Does anyone know of generalizations of pcf theory where we might consider products of the form:

$$\aleph_1 \times (\aleph_2 \times \aleph_2) \times (\aleph_3 \times \aleph_3 \times \aleph_3) \dots$$

$$(\aleph_1 \times \aleph_2 \times \dots) \times (\aleph_1 \times \aleph_2 \times \dots) \times \dots$$

or, more abstractly:

$$P_1 \times P_2 \times P_3 \times \dots$$

where each $P_i$ is a $\mathrm{cof}(P_i)$-directed partial order. My motivation is that I'm interested in what the relationship between $$\max\mathrm{pcf}\langle \aleph_1,\aleph_2,\aleph_2,\aleph_3,\aleph_3,\aleph_3,\dots\rangle$$ and $$\max\mathrm{pcf}\langle\aleph_n\rangle_{0 < n < \omega }$$ might be because it might help me understand the relationship between $$\max \mathrm{pcf} \langle \aleph_n \rangle_{0 < n <\omega }$$ and $$\max \mathrm{pcf} \langle \aleph_{2n} \rangle_{0 < n <\omega }$$

Claim: The $\mathrm{pcf}$ structure on $(\aleph_1 \times \aleph_2 \times \dots) \times (\aleph_1 \times \aleph_2 \times \dots) \times \dots$ gives nothing new.

Proof: For notational convenience, let $A : \omega \to \mathrm{Reg}$ be defined by $A(n) = \aleph_n$ and let $B : \omega \cdot \omega \to \mathrm{Reg}$ be defined by $B(\omega\cdot m + n) = \aleph_n$. Define

$$\mathrm{pcf}(A) = \{\mathrm{cf}(\Pi_{n<\omega}A(n)/U)\ :\ U \in \beta \omega\}$$ $$\mathrm{pcf}(B) = \{\mathrm{cf}(\Pi_{\alpha<\omega\cdot\omega}B(\alpha)/U)\ :\ U \in \beta (\omega\cdot\omega)\}$$

Where $\beta X$ denotes the set of all ultrafilters on $X$. It's not hard to see that $\mathrm{pcf}(A) \subseteq \mathrm{pcf}(B)$ and since $\mathrm{pcf}(A)$ is an interval of regular cardinals, it suffices to show that $\max \mathrm{pcf}(B) = \max \mathrm{pcf}(B)$. We know that we can find an everywhere-pointwise-dominating family on $\Pi A$ of size $\lambda := \max\mathrm{pcf}(A)$. If we can find a dominating family on $\Pi B$ of that same size, we'll be done.

So, given a dominating family $\mathcal{F}$ on $\Pi A$ of size $\lambda$, just let $\mathcal{F}^\ast \subseteq \Pi B$ consist of functions of the form: $$f^\ast (\omega\cdot m + n) = f(n)$$ for each $f \in \mathcal{F}$. Now given $g \in \Pi B$, define: $$g'(\omega\cdot m + n) = \sup_{m' \in \omega}g(\omega\cdot m' + n)$$ We see that $g' \geq g$ everywhere pointwise, and $g' \in \Pi B$ since we're always taking countable suprema within uncountable regular cardinals. Since $g'$ has the same value at any of its coordinates that correspond to the same $\aleph_n$, it's clear that there's some $f \in \mathcal{F}$ such that $f^\ast \in \mathcal{F}^\ast$ dominates $g'$ everywhere, and hence $g$ everywhere.


1 Answer 1


I don't know if anyone has looked at such things systematically, but I know Shelah has made use of structures of this form at various times. The examples which follow are just what I can remember off-hand; I know there's more buried in his work, but this is where I remembered seeing such a construction:

  1. Clause $(\gamma)$ on page 1641 of [Sh:589]:

Applications of PCF theory. J. Symbolic Logic 65 (2000), no. 4, 1624–1674.

(It's available on JSTOR here; the Arxiv version looks like an older iteration)

  1. The second proof of his Revised GCH theorem in [Sh:460], in particular Claim 2.6.

The generalized continuum hypothesis revisited. Israel J. Math. 116 (2000), 285–321.

(Available on SpringerLink here, again the Arxiv version is a bit dated)

I don't think that you get anything really new as far as pcf theory by doing this. Rather, it's just that sometimes such a structure is the right way to organize things.


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