## Proof of Claim 3.3 continued…

In the last post I told you what I am going to do, and in this post I want to formulate things as a stand-alone lemma:

Lemma 1 Let ${\mu}$ be a singular cardinal, and suppose

$\displaystyle \aleph_0<{\rm cf}(\sigma)\leq\sigma\leq{\rm cf}(\mu)<\theta\leq\kappa<\mu.$

Further suppose

$\displaystyle \mathcal{F}\subseteq\prod({\sf Reg}\cap(\kappa,\mu)) \ \ \ \ \ (1)$

has the property that whenever ${\mathfrak{a}}$ is a subset of ${{\sf Reg}\cap(\kappa,\mu)}$ of cardinality less than ${\theta}$, for any ${g\in\prod\mathfrak{a}}$ there is a family ${\mathcal{F}_0\subseteq \mathcal{F}}$ of cardinality ${<\sigma}$ such that

$\displaystyle (\forall\eta\in\mathfrak{a})(\exists f\in \mathcal{F}_0)\bigl[g(\eta)

Then given ${Y\in [\mu]^{<\theta}}$, there is a set ${H\subseteq\mathcal{F}}$ of cardinality ${<\sigma}$ such that

$\displaystyle Y\subseteq\mu\cap{\rm Sk}_{\mathfrak{A}}(\kappa+1\cup\{\mu\}\cup H), \ \ \ \ \ (3)$

where ${\mathfrak{A}=\langle H(\chi),\in, <_\chi\rangle}$ is “as usual”.

I don’t know if I’ll give the entire proof in this post, but I’ll at least get started.

So let ${Y}$ be a subset of ${\mu}$ of cardinality less than ${\theta}$. We now define (by recursion on ${k<\omega}$) objects ${N_k}$ and ${H_k}$ such that

1. each ${N_k}$ is an elementary submodel of ${\mathfrak{A}}$ of cardinality less than ${\theta}$, and
2. each ${H_k}$ is a subset of ${\mathcal{F}}$ of cardinality less than ${\sigma}$.

The construction starts by setting

$\displaystyle N_0:={\rm Sk}_{\mathfrak{A}}(Y\cup\{\mu\}). \ \ \ \ \ (4)$

Now assume that we are given the model ${N_k}$. We assume that our construction has achieved ${|N_k|<\theta}$ and so

$\displaystyle \mathfrak{a}_k:=N_k\cap{\sf Reg}\cap (\kappa,\mu) \ \ \ \ \ (5)$

is of cardinality ${<\theta}$. We let ${g_k}$ be the characteristic function of ${N_k}$ in ${\mathfrak{a}_k}$, i.e.,

$\displaystyle g_k\in\prod\mathfrak{a}_k, \ \ \ \ \ (6)$

and

$\displaystyle g_k(\eta)=\sup(N_k\cap\eta) \ \ \ \ \ (7)$

for each ${\eta\in \mathfrak{a}_k}$. (Notice that this definition makes sense, as ${|N_k|<\theta\leq\kappa<\eta}$.)

By our hypothesis, there is a family ${H_k\subseteq \mathcal{F}}$ of cardinality ${<\sigma}$ such that

$\displaystyle (\forall\eta\in\mathfrak{a}_k)(\exists f\in H_k)\left[g_k(\eta)

We now define

$\displaystyle N_{k+1}={\rm Sk}_{\mathfrak{A}}(N_k\cup H_k), \ \ \ \ \ (9)$

and the construction continues. (Note that ${N_{k+1}}$ is still of cardinality less than ${\theta}$, as ${H_k}$ had cardinality less than ${\sigma}$.)

In the end, let us define

$\displaystyle N:=\bigcup_{k<\omega}N_k, \ \ \ \ \ (10)$

and

$\displaystyle H:=\bigcup_{k<\omega}H_k. \ \ \ \ \ (11)$

At this point, our assumption that ${\sigma}$ has uncountable cofinality becomes important, as this implies

$\displaystyle |H|<\sigma. \ \ \ \ \ (12)$

In our next post, we’ll show that this ${H}$ does the job.