Ignore that last post…

September 21, 2011 at 14:25 | Posted in Uncategorized | Leave a comment

Actually, I think I lied to you in the last post, at least as far as our strategy is concerned. I’ll outline another route to the result that is closer to what Shelah does; it’s actually shorter than what I had in mind originally.

Recalling our situation, {\mu} is a singular cardinal, and {\mathfrak{A}=\langle H(\chi), \in, <_\chi\rangle} is as usual. We have also assumed {\aleph_0<{\rm cf}(\sigma)\leq\sigma\leq{\rm cf}(\mu)<\theta\leq\kappa<\mu}. Let us assume that {\mathcal{F}} is a family of functions as required in the definition of {\lambda_2}, i.e., a collection of functions that are “{\sigma}-cofinal in any subproduct of size {<\theta} constructed from {\prod({\sf Reg}\cap(\kappa,\mu)}” (see previous posts for a more precise definition, but things should be clear from the construction).

Given a finite subset {F} of {\mathcal{F}}, we define

\displaystyle  A_F:=\mu\cap{\rm Sk}_{\mathfrak{A}}(\kappa+1\cup\{\mu\}\cup F), \ \ \ \ \ (1)

and define

\displaystyle  \mathcal{P}=\{A_F:F\in [\mathcal{F}]^{<\aleph_0}\}. \ \ \ \ \ (2)

Note that each {A_F} is a subset of {\mu} of cardinality {\kappa}, so

\displaystyle  \mathcal{P}\subseteq [\mu]^\kappa. \ \ \ \ \ (3)

Also, it should be clear that

\displaystyle  |\mathcal{P}|\leq |[\mathcal{F}]^{<\aleph_0}|=|\mathcal{F}|, \ \ \ \ \ (4)

so we will be finished if we can prove that any {Y\in [\mu]^{<\theta}} can be covered by a union of fewer than {\sigma} sets drawn from {\mathcal{P}}.

How will we do this? Given {Y\in [\mu]^{<\theta}}, we will produce a family {H\subseteq \mathcal{F}} satisfying the following two properties:

\displaystyle  |H|<\sigma, \ \ \ \ \ (5)


\displaystyle  Y\subseteq\mu\cap{\rm Sk}_\mathfrak{A}(\kappa+1\cup\{\mu\}\cup H). \ \ \ \ \ (6)

This suffices, because

\displaystyle  \mu\cap{\rm Sk}_\mathfrak{A}(\kappa+1\cup\{\mu\}\cup H)=\bigcup\{A_F: F\in [H]^{<\aleph_0}\}, \ \ \ \ \ (7)

and since {|[H]^{<\aleph_0}|<\sigma}, we will have shown that {Y} can be covered by a union of fewer than {\sigma} sets from {\mathcal{P}}.

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