## A technical result (continued)

The following proposition can be viewed as a continuation of the technical result from the last post. What I’m trying to do is to look at ${{\rm pp}_{\Gamma(\theta,\sigma)}(\mu)}$ from different points of view so that we get a better picture of what’s going on.

[Warning: This post may be revised as I haven’t had enough time to proofread it properly]

Proposition 1 Let ${\sigma\leq{\rm cf}(\mu)<\theta<\mu}$ be cardinals, with ${\sigma}$ and ${\theta}$ regular. Then the following two statements are equivalent for a cardinal ${\kappa}$:

1. ${\kappa\in{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu)}$
2. there is a progressive set of regular cardinals ${\mathfrak{a}}$ cofinal in ${\mu}$ such that
• ${|\mathfrak{a}|<\theta}$,
• ${\kappa\in{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})}$, and
• ${\mathfrak{b}_{\kappa}[\mathfrak{a}]}$ is cofinal in ${\mathfrak{a}}$.

Proof: Assume first that ${\kappa\in{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu)}$, and let ${\mathfrak{a}}$ and ${I}$ witness this fact. These objections ${\mathfrak{a}}$ and ${I}$ put ${\kappa}$ into ${{\rm pcf}_{\sigma\text{-complete}}(\mu)}$, so we need only verify that the generator ${\mathfrak{b}_{\kappa}[\mathfrak{a}]}$ is unbounded in ${\mathfrak{a}}$. This follows easily, though, as ${I}$ contains all the bounded subsets of ${\mathfrak{a}}$.

Now suppose that ${\mathfrak{a}}$ is a progressive set of regular cardinal cofinal in ${\mu}$ and of cardinality less than ${\theta}$, and that ${\kappa}$ is in ${{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})}$ with ${\mathfrak{b}_{\kappa}[\mathfrak{a}]}$ unbounded in ${\mathfrak{a}}$. Let ${J}$ be the ${\sigma}$-complete ideal generated by ${J_{<\kappa}[\mathfrak{a}]}$ together with the bounded subsets of ${\mathfrak{a}}$.

Our assumptions imply that ${\mathfrak{b}_\kappa[\mathfrak{a}]\notin J}$ (since ${\kappa}$ is in ${{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})}$ we know ${\mathfrak{b}_{\kappa}[\mathfrak{a}]}$ cannot be covered by fewer than ${\sigma}$ sets from ${J_{<\kappa}[\mathfrak{a}]}$). Basic pcf-theory tells us

$\displaystyle {\rm tcf}\left(\prod\mathfrak{b}_\kappa[\mathfrak{a}], <_{J_{<\kappa}[\mathfrak{a}]}\right)=\kappa, \ \ \ \ \ (1)$

and it follows

$\displaystyle \kappa={\rm tcf}\left(\prod\mathfrak{b}_{\kappa}[\mathfrak{a}], <_J\right) \ \ \ \ \ (2)$

as well.

But now we see that ${\mathfrak{b}_\kappa[\mathfrak{a}]}$ and ${J}$ witness that ${\kappa}$ is a member of ${{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu)}$, as was claimed. $\Box$

The following corollary is an application of the above result; it is the main point of this post, and will be important in the course of later discussions.

Corollary 2 Let ${\sigma\leq{\rm cf}(\mu)<\theta<\mu}$ be cardinals, with ${\sigma}$ and ${\theta}$ regular. If ${\mathfrak{a}}$ is a progressive set of regular cardinals cofinal in ${\mu}$ with ${|\mathfrak{a}|<\theta}$, then there is a ${\xi<\mu}$ such that

$\displaystyle \sup{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a}\setminus\xi)\leq{\rm pp}_{\Gamma(\theta,\sigma)}(\mu). \ \ \ \ \ (3)$

Proof: Let ${\mathfrak{a}}$ be a set of cardinals as in the corollary, and define

$\displaystyle \tau=({\rm pp}_{\Gamma(\theta,\sigma)}(\mu))^+. \ \ \ \ \ (4)$

Let ${J}$ be the ${\sigma}$-complete ideal on ${\mathfrak{a}}$ generated by ${J_{<\tau}[\mathfrak{a}]}$ together with the bounded subsets of ${\mathfrak{a}}$. We claim that ${J}$ is not a proper ideal, i.e.,

$\displaystyle \mathfrak{a}\in J. \ \ \ \ \ (5)$

Suppose this fails. Let ${\kappa\in{\rm pcf}(\mathfrak{a})}$ be least such that ${\mathfrak{b}_\kappa[\mathfrak{a}]\notin J}$. Then

$\displaystyle J_{<\kappa}[\mathfrak{a}]\subseteq J, \ \ \ \ \ (6)$

and since ${J}$ is ${\sigma}$-complete, it follows that

$\displaystyle \kappa\in{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a}). \ \ \ \ \ (7)$

Note also that

$\displaystyle J_{<\tau}[\mathfrak{a}]\subseteq J\Longrightarrow \tau\leq \kappa, \ \ \ \ \ (8)$

and hence

$\displaystyle {\rm pp}_{\Gamma(\theta,\sigma)}(\mu)<\kappa. \ \ \ \ \ (9)$

Finally, ${J}$ extends the bounded ideal on ${\mathfrak{a}}$ and so we know that ${\mathfrak{b}_{\kappa}[\mathfrak{a}]}$ must be unbounded in ${\mathfrak{a}}$. Given our proposition, this would imply

$\displaystyle \kappa\leq{\rm pp}_{\Gamma(\theta,\sigma)}(\mu), \ \ \ \ \ (10)$

Thus, ${J}$ is not a proper ideal. Decoding what this means, we find that there exists a ${\xi<\mu}$ such that ${\mathfrak{a}\setminus\xi}$ can be covered by a union of fewer than ${\sigma}$ sets from ${J_{<\tau}[\mathfrak{a}]}$, and this implies
$\displaystyle {\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a}\setminus\xi)\subseteq\tau. \ \ \ \ \ (11)$
But ${{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})}$ is a collection of cardinals, and given the definition of ${\tau}$, the conclusion of the corollary is immediate. $\Box$