Pedagogical Remarks

September 28, 2011 at 15:21 | Posted in Uncategorized | Leave a comment

In this post, I want to take a stab at “pedagogically reorganizing” the preceding proof. This post should be considered tentative, as I’m just trying to work out what the best definitions should be.

Definition 1 Let {\mu} be a singular cardinal, with {\mathfrak{A}} as usual. Suppose {N} is an elementary submodel of {\mathfrak{A}}, {H} is a family of functions in {\prod(\mu\cap{\sf Reg})}, and {\kappa<\mu}. We say {H} supports {N} beyond {\kappa} if

  • {H\subseteq N\cap\prod(\mu\cap{\sf Reg})}, and
  • for every {\eta\in N\cap (\kappa,\mu)\cap{\sf Reg}} and {\alpha\in N\cap\eta}, there is an {f\in H} such that {\alpha<f(\eta)}.

If I’ve done things right, the following should be true.

Theorem 2 Let {\mu} be a singular cardinal, and assume {\sigma\leq{\rm cf(\mu)}<\theta\leq\kappa<\mu} with {\sigma} infinite. If {\mathcal{F}\subseteq\prod(\mu\cap {\sf Reg})} has the property that for every {Y\in [\mu]^{<\theta}}, there is an elementary submodel {N} of {\frak{A}} such that

  • {Y\cup \{\kappa,\mu\}\subseteq N},
  • {|N|=\kappa}, and
  • {N} is supported past {\kappa} by a subset of {\mathcal{F}} of cardinality less than {\sigma},

then

\displaystyle  {\rm cov}(\mu,\kappa^+,\theta,\sigma)\leq |\mathcal{F}|. \ \ \ \ \ (1)

Proof: Assume the above. Given a finite subset {F} of {\mathcal{F}}, let

\displaystyle  A_F:=\mu\cap{\rm Sk}_{\mathfrak{A}}(\kappa+1\cup\{\mu\}\cup F), \ \ \ \ \ (2)

and define

\displaystyle  \mathcal{P}:=\{A_F: F\in[\mathcal{F}]^{<\aleph_0}\}. \ \ \ \ \ (3)

Clearly {\mathcal{P}} and {\mathcal{F}} are of the same cardinality (our hypotheses certainly imply {\mathcal{F}} is infinite!). Since {\mathcal{P}\subseteq [\mu]^\kappa}, we will be done provided we show that every {Y\in [\mu]^{<\theta}} can be covered by a union of fewer than {\sigma} sets in {\mathcal{P}}.

Given {Y}, let {N} be an elementary submodel as in the assumptions of the theorem, and let {H} be a subset of {\mathcal{F}} such that

  • {|H|<\sigma}, and
  • {H} supports {N} beyond {\kappa}.

Let us define

\displaystyle  N^+:={\rm Sk}_{\mathfrak{A}}(\kappa+1\cup\{\mu\}\cup H), \ \ \ \ \ (4)

and note that Lemma 1 in the preceding post tells us

\displaystyle  N\cap\mu\subseteq N^+. \ \ \ \ \ (5)

But {N^+\cap\mu} is the union of fewer than {\sigma} sets from {\mathcal{P}}, as

\displaystyle  N^+\cap\mu=\bigcup\{A_F: F\in [H]^{<\aleph_0}\} \ \ \ \ \ (6)

and so we’re done.

\Box

What’s the point? Well, the above emphasizes that we can bound covering numbers by the sizes of “supportive” families of functions in {\prod(\mu\cap{\sf Reg})}. This is “what’s going on” in this section of The Book, and I think the proofs Shelah gives are best viewed as ways of constructing “supportive families”. We’ll see how it goes when I look at the next result in the chapter…

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