Chugging along

September 12, 2011 at 16:35 | Posted in Uncategorized | Leave a comment

Keeping the notation the same as in the last post, we begin our quest to prove {\lambda_1\leq\lambda_2}. The proof is not surprising. Start by letting

\displaystyle  F\subseteq\prod({\sf Reg}\cap(\kappa,\mu)) \ \ \ \ \ (1)

be a family of cardinality {\lambda_2} “covering all small products”. We let {\chi} be a sufficiently large regular cardinal, and let {M} be an elementary submodel of {H(\chi)} containing the relevant parameters, with

\displaystyle  F\subseteq M. \ \ \ \ \ (2)

Our plan is to prove that

\displaystyle  \mathcal{P}:=M\cap [\mu]^\kappa \ \ \ \ \ (3)

is a family satisfying the needed “covering requirements”. Said another way, we will prove that any element of {[\mu]^{<\theta}} can be covered by a union of fewer than {\sigma} sets from {\mathcal{P}}. Since

\displaystyle  |\mathcal{P}|=|M|=\lambda_2, \ \ \ \ \ (4)

this will establish {\lambda_1\leq \lambda_2}.

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