Easy Direction

August 29, 2011 at 14:07 | Posted in Uncategorized | Leave a comment

My daughter was born last week, so this post has been written by someone for whom sleep is but a fond memory…

Anyway, picking up where we left off, let us define

\displaystyle  \lambda_1:={\rm cov}(\mu,\kappa^+,\theta,\sigma) \ \ \ \ \ (1)

and

\displaystyle  \lambda_2={\rm cf}^\sigma_{<\theta}\left(\prod({\sf Reg}\cap(\kappa,\mu))\right). \ \ \ \ \ (2)

We’ll dispose of the easy direction first, and show

\displaystyle  \lambda_2\leq\lambda_1. \ \ \ \ \ (3)

Thus, let {\mathcal{P}\subseteq [\mu]^\kappa} be a family of cardinality {\lambda_1} with the property that any set in {[\mu]^{<\theta}} is covered by a union of fewer than {\sigma} sets in {\mathcal{P}}.

For each {A\in\mathcal{P}}, we define a function {f_A} with domain {{\sf Reg}\cap(\kappa,\mu)} by setting

\displaystyle  f_A(\eta)=\sup(A\cap\eta)+1 \ \ \ \ \ (4)

whenever {A\cap\eta\neq\emptyset}; otherwise, set {f_A(\eta)} equal to {0}.

Since elements of {\mathcal{P}} have cardinality {\kappa}, it is clear that each {f_A} is an element of {\prod({\sf Reg}\cap(\kappa,\mu))}. We claim that the collection {\mathcal{A}=\{f_A:A\in\mathcal{P}\}} witnesses {\lambda_2\leq\lambda_1}.

This should follow easily from the definitions involved via a proof by contradiction: if not, then there is a set {B\subseteq {\sf Reg}\cap(\kappa,\mu)} of cardinality less than {\theta} and a function {g\in\prod B} such that {g} cannot be majorized by taking the supremum of any collection of fewer than {\sigma} functions in {\mathcal{A}}.

By the definition of {\mathcal{P}}, we can find {\sigma_0<\sigma} and {\{A_\alpha:\alpha<\sigma_0\}\subseteq\mathcal{P}} such that

\displaystyle  {\rm ran}(g)\subseteq\bigcup_{\alpha<\sigma_0}A_\alpha. \ \ \ \ \ (5)

For each {\alpha<\sigma_0}, let {g_\alpha} denote the function {f_{A_\alpha}}. Given {\eta\in B}, we know {g(\eta)} lies in {A_\alpha} for some {\alpha<\sigma_0}, hence

\displaystyle  g(\eta)\leq\sup(A_\alpha\cap\eta)<g_\alpha(\eta). \ \ \ \ \ (6)

Thus, for each {\eta\in B} there is an {\alpha<\sigma_0} such that {g(\eta)<g_\alpha(\eta)}, and since each {g_\alpha} is in {\mathcal{A}}, we have a contradiction.

Back!

August 3, 2011 at 21:36 | Posted in Uncategorized | Leave a comment

Well, I found a little time to work on the blog. The plan is to keep working through the 3rd section of the last chapter of The Book ([Sh:400] in Shelah’s nomenclature). Our last substantive postings had to do with Conclusion 3.2A, so I guess it’s time to move on to Claim 3.3. We’ll give a direct quote of the result from The Book, and then formulate what we’ll actually prove.

Proposition 1 (Claim 3.3) If {\aleph_0<{\rm cf}(\sigma)\leq\sigma<\theta\leq\mu<\lambda} then

\displaystyle  {\rm cov}(\lambda,\mu^+,\theta,\sigma)={\rm cf}^\sigma_{<\theta}\left(\prod({\sf Reg}\cap(\mu,\lambda))\right)+\lambda. \ \ \ \ \ (1)

We’ve been concentrating on arithmetic at singular cardinals, so I’m going to restrict myself to this case. I’m also going to switch some notation around to remain consistent with previous postings. Taking all this into account, we will therefore give a proof of the following:

Proposition 2 Suppose {\mu} is singular, and

\displaystyle  \aleph_0<{\rm cf}(\sigma)\leq\sigma\leq{\rm cf}(\mu)<\theta\leq\kappa<\mu. \ \ \ \ \ (2)

Then

\displaystyle  {\rm cov}(\mu,\kappa^+,\theta,\sigma)={\rm cf}^\sigma_{<\theta}\left(\prod({\sf Reg}\cap(\kappa,\mu))\right). \ \ \ \ \ (3)

Recalling what these things mean:

The left side is the minimum cardinality of a family {\mathcal{P}\subseteq [\mu]^\kappa} such that every member of {[\mu]^{<\theta}} can be covered by a union of fewer than {\sigma} sets from {\mathcal{P}}.

The right side is the minimum cardinality of a family {\mathcal{F}} of functions in {\prod({\sf Reg}\cap(\kappa,\mu))} whose restrictions are {<\sigma}-cofinal in any “subproduct” concentrating on sets of size less than {\theta}.

For a particular instance of this result, we have for example the following:

\displaystyle  {\rm cov}(\aleph_{\omega_1},\aleph_3,\aleph_2,\aleph_1)={\rm cf}^{\aleph_1}_{<\aleph_2}\left(\prod({\sf Reg}\cap(\aleph_2,\aleph_{\omega_1}))\right). \ \ \ \ \ (4)

Now the above is an equation only a mother would love, but it fits in nicely with Shelah’s project: recall that the goal of the “cov vs. pp” material is to show that much of cardinal arithmetic can be expressed in “pcf language”. The covering number is question is a bit obscure, but it certainly falls under the purview of cardinal arithmetic, while the cofinality number on the other side of the equation is certainly something defined in “pcf language”.

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