## Pedagogical Remarks

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.

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…