A quick thought

April 26, 2011 at 11:13 | Posted in Uncategorized | Leave a comment

This particular post is tangential to the task at hand. I just want play around with some of the ideas in yesterday’s posts to sharpen them a bit, and maybe prove a fact or two. I’m still not sure if I’m making things too difficult here by bringing in all the pcf stuff; it may be the case that these sorts of facts are well-known and provable by other methods. And once again: I’m writing these entries quickly, just trying to get ideas down, and there’s always the possibility I’m making silly mistakes somewhere. But that’s the point of a blog — I can work out things as I go along, and make corrections where I need to.

In the statement of the following theorem, {<^*} refers to the natural “modulo a bounded set” order, i.e. for {f} and {g} in {\prod\mathfrak{a}}, we have {f<^* g} if and only if the set of {\theta} for which {g(\theta)\leq f(\theta)} is bounded in {\mathfrak{a}}.

Theorem 1 Let {\mathfrak{a}} be a progressive set of regular cardinals. Then there is a {\xi<\sup\mathfrak{a}} such that

\displaystyle  \max{\rm pcf}(\mathfrak{a}\setminus\xi)={\rm cf}\left(\prod\mathfrak{a}, <^*\right). \ \ \ \ \ (1)

Recall that

\displaystyle  \max{\rm pcf}(\mathfrak{a}\setminus\xi)={\rm cf}\left(\prod (\mathfrak{a}\setminus\xi), <\right), \ \ \ \ \ (2)

where “{f<g}” means “{f(\theta)<g(\theta)} for all {\theta\in\mathfrak{a}}”, and so really the theorem gives us a {\xi<\sup\mathfrak{a}} such that

\displaystyle  {\rm cf}\left(\prod (\mathfrak{a}\setminus\xi), <\right)={\rm cf}\left(\prod\mathfrak{a}, <^*\right). \ \ \ \ \ (3)

Proof: It is clear that

\displaystyle  {\rm cf}\left(\prod\mathfrak{a}, <^*\right)\leq {\rm cf}\left(\prod (\mathfrak{a}\setminus\xi), <\right) \ \ \ \ \ (4)

for any {\xi<\sup\mathfrak{a}}, as a cofinal set in the right hand partial order yields a cofinal set in the left hand one.

Now let us define

\displaystyle  \tau:={\rm cf}\left(\prod\mathfrak{a}, <^*\right)^+ \ \ \ \ \ (5)

and consider the ideal {J} on {\mathfrak{a}} generated by {J_{<\tau}[\mathfrak{a}]} together with the bounded subsets of {\mathfrak{a}}.

We claim that {\mathfrak{a}\in J}. Why? Well, assume {J} is a proper ideal on {\mathfrak{a}}, and let {D} be an ultrafilter on {\mathfrak{a}} disjoint to {J}. Since {J_{<\tau}[\mathfrak{a}]\subseteq J}, we know

\displaystyle  {\rm cf}\left(\prod\mathfrak{a}, <^*\right)<\tau\leq{\rm cf}\left(\prod\mathfrak{a}, <_D\right). \ \ \ \ \ (6)

But {D} is also disjoint to the bounded ideal, so Conclusion 3.2 on page 62 of Cardinal Arithmetic tells us, we have

\displaystyle  {\rm cf}\left(\prod\mathfrak{a}, <_D\right)\leq{\rm cf}\left(\prod\mathfrak{a}, <^*\right) \ \ \ \ \ (7)

and we have a contradiction.

So, it follows that there is a {\xi<\sup(\mathfrak{a})} such that {\mathfrak{a}\setminus\xi\in J_{<\tau}[\mathfrak{a}]}. But

\displaystyle  \mathfrak{a}\setminus\xi\in J_{<\tau}[\mathfrak{a}]\Longleftrightarrow\max{\rm pcf}(\mathfrak{a}\setminus\xi)<\tau, \ \ \ \ \ (8)

and so

\displaystyle  \max{\rm pcf}(\mathfrak{a}\setminus\xi)\leq{\rm cf}\left(\prod\mathfrak{a}, <^*\right) \ \ \ \ \ (9)

as desired. \Box

One obvious corollary of the above is the following result on scales.

