Conclusion 3.2A (part 1)

June 16, 2011 at 13:28 | Posted in Uncategorized | Leave a comment

In this post, we want to look at the next result in The Book: Conclusion 3.2A on page 378. What I will do is state this exactly as it appears in the book, and then reformulate it in conjunction with providing some commentary. I’ll attempt to prove it in a future post.

Theorem 1 (Conclusion 3.2A page 378) If {\lambda=\aleph_{\xi(*)+\zeta(*)}}, {\zeta(*)<\aleph_{\xi(*)}} a limit ordinal, then:

  1. for {i_0<\zeta(*)} large enough,

    \displaystyle  {\rm pp}(\lambda)=^+{\rm cf}_{{\rm cf}[\zeta(*)]}\left(\prod\{\aleph_{\xi(*)+i+1}: i_0<i<\zeta(*)\}\right). \ \ \ \ \ (1)

  2. if {{\rm cf}[\zeta(*)]\leq\kappa\leq |\zeta(*)|^+} then for {i_0<\zeta(*)} large enough

    \displaystyle  {\rm pp}_\kappa(\lambda)=^+{\rm cf}_\kappa\left(\{\aleph_{\xi(*)+i+1}:i_0<i<\zeta(*)\}\right). \ \ \ \ \ (2)

I’m going to concentrate on the first part of the above; here’s my attempt at reformulating it.

Theorem 2 (Conclusion 3.2A (i), reformulated) Suppose {\mu} is a singular cardinal of cofinality {\kappa} with {\mu<\aleph_\mu}. Then for some {\tau<\mu},

\displaystyle   {\rm pp}(\mu)=^+{\rm cf}_{\kappa}\left(\prod(\tau,\mu)\cap{\sf Reg}\right). \ \ \ \ \ (3)

The right-hand side of the above is not something we’ve explicitly defined yet, but it’s a reasonable variant of the {{\rm cf}^\sigma_{<\theta}(\prod\mathfrak{a})} notation we’ve been working with. The notation

\displaystyle   {\rm cf}_\kappa\left(\prod\mathfrak{a}\right) \ \ \ \ \ (4)

refers to the minimum cardinality of a family {F\subseteq\prod\mathfrak{a}} with the property that for any {\mathfrak{b}\subseteq\mathfrak{a}} of cardinality {\kappa}, the collection

\displaystyle  \{f\upharpoonright\mathfrak{b}:f\in F\} \ \ \ \ \ (5)

is cofinal in {(\prod\mathfrak{b}, <)}.

This definition is robust under small modifications: it is also the minimum cardinality of a family {F\subseteq\prod\mathfrak{a}} with the property that for any {\mathfrak{b}\subseteq\mathfrak{a}} of cardinality at most {\kappa} and any {g\in\prod\mathfrak{a}}, there is a finite set {F_0\subseteq F} such that

\displaystyle  g(\eta)<\max\{f(\eta):f\in F_0\}\text{ for all }\eta\in\mathfrak{b}. \ \ \ \ \ (6)

In other words,

\displaystyle  {\rm cf}_\kappa\left(\prod\mathfrak{a}\right)={\rm cf}_{<\kappa^+}^{\aleph_0}\left(\prod\mathfrak{a}\right), \ \ \ \ \ (7)

and we are back with the sorts of cardinals we’ve been considering of late.

There is something else implicit in this theorem that I want to bring out: it shows us that if {\mu} is a singular cardinal that isn’t a fixed point, then {{\rm pp}(\mu)} can be calculated just by looking at structure

\displaystyle  \left(\prod(\mu\cap{\sf Reg}), <\right). \ \ \ \ \ (8)

In this situation {{\rm pp}(\mu)} is the minimum cardinality of a family of functions in {\prod(\mu\cap{\sf Reg})} with the property that for some {\tau<\mu}, for any {\mathfrak{b}\subseteq(\tau,\mu)\cap{\sf Reg}} of cardinality {{\rm cf}(\mu)} the family

\displaystyle  \{f\upharpoonright\mathfrak{b}:f\in F\} \ \ \ \ \ (9)

is cofinal in {\prod\mathfrak{b}}. Roughly speaking, if {\mu} is singular and not a fixed point, then {{\rm pp}(\mu)} is the minimum cardinality of a family of functions in {\prod\mu\cap{\sf Reg}} which can cover all products of size {{\rm cf(\mu)}} drawn from some fixed tail of {\mu\cap{\sf Reg}}. That seems like it is worth knowing…

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