Tuesday, July 01, 2014

Characteristic functions

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

Definition 1
Let $N$ be an elementary substructure of $H(\chi)$ for some sufficiently large regular $\chi$.
The characteristic function of $N$ is defined by

$\ch_N(\theta)=\sup(N\cap\theta)$

whenever $\theta$ is a regular cardinal.

We've seen in previous blog posts how important the characteristic functions of models are when connecting pcf theory to classical cardinal arithmetic.

We're going to be looking at obtaining refinements of the following cornerstone result of pcf theory:



Suppose $A$ is a progressive set of regular cardinals and let $\lambda=\max\pcf(A)$.   If

$$|A|<\kappa=\cf(\kappa)<\min(A)$$

then there is a family $F\subseteq\prod A$ of size $\lambda$ such that  $\ch_N\restr A\in F$ whenever $N$ is a $\kappa$-presentable elementary substructure of $H(\chi)$.


[This is essentially Lemma 3.4 on page 63 of Cardinal Arithmetic, but it is also an immediate consequence of Corollary 5.9 of Abraham-Magidor]


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