Wednesday, July 20, 2016

On a theorem of Gitik and Shelah II: Blanket Assumptions


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

Assumptions for the next few posts are:


  • $\kappa$ weakly compact
  • $\mu>2^\kappa$ is singular of cofinality $\kappa$
  • $A\subseteq\mu$ is a set of regular cardinals
  • $A$ is unbounded in $\mu$ of cardinality $\kappa$
  • $J$ is a $\kappa$-complete ideal on $A$ extending the bounded ideal
  • $\lambda=\tcf(\prod A/ J)$
Since $J$ extends the bounded ideal, we may as well assume

  • $(2^\kappa)^+<\min(A)$ 
Of course this implies  $|A|<\min(A)$, and so without loss of generality $A$ is progressive.

Let $D$ be an ultrafilter on $A$ disjoint to $J$.  Then $\cf(\prod A/D)=\lambda$ and hence $\lambda\in \pcf(A)$.  Furthermore, if $B_\lambda[A]$ is a pcf generator for $\lambda$ then $B_\lambda[A]$ must be in $D$ and therefore unbounded in $A$.

It follows that $\tcf(\prod B_\lambda[A]/J)=\lambda$ as well, and so we may assume (by passing to $B_\lambda[A]$ if necessary) that

  • $\lambda=\max\pcf(A)$

Our assumption that $2^\kappa<\min(A)$ tells us that

(1)  $\pcf(\pcf(A))=\pcf(A)$, and

(2)  $|\pcf(A)|\leq 2^\kappa<\min(A)=\min(\pcf(A))$


and so $\pcf(A)$ has a transitive set of generators, that is, a sequence $\langle B_\theta:\theta\in\pcf(A)\rangle$ such that

  • $B_\theta\subseteq \pcf(A)$
  • $B_\theta$ is a generator for $\theta$ in $\pcf(\pcf(A))=\pcf(A)$, and
  • $\tau\in B_\theta\Longrightarrow B_\tau\subseteq B_\theta$.




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