Proof of Claim 3.3 continued…

September 21, 2011 at 15:05 | Posted in Uncategorized | Leave a comment

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)<f(\eta)]. \ \ \ \ \ (2)

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)<f(\eta)\right]. \ \ \ \ \ (8)

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.

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