On ADS (Part 3)

February 9, 2012 at 11:40 | Posted in Uncategorized | Leave a comment

Continuing the construction, let us assume that {\mu} is singular of countable cofinality and {{\rm cov}(\mu,\mu,\aleph_1,2)>\mu^+}. For each {\beta<\mu^+}, let {\langle A^\beta_n:n<\omega\rangle} be a sequence of sets such that

  • {A^\beta_0=\emptyset},
  • {\beta=\bigcup_{n<\omega}A^\beta_n},
  • {|A^\beta_n|<\mu} for all {n<\omega}, and
  • {A^\beta_n\subseteq A^\beta_{n+1}}.

By induction on {\alpha<\mu^+} we choose sets {x_\alpha\in[\mu]^{\aleph_0}} such that for no {\beta<\mu^+} and {n<\omega} is {x_\alpha} a subset of {\bigcup\{x_\gamma:\gamma\in A^\beta_n\cap\alpha\}}.

Notice that each set of the form {\bigcup\{x_\gamma:\gamma\in A^\beta_n\cap\alpha\}} is of cardinality less than {\mu}, and there are {\mu^+} sets of this form. Our assumption that {{\rm cov}(\mu,\mu,\aleph_1,2)} is greater than {\mu^+} tells us that a suitable {x_\alpha} can always be found.

Let {\eta_\alpha:\omega\rightarrow x_\alpha} be a bijection, and let {A_\alpha:=\{\eta_\alpha\upharpoonright\ell:\ell<\omega\}}. We will show that the collection {\{A_\alpha:\alpha<\mu^+\}} will witness {{\rm ADS}_\mu}.

What this amounts to is that for each {\beta<\mu^+}, we need a function {h_\beta:\beta\rightarrow\omega} so that

\displaystyle  \Delta(\alpha,\gamma)\leq\max\{h_\beta(\alpha),h_\beta(\gamma)\} \ \ \ \ \ (1)

for all {\alpha,\gamma<\beta}, where

\displaystyle  \Delta(\alpha,\gamma)=\text{ least }\ell\text{ such that }\eta_\alpha(\ell)\neq\eta_\gamma(\ell). \ \ \ \ \ (2)

Lemma 1 For each {n<\omega}, the family {\{x_\alpha:\alpha\in A^\beta_n\}} has a transversal (i.e., a one-to-one choice function) {f^\beta_n}.

Proof: We define {f^\beta_n\upharpoonright (A^\beta_n\cap\alpha)} by induction on {\alpha}. This is trivial for {\alpha=0} and {\alpha} limit. If {\alpha=\gamma+1}, then {x_\gamma} is not a subset of {\bigcup\{x_\epsilon:\epsilon\in A^\beta_n\cap\gamma\}} so {f^\beta_n(\gamma)} can be defined easily. \Box

Armed with the preceding lemma, we define a function {k_\beta:\beta\rightarrow\omega} as follows:

Given {\alpha<\beta}, we know {\alpha\in A^\beta_{n+1}\setminus A^\beta_n} for some unique {n<\omega}. We define {k_\beta(\alpha)} to be the unique {k<\omega} such that {f^\beta_n(\alpha)\eta_\alpha(k)}.

In English, {k_\beta(\alpha)} answers the question “when does {\eta_\alpha} enumerate the value of {f^\beta_n(\alpha)}?”.

Our goal is to show that the function {k_\beta} almost works, and then show how to modify {k_\beta} to a function {h_\beta} that actually does the job.

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