Wednesday, July 02, 2014

Technical Lemma

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

In this post, I want to fulfill a promise by stating a lemma that is both a more technical version of Lemma 5.8 of Abraham-Magidor and a simplified version of something from Cardinal Arithmetic.   We are modifying the very nice work of Abraham and Magidor so that we can push through the result of Shelah that we will need.   The proof of the lemma will occupy subsequent posts.

Main Technical Lemma
Assume the following:

  •  $A$ is a progressive set of regular cardinals satisfying $|A|^+<\min(A)$
  •   $\kappa$ is a regular cardinal satisfying $|A|<\kappa<\min(A)$
  • $N$ is a $\kappa$-presentable elementary submodel of $H(\chi)$ for some sufficiently large regular $\chi$
  • $\langle N_\alpha:\alpha<\kappa\rangle$ witnesses the $\kappa$-presentability of $N$
  • $\lambda=\max\pcf(A)$
  • $\bar{f}=\langle f_\xi:\xi<\lambda\rangle$ is a minimally obedient universal sequence for $\lambda$
  • $\{\lambda, \bar{f}, A\}\in N_0$
Furthermore, for $\alpha<\beta<\kappa$ let us define

$b(\alpha,\beta)=\{a\in A: \sup(N_\alpha\cap a)\leq f_{\sup(N_\beta\cap\lambda)}(a)\}.$

Then there is a club $C\subseteq\kappa$ such that for any $\alpha<\beta$ in $C$,

  1. $b(\alpha,\beta)$ is a pcf-generator for $\lambda$, and
  2. $\ch_N\restr b(\alpha,\beta)=f_{\sup(N\cap\lambda)}\restr b(\alpha,\beta)$


This is a souped up version of Lemma 5.8 from Abraham-Magidor, and it is also a close relative of Claim 1.2 on page 315 of Cardinal Arithmetic.

Note for future reference that each set $b(\alpha,\beta)$ is in $N$, as they are definable from parameters available in $N$.


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