\DeclareMathOperator{\ch}{Ch}
\DeclareMathOperator{\PP}{pp}
\DeclareMathOperator{\Sk}{Sk}
\)

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