## 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 $\langle f_\xi:\xi<\delta\rangle$