[Sh:430.1-2] Framework for proof

January 26, 2011 at 18:46 | Posted in Cardinal Arithmetic, [Sh:430] | Leave a comment

Let us assume that {\mu} is a singular cardinal with {\rm{pp}(\mu)=\mu^+}. We will prove that there is a family {\mathcal{P}\subseteq [\mu]^{<\mu}} of cardinality {\mu^+} such that every member of {[\mu]^{\rm{cf}(\mu)}} is a subset of some member of {\mathcal{P}}.

Fix a sufficiently large regular cardinal {\chi}; we will be working with elementary submodels of the structure {\langle H(\chi),\in, <_\chi\rangle} where {<_\chi} is some appropriate well-ordering of {H(\chi)} used to give us definable Skolem functions.

Let {\mathfrak{M}=\langle M_\alpha:\alpha<\mu^+\rangle} be a {\mu^+}-approximating sequence, that is, {\mathfrak{M}} is a {\in}-increasing and continuous sequence of elementary submodels of {H(\chi)} such that for each {\alpha<\mu^+},

  • {\mu\in M_0},
  • {M_\alpha} has cardinality {\mu},
  • {M_\alpha\cap\mu^+} is an initial segment of {\mu^+} (so {\mu+1\subseteq M_\alpha\cap\mu^+}), and
  • {\langle M_\beta:\beta<\alpha\rangle\in M_{\alpha+1}}.

Let {M^*} denote {\bigcup_{\alpha<\mu^+} M_\alpha}. Then {M^*} is an elementary submodel of {H(\chi)} of cardinality {\mu^+} containing {\mu^+} as both an element and subset. Define

\displaystyle  \mathcal{P}=M^*\cap [\mu]^{<\mu}. \ \ \ \ \ (1)

Clearly {\mathcal{P}\subseteq [\mu]^{<\mu}} and {|\mathcal{P}|=\mu^+}, so we need only verify that for any {a\in [\mu]^{\rm{cf}(\mu)}}, there is a {b\in \mathcal{P}} with {a\subseteq b}.

In broad terms, this will be done via an “{I[\lambda]} argument” with {\lambda=\mu^+}, but we’ll fill in the details in further posts.

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