The Story of Generators 1

Introduction

I’m not sure exactly where things left off, so I’ll just begin with a series of posts on generators for pcf. I will also try to keep the posts short so that I can ease back into the routine of writing them.

Let us assume that ${A}$ is a progressive set of regular cardinals, and ${\lambda\in{\rm pcf}(A)}$. I will assume we know already some of the basics about the ideal ${J_{<\lambda}[A]}$ and I will use these facts without much comment.

What we want to look at is the existence of a generator for ${\lambda}$: we will sketch the proof that the ideal ${J_{\leq\lambda}[A]}$ is generated over ${J_{<\lambda}[A]}$ by a single set ${B_\lambda}$. And I do mean “sketch”, as the details are worked out nicely in Section 4 of the Abraham/Magidor article in the Handbook of Set Theory.

Universal Sequences

Definition 1 Suppose ${\lambda\in{\rm pcf}(A)}$. A sequence ${\bar{f}=\langle f_\alpha:\alpha<\lambda\rangle}$ of functions in ${\prod A}$ is a universal sequence for ${\lambda}$ if

• ${\bar{f}}$ is ${<_{J_{<\lambda}[A]}}$-increasing, and

• ${\bar{f}}$ is cofinal in ${\prod A/D}$ whenever ${D}$ is an ultrafilter on ${A}$ with ${\lambda={\rm cf}(\prod A/D)}$.

Universal sequences are tightly related to the existence of generators for pcf, as we shall see. I want to point out the following result:

Theorem 2 Suppose ${A}$ is a progressive set of regular cardinals and ${\lambda\in{\rm pcf}(A)}$ Then the following statements are equivalent:

1. There is a universal sequence for ${\lambda}$.
2. There is a family ${F\subseteq \prod A}$ such that for any ultrafilter ${D}$, if ${\prod A/D}$ has cofinality ${\lambda}$ then ${F}$ remains unbounded in ${\prod A/D}$.
3. There is a family ${\langle B_\alpha:\alpha<\lambda\rangle}$ of subsets of ${A}$ such that
• ${\alpha<\beta\Longrightarrow B_\alpha\subseteq B_\beta}$ modulo ${J_{<\lambda}[A]}$, and
• ${J_{\leq\lambda}[A]}$ is the ideal generated by ${J_{<\lambda}[A]}$ together with the sets ${\{B_\alpha:\alpha<\lambda\}}$.

The above is basically Fact 2.2 on page 13 of Cardinal Arithmetic, and it follows quite easily from the work done in the first section of the book. (The 3rd statement says, in the notation of the book, that ${\lambda}$ is semi-normal.) Abraham and Magidor develop basic pcf theory in a slightly different order, and deriving the above result from the material they present prior to defining universal sequences is a bit more difficult.

Of course, the main point is the following result:

Theorem 3 If ${A}$ is a progressive set of regular cardinals, then every ${\lambda\in{\rm pcf}(A)}$ has a universal sequence.

Again, we will not prove the above as this is Theorem 4.2 of the Abraham-Magidor article, and their proof is quite clear.

What have we learned from the above? We have outlined the first steps towards proving that generators exist. Putting the two theorems presented here together, we see that if ${\lambda\in{\rm pcf}(A)}$, then ${J_{\leq\lambda}[A]}$ is pretty simply generated over ${J_{<\lambda}[A]}$.

In our next post, we’ll see how “tuning up” a universal sequence leads to the existence of generators.