Tuesday, June 11, 2013

Final Inequality 1

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

Recall
  • $\cf(\lambda)<\theta=\cf(\theta)<\lambda$, 
  • $\lambda(1):=\cf_{<\theta}(\prod(\lambda\cap\REG), <_{J^\bd_\lambda})$, and
  • $\lambda(2)$ is the minimum cardinality of a family $\mathcal{P}\subseteq[\lambda(1)]^{<\lambda}$ such that for any $B\in[\lambda(1)]^\theta$ there is an $A\in\mathcal{P}$ with $A\cap B$ infinite.

We wish to prove $\cov(\lambda,\lambda,\theta,2)\leq\lambda(2)$.  This requires us to produce a family
$\mathcal{P}^*\subseteq[\lambda]^{<\lambda}$ such that
  • $|\mathcal{P}^*|\leq\lambda(2)$, and
  • for any $B\in [\lambda]^{<\theta}$ there is an $A\in\mathcal{P}^*$ with $B\subseteq A$.
We start by fixing a family $\mathcal{F}\subseteq \prod(\lambda\cap\REG)$ witnessing the definition of $\lambda(1)$, say $\mathcal{F}=\{f_\alpha:\alpha<\lambda(1)\},$ and let $\mathcal{P}\subseteq[\lambda]^{<\lambda}$ be as in the definition of $\lambda(2)$.





To find $\mathcal{P}^*$, let us assume $\chi$ is a regular cardinal much larger than any of the cardinals considered above, and let (as usual) $\mathfrak{A}$ denote the structure $\langle H(\chi),\in, <_\chi)$,where $<_\chi$ is some well-ordering of $H(\chi)$.  We will use $\Sk_\mathfrak{A}(B)$ to denote the Skolem hull of $B$ in the structure $\mathfrak{A}$.


Ad hoc definition:
Let us say that an elementary submodel $M$ of $\mathfrak{A}$ is acceptable if  
$M=\Sk_{\mathfrak{A}}(B\cup p\cup\mu+1)$
where
  • $B$ is an initial segment of some member of $\mathcal{P}$, 
  • $p=\{\theta,\lambda,\lambda(1),\lambda(2), \mathcal{F}, \mathcal{P},...\}$, and
  • $\mu<\lambda$
(We have been sloppy with $p$.  The intent is that $p$ encodes the parameters associated with the preceding discussion.)


If $M$ is an acceptable model, then $|M|<\lambda$.  Also note that there are at most $\lambda(2)=|\mathcal{P}|$ acceptable models.

We define $\mathcal{P}^*\subseteq[\lambda]^{<\lambda}$ to be all sets of the form $M\cap\lambda$ where $M$ is an acceptable model.

It remains to show that $\mathcal{P}^*$ has the required covering property.  We will do this in the next couple of posts.





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