Strategy and assumptions

May 10, 2011 at 16:21 | Posted in Uncategorized | Leave a comment

In this post, we’ll lay out the notation and strategy for the proof.

Let us assume {\sigma}, {\theta}, {\mu}, and {\lambda} are cardinals with

\displaystyle  \aleph_0<\sigma\leq{\rm cf}(\mu)<\theta\leq\mu, \ \ \ \ \ (1)

such that

\displaystyle  {\rm tcf}(\prod\mathfrak{a}, <_I)<\lambda \ \ \ \ \ (2)

whenever {\mathfrak{a}} and {I} satisfy

  • {\mathfrak{a}} is a progressive set of regular cardinals with {\sup(\mathfrak{a})=\mu} and {|\mathfrak{a}|<\theta},
  • {I} is a {\sigma}-complete ideal on {\mathfrak{a}} extending the bounded ideal, and
  • the structure {(\prod\mathfrak{a}, <_I)} has a true cofinality.

To save ourselves some time later, let us agree to call such {\mathfrak{a}} and {I} relevant.

Now let {\chi} be a sufficiently large regular cardinal, and let {M} be an elementary submodel of {\langle H(\chi),\in, <_\chi\rangle} satisfying

  • {|M|<\lambda},
  • {\{\sigma,\theta,\mu, \lambda\}\in M}, and
  • {M\cap\lambda} is an ordinal of cofinality greater than {\theta}.

Our goal is to show that the collection

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

witnesses {{\rm cov}(\mu,\mu,\theta,\sigma)\leq |M|}, i.e., that every set in {[\mu]^{<\theta}} can be covered by a union of fewer than {\sigma} sets from {\mathcal{P}}.

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