Tuesday, July 01, 2014

Club obedient sequences I

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

We will follow Abraham-Magidor for a little bit.

Assume the following:

  • $A$ is a progressive set of regular cardinals
  • $\kappa$ is a regular cardinal with $|A|<\kappa<\min(A)$,
  • $\langle f_\xi:\xi<\delta\rangle$ is a sequence of functions in $\prod A$ with $\cf(\delta)=\kappa$.

Given a club $E\subseteq\delta$ of order-type $\kappa$, we define

$h_E = \sup\{f_\xi:\xi\in E\}$.


Note the following:
  • $h_E$ is in $\prod A$ as $\kappa<\min(A)$,
  • if $C\subseteq E$ are clubs in $\delta$, then $h_C\leq h_E$ in $\prod A$, and
  • there is a club $C\subseteq \delta$ of order-type $\kappa$ such that $h_C\leq h_E$ for any club $E\subseteq \delta$.
The last point follows by an easy proof by contradiction. If there is no such club $C$, then we can build a decreasing sequence $\langle E_\alpha:\alpha<|A|^+\rangle$ of clubs in $\delta$ such that
  • $E_\alpha=\bigcap_{\beta<\alpha}E_\beta$ if $\alpha$ is a limit ordinal, and
  • $h_{E_{\alpha}}\nleq h_{E_{\alpha+1}}$
For each $\alpha<|A|^+$, we can find $a_\alpha\in A$ such that

$h_{E_{\alpha+1}}(a_\alpha)< h_{E_\alpha}(a_\alpha)$

and then stabilize to find a single $a\in A$ such that 

$h_{E_{\alpha+1}}(a)< h_{E_{\alpha}}(a)$

for unboundedly many indices $\alpha$.

Since the sequence of clubs $E_\alpha$ is decreasing, we get an infinite decreasing sequence of ordinals. Contradiction.


Now suppose  $J$ is an ideal over $A$, and the sequence $\langle f_\xi:\xi<\delta\rangle$ is $<_J$-increasing.  In this case, if $C$ is the "minimal" club as above, then 

$\xi\in C\Longrightarrow f_\xi\leq f_C$,

and hence $f_C$ is a $<_J$-upper bound for the sequence $\langle f_\xi:\xi<\delta\rangle$.

The function $f_C$ is called the minimal club-obedient bound of $\langle f_\xi:\xi<\delta\rangle$





No comments: