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Let $A$ be a non-empty set of regular cardinals such that $\vert A\vert <\text{min}\ A$, and $\{\nu_i\mid i<i_0\}\subseteq \text{pcf}\ A$ be a strict increasing sequence having limit length $i_0$.Then $\mathcal{J}_{<\nu_{i_0}:=\text{sup}\{\nu_i\mid i<i_0\}}A=\bigcup\{\mathcal{J}_{<\nu_i}A\mid i<i_0\}$ hence there is $\nu\in\text{pcf}\ A$ such that $\nu\geq\nu_{i_0}$.

But in this case does $\nu_{i_0}\text{is regular}\Rightarrow\nu_{i_0}\in\text{pcf}\ A$ hold? or is there a counter-example in higher axioms than ZFC? I see when the special case, $A$ is an interval, this holds true.

[When $A$ denotes a set of regulars, $\mathcal{J}_{<\nu}A:=\{B\subseteq A\mid (\forall D\ \text{ultrafilter on}\ A), B\in D\Rightarrow \text{cf}\ \prod A/{D}<\nu\}$.]

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This is open. In fact, the situation you describe is related to one of the most important open problems in pcf theory:

Can there can be a progressive set of regular cardinals $A$ (that is, $|A|<\min(A)$) with the property that $pcf(A)$ contains a regular limit point?

This is essentially equivalent to the question of whether $pcf(pcf(A))=pcf(A)$ for all sets of regular cardinals $A$ satisfying $|A|<\min(A)$: $pcf(pcf(A))=pcf(A)$ if $pcf(A)$ fails to have a regular (=weakly inaccessible) limit point, and if there is an example where pcf DOES have a weakly inaccessible limit point, then one can force over this model to get a progressive set $A$ for which $pcf(pcf(A))\neq pcf(A)$.

Shelah talks a bit about this in his paper [Sh:666], as well as in the Analytical Guide appended to his book Cardinal Arithmetic.

Back to your question:

If $A$ is an interval of regular cardinals, then it is known $|pcf(A)|<|A|^{+4}$, so that $pcf(A)$ can never have a regular limit point in this situation, and the special case you mention in your question will never arise.

In the more general case, what you are asking is one of an entire family of so-called compactness questions for pcf theory that are still very much open, and seemingly still out of reach.

[Sh:666] Shelah, Saharon, On what I do not understand (and have something to say). I, Fundam. Math. 166, No. 1-2, 1-82 (2000). ZBL0966.03044.

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  • $\begingroup$ What is the idea used to force the existence of progressive $A$ such that $\text{pcf}\ \text{pcf}\ A\not=\text{pcf}\ A$ from a progressive $B$ with a regular limit point? and where the proof can be found? $\endgroup$
    – Sho Banno
    Commented Apr 17, 2020 at 21:50
  • $\begingroup$ I do not have access to all of my materials because I am working from home, but there is discussion in the paper I referenced above, see Conjecture 1.10 and the subsequent pages. The idea is that if $\kappa$ is a weakly inaccessible limit point of $pcf(A)$, then you can force (via a mild forcing) a function $f\in\prod pcf(A)\cap\kappa$ dominating the ground model elements of the product. $\endgroup$
    – Todd Eisworth
    Commented Apr 18, 2020 at 2:12

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