Wednesday, July 02, 2014

Club Obedient Sequences II

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

Continuing with some preliminary material from Abraham-Magidor:

Definition
Suppose that $\lambda\in\pcf(A)$.  A $<_{J_<\lambda}$-increasing sequence $\bar{f}=\langle f_\xi:\xi<\lambda\rangle$ in $\prod A$ is a universal sequence for $\lambda$ if $\bar{f}$ is cofinal in $\prod A/D$ for every ultrafilter $D$ on $A$ satisfying $\lambda = \cf(\prod A/D)$.


We discussed universal sequences on our old blog a bit during the series of posts on existence of pcf generators.  The important fact is that they exist whenever $A$ is a progressive set of regular cardinals and $\lambda\in\pcf(A)$.  [Theorem 4.2 on page 1180 of Abraham-Magidor.]

Definition
Suppose that $\lambda\in\pcf(A)$ and $\bar{f}=\langle f_\xi:\xi<\lambda\rangle$ is a universal sequence for $\lambda$.  Suppose that $\kappa$ is a regular cardinal with $|A|<\kappa<\min(A)$ (so we need $|A|^+<\min(A)$).  We say that $f$ is minimally obedient (at cofinality $\kappa$) if for every $\delta<\kappa$ of cofinality $\kappa$, $f_\delta$ is the minimal club-obedient bound of $\langle f_\xi:\xi<\delta\rangle$.
We say $\bar{f}$ is minimally obedient if $|A|^+<\min(A)$ and $\bar{f}$ is minimally obedient at cofinality $\kappa$ for every regular $\kappa$ satisfying $|A|<\kappa<\min(A)$.


If $|A|^+<\min(A)$ and $\lambda\in\pcf(A)$, then minimally obedient universal sequences exist, as one need only modify a given universal sequence in a straightforward way.  The details are spelled out on page 1192 of Abraham-Magidor.

One should view minimal obedience as a form of continuity for the sequence $\langle f_\xi:\xi:<\lambda\rangle$.  In fact, Shelah's name for this condition is $^b$-continuity.

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