The main proof, part 1.

February 14, 2011 at 12:13 | Posted in Uncategorized | Leave a comment

General results on the approachability ideal {I[\mu^+]} tell us there is a sequence

\displaystyle  \bar{C}=\langle C_\alpha:\alpha<\mu^+\rangle \ \ \ \ \ (1)

such that

  • {C_\alpha} is a closed (not necessarily unbounded!) subset of {\alpha},
  • {C_\alpha} has order-type at most {\kappa^+}, where {\kappa={\rm cf}(\mu)},
  • {\beta\in{\rm nacc}(C_\alpha)\Longrightarrow C_\beta= C_\alpha\cap\beta}, and
  • the set of {\delta<\mu^+} of cofinality {\kappa^+} with {\delta=\sup(C_\delta)} is stationary.

The above is a result from [Sh:420]; it says that there is a stationary subset of {S^{\mu^+}_{\kappa^+}} in the ideal {I[\mu^+]}. The details of this are worked out in my Handbook article — see Theorem 3.18 and Theorem 3.7.

\bigskip

Without loss of generality, the sequence {\bar{C}} lies in the model {M_0}, hence in each {M_\alpha} and {M^*} as well. Let {\langle \mu_i:i<\kappa\rangle} be the {<_\chi}-least continuous and increasing sequence of cardinals cofinal in {\mu} satisfying {\kappa<\mu_0}; note this means that the sequence is going to be in every elementary submodel of the structure {\mathfrak{A}}.

Moving back to the main proof, let us fix {a\in [\mu^+]^{{\rm cf }\mu}}; recall that we want to show {a} is a subset of some element {M^*\cap [\mu]^{<\mu}}, and the preceding post gives us our strategy.

For each {\alpha<\mu^+} and {i<\kappa}, we define

\displaystyle  \mathcal{N}(\alpha,i)={\rm Sk}_{\mathfrak{A}}(\mu_i\cup \{f_\xi:\xi\in C_\alpha\}), \ \ \ \ \ (2)

and

\displaystyle  \mathcal{N}^+(\alpha, i)={\rm Sk}_{\mathfrak{A}}(\mu_i\cup \{f_\xi:\xi\in C_\alpha\}\cup a). \ \ \ \ \ (3)

We note the following:

  • {\mathcal{N}(\alpha,i)\subseteq\mathcal{N}^+(\alpha, i)\subseteq M_{\alpha+1}\subseteq M^*},
  • {\mathcal{N}(\alpha, i)\in M_{\alpha+1}}, and
  • both {\mathcal{N}(\alpha,i)} and {\mathcal{N}^+(\alpha,i)} have cardinality {\mu_i}.

Next, we define

\displaystyle  \mathfrak{a}_{\alpha, i}=\mathcal{N}(\alpha,i)\cap\mu\cap{\sf Reg}\setminus{\mu_i^{++}}, \ \ \ \ \ (4)

and let {g_{\alpha, i}} be the characteristic function of {\mathcal{N}^+(\alpha, i)} (note the “+”!) in {\mathfrak{a}_{\alpha, i}}, i.e., the domain of {g_{\alpha, i}} is {\mathfrak{a}_{\alpha, i}}, and for {\theta\in\mathfrak{a}_{\alpha, i}}, we have

\displaystyle  g_{\alpha, i}(\theta)=\sup(\mathcal{N}^+(\alpha,i)\cap\theta)<\theta. \ \ \ \ \ (5)

The function {g_{\alpha, i}} lies in {\prod\mathfrak{a}_{\alpha, i}}, but there is no reason to believe it is an element of {M^*}. Note, however, that {\mathfrak{a}_{\alpha, i}} IS an element of {M^*}. It is also a progressive set of regular cardinals cofinal in {\mu}, and therefore by earlier work, we know

\displaystyle  \langle f_\beta\upharpoonright\mathfrak{a}_{\alpha, i}:\beta<\mu^+\rangle \ \ \ \ \ (6)

is a scale in {\mathfrak{a}_{\alpha, i}} modulo the ideal of bounded subsets of {\mathfrak{a}_{\alpha, i}}.

Since {\prod(\mu\cap{\sf Reg})} is {\mu^+}-directed modulo the ideal of bounded subsets of {\mu}, it follows that for each {\alpha<\mu^+}, there exists a {\gamma(\alpha)<\mu^+} such that

\displaystyle  (\forall i<\kappa)\left(g_{\alpha, i}< f_{\gamma(\alpha)}\upharpoonright\mathfrak{a}_{\alpha, i}\mod J^{\rm bd}[\mathfrak{a}_{\alpha, i}]\right) \ \ \ \ \ (7)

The function mapping {\alpha} to {\gamma(\alpha)} need not be in {M^*}. Nevertheless, the set of {\delta<\mu^+} is closed unbounded in {\mu^+}, and so we can find a {\delta} of cofinality {\kappa^+} such that

  • {\delta} is closed under the map {\alpha\mapsto\gamma(\alpha)}, and
  • {C_\delta} is closed unbounded in {\delta} (of order-type {\kappa^+}).

We’ll end this post on this cliff-hanger…

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