Tuesday, May 28, 2013

A chain of inequalities

\(
\def\pp{\rm{pp}}
\def\pcf{\rm{pcf}}
\def\cov{\rm{cov}}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\)

I spent the weekend reading the very last section of Shelah's Cardinal Arithmetic, and I want to try my hand at writing a reasonable exposition of it.

The main concern he has is the following problem::

Question:
Suppose $\lambda$ is a singular cardinal.  Is $\pp(\lambda)=\cov(\lambda,\lambda,(\cf(\lambda))^+,2)$?

He discusses this question quite a bit in the Analytical Guide for Cardinal Arithmetic.  The "interesting" case is when $\lambda$ is a fixed point of cofinality $\aleph_0$.

In this post, I want to gather together a few inequalities culled from the book and try to build a coherent picture of what's going on.  I'm going to leave some things undefined for now just to get the statement out.
Over my next few posts I'll try to define everything and provide proofs for each of the inequalities.

__________
Theorem
Suppose $\cf(\lambda)<\theta=\cf(\theta)<\lambda$.

Define
$$\lambda(1)=\cf_{<\theta}(\prod(\lambda\cap\REG),<_{J^\bd_{\lambda}})$$

and let $\lambda(2)$ be the minimum cardinality of a family $\mathcal{P}\subseteq[\lambda(1)]^{<\lambda}$ such that for any infinite $B\in[\lambda(1)]^{\theta}$, there is an $A\in \mathcal{P}$ with $A\cap B$ infinite.

Then

$$\pp_{<\theta}(\lambda)\leq\lambda(1)\leq\cov(\lambda,\lambda,\theta, 2)\leq\lambda(2)\leq\cov(\lambda(1),\lambda,\theta^+,\theta).$$
__________

The main case of interest is when $\theta=\cf(\lambda)=\aleph_0$.  In this situation, we get:
$$\lambda(1)=\cf_{\aleph_0}(\prod(\lambda,\cap\REG, <_{J^\bd_{\lambda}}),$$

and 

$$\lambda(2)=\min\{|\mathcal{P}|:\mathcal{P}\subseteq[\lambda(1)]^{<\lambda}\wedge(\forall B\in [\lambda(1)]^{\aleph_1})(\exists A\in\mathcal{P})[|A\cap B|\geq\aleph_0]\},$$

while the conclusion is

$$\pp(\lambda)\leq\lambda(1)\leq\cov(\lambda,\lambda,\aleph_1, 2)\leq\lambda(2)\leq\cov(\lambda(1),\lambda,\aleph_2,\aleph_1).$$




1 comment:

Alberto Levi said...

My name is Alberto Levi, I am doing a post-doc in Universidade de São Paulo (Brazil), and I appreciate your blog.
We know that
pp(λ)=cov(λ,λ,(cf(λ))+,2)
for all singular cardinals that are not fixed points. In the Shelah's book "Cardinal Arithmetic", the question (beta) of section 14.7 of the Analytical Guide, asks if the equality holds for all the cardinals with countable cofinality (including fixed points).
My question is: does the equality hold for the fixed points of uncountable cofinality? And in the case of an affirmative answer to the question (beta) of Shelah's book?
Thank you for your attention.