Corollary 2 Suppose {\mu} is singular, and {\langle \mu_i:i<{\rm cf}(\mu)\rangle} is an increasing sequence of regular cardinals carrying a scale of length {\mu^+}. Then there is an {i_0<{\rm cf}(\mu)} such that {\langle \mu_i:i_0\leq i<{\rm cf}(\mu)\rangle} carries a scale {\langle f_\alpha:\alpha<\mu^+\rangle} with the property that for any {g\in\prod_{i_0\leq i<{\rm cf}(\mu)}\mu_i}, there is an {\alpha<\mu^+} such that {g(i)<f(i)} whenever {i_0\leq i<{\rm cf}(\mu)}.

The above is easy to arrange by brute force if {\mu} is of countable cofinality (as the increasing sequence of regular cardinals has {\max{\rm pcf}} equal to {\mu^+}) but I don’t see immediately that the brute force argument generalizes to {\mu} of uncountable cofinality, where {\max{\rm pcf}} could be large by virtue of an initial segment of {\langle \mu_i:i<{\rm cf}(\mu)\rangle}.

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A technical result (continued)

April 25, 2011 at 17:01 | Posted in Uncategorized | Leave a comment

The following proposition can be viewed as a continuation of the technical result from the last post. What I’m trying to do is to look at {{\rm pp}_{\Gamma(\theta,\sigma)}(\mu)} from different points of view so that we get a better picture of what’s going on.

[Warning: This post may be revised as I haven’t had enough time to proofread it properly]

Proposition 1 Let {\sigma\leq{\rm cf}(\mu)<\theta<\mu} be cardinals, with {\sigma} and {\theta} regular. Then the following two statements are equivalent for a cardinal {\kappa}:

  1. {\kappa\in{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu)}
  2. there is a progressive set of regular cardinals {\mathfrak{a}} cofinal in {\mu} such that
    • {|\mathfrak{a}|<\theta},
    • {\kappa\in{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})}, and
    • {\mathfrak{b}_{\kappa}[\mathfrak{a}]} is cofinal in {\mathfrak{a}}.

Proof: Assume first that {\kappa\in{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu)}, and let {\mathfrak{a}} and {I} witness this fact. These objections {\mathfrak{a}} and {I} put {\kappa} into {{\rm pcf}_{\sigma\text{-complete}}(\mu)}, so we need only verify that the generator {\mathfrak{b}_{\kappa}[\mathfrak{a}]} is unbounded in {\mathfrak{a}}. This follows easily, though, as {I} contains all the bounded subsets of {\mathfrak{a}}.

Now suppose that {\mathfrak{a}} is a progressive set of regular cardinal cofinal in {\mu} and of cardinality less than {\theta}, and that {\kappa} is in {{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})} with {\mathfrak{b}_{\kappa}[\mathfrak{a}]} unbounded in {\mathfrak{a}}. Let {J} be the {\sigma}-complete ideal generated by {J_{<\kappa}[\mathfrak{a}]} together with the bounded subsets of {\mathfrak{a}}.

Our assumptions imply that {\mathfrak{b}_\kappa[\mathfrak{a}]\notin J} (since {\kappa} is in {{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})} we know {\mathfrak{b}_{\kappa}[\mathfrak{a}]} cannot be covered by fewer than {\sigma} sets from {J_{<\kappa}[\mathfrak{a}]}). Basic pcf-theory tells us

\displaystyle  {\rm tcf}\left(\prod\mathfrak{b}_\kappa[\mathfrak{a}], <_{J_{<\kappa}[\mathfrak{a}]}\right)=\kappa, \ \ \ \ \ (1)

and it follows

\displaystyle  \kappa={\rm tcf}\left(\prod\mathfrak{b}_{\kappa}[\mathfrak{a}], <_J\right) \ \ \ \ \ (2)

as well.

But now we see that {\mathfrak{b}_\kappa[\mathfrak{a}]} and {J} witness that {\kappa} is a member of {{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu)}, as was claimed. \Box

The following corollary is an application of the above result; it is the main point of this post, and will be important in the course of later discussions.

