## Conclusion 3.2A (part 1)

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

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’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…