[FOM] PCF Theory

[Amit Gupta]

The "iff" is simply saying

X ⊆ A  and pcf(X) ⊆ λ iff for every ultrafilter D over A such that X ∈ D,
cf(Product A/D) < λ

I assume you're working with the definition:

pcf(X) = {cf(Product X/D) | D is an ultrafilter on X}

(-->).  Suppose X is a subset of A, and pcf(X) is contained in λ, i.e. for
every ultrafilter D on X, cf(Prod X/D) < λ.  Now let D be an ultrafilter
for A such that X ∈ D.  It's not hard to see that E = {X \cap S | S ∈ D} is
an ultrafilter on X, and so cf(Product X/E) < λ.  We just need to show that
cf(Product X/E) = cf(Product A/D).  But it's not hard to see that the
function Product A/D --> Product X/E defined by [f]_D \mapsto [f|X]_E is an
isomorphism of partial orders.

(<--).  We prove the constrapositive.  Suppose pcf(X) is not a subset of
λ.  Then by definition, there's some D on X such that cf(Product X/D) = \mu
for some \mu greater than or equal to λ.  One can easily extend D to an
ultrafilter on A: E = {S ⊆ A | S \cap X ∈ D}.  And it's not hard to see
that cf(Product A/E) = \mu, again, because Product X/D and Product A/E are
isomorphic.

Amit


> ---------- Forwarded message ----------
> From: pax0 at seznam.cz
> To: "Foundations of Mathematics" <fom at cs.nyu.edu>
> Date: Tue, 06 Dec 2011 17:18:42 +0100 (CET)
> Subject: [FOM] PCF theory
> I'm trying to understand some pcf theory --
> the equivalence of the sentence after "In plain words," with (1):
> >Let A be a set of regular cardinals. For any cardinal λ define
> >
> >(1)  J<λ[A] = {X ⊆ A | pcf(X) ⊆ λ}.
> >
> >In plain words, X ∈ J<λ[A] iff for every ultrafilter D over A such that X
> ∈ D,
> >cf(Product A/D) < λ. That is, X “forces” the cofinalities of its
> ultraproducts to
> >be below λ.
>
> One time we are taking ultrafilter D over A containing X and in (1) over X
> in pcf(X).
> Can someone explain both directions in "iff" in some detail?
> Thank you Jan Pax
>
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