Corollary 2 Let {\sigma\leq{\rm cf}(\mu)<\theta<\mu} be cardinals, with {\sigma} and {\theta} regular. If {\mathfrak{a}} is a progressive set of regular cardinals cofinal in {\mu} with {|\mathfrak{a}|<\theta}, then there is a {\xi<\mu} such that

\displaystyle  \sup{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a}\setminus\xi)\leq{\rm pp}_{\Gamma(\theta,\sigma)}(\mu). \ \ \ \ \ (3)

Proof: Let {\mathfrak{a}} be a set of cardinals as in the corollary, and define

\displaystyle  \tau=({\rm pp}_{\Gamma(\theta,\sigma)}(\mu))^+. \ \ \ \ \ (4)

Let {J} be the {\sigma}-complete ideal on {\mathfrak{a}} generated by {J_{<\tau}[\mathfrak{a}]} together with the bounded subsets of {\mathfrak{a}}. We claim that {J} is not a proper ideal, i.e.,

\displaystyle  \mathfrak{a}\in J. \ \ \ \ \ (5)

Suppose this fails. Let {\kappa\in{\rm pcf}(\mathfrak{a})} be least such that {\mathfrak{b}_\kappa[\mathfrak{a}]\notin J}. Then

\displaystyle  J_{<\kappa}[\mathfrak{a}]\subseteq J, \ \ \ \ \ (6)

and since {J} is {\sigma}-complete, it follows that

\displaystyle  \kappa\in{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a}). \ \ \ \ \ (7)

Note also that

\displaystyle  J_{<\tau}[\mathfrak{a}]\subseteq J\Longrightarrow \tau\leq \kappa, \ \ \ \ \ (8)

and hence

\displaystyle  {\rm pp}_{\Gamma(\theta,\sigma)}(\mu)<\kappa. \ \ \ \ \ (9)

Finally, {J} extends the bounded ideal on {\mathfrak{a}} and so we know that {\mathfrak{b}_{\kappa}[\mathfrak{a}]} must be unbounded in {\mathfrak{a}}. Given our proposition, this would imply

\displaystyle  \kappa\leq{\rm pp}_{\Gamma(\theta,\sigma)}(\mu), \ \ \ \ \ (10)

yielding a contradiction.

Thus, {J} is not a proper ideal. Decoding what this means, we find that there exists a {\xi<\mu} such that {\mathfrak{a}\setminus\xi} can be covered by a union of fewer than {\sigma} sets from {J_{<\tau}[\mathfrak{a}]}, and this implies

\displaystyle  {\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a}\setminus\xi)\subseteq\tau. \ \ \ \ \ (11)

But {{\rm pcf}_{\sigma\text{-complete}}(\mathfrak{a})} is a collection of cardinals, and given the definition of {\tau}, the conclusion of the corollary is immediate. \Box

A technical result

April 25, 2011 at 15:36 | Posted in Uncategorized | Leave a comment

This post is concerned with a cosmetic reformulation {{\rm pp}_{\Gamma(\theta,\sigma)}(\mu)}; I may add the proof later on when I get some time (mainly to make sure it’s all formulated exactly right), but the argument is a straightforward one.

Anyway, let us suppose {\sigma} and {\theta} are regular cardinals, and {\sigma\leq{\rm cf}(\mu)<\theta<\mu}.

We define the {{\rm PP}^0_{\Gamma(\theta,\sigma)}(\mu)} to consist of all cardinals {\kappa} for which there is a cardinal {\tau<\theta}, a {\sigma}-complete ideal {I} on {\tau}, and a sequence {\langle \mu_i:i<\tau\rangle} of regular cardinals below {\mu} such that

  • {\{i<\tau:\mu_i<\epsilon\}\in I} for each {\epsilon<\mu}, and
  • {\kappa={\rm tcf}\left(\prod_{i<\tau}\mu_i, <_I\right)}.

Notice that the above definition is tailored to guarantee

\displaystyle  {\rm pp}_{\Gamma(\theta,\sigma)}(\mu)=\sup{\rm PP}^0_{\Gamma(\theta,\sigma)}(\mu). \ \ \ \ \ (1)

On the other hand, the set {{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu)} consists of all cardinals {\kappa} such that

\displaystyle  \kappa={\rm tcf}\left(\prod\mathfrak{a}, <_I\right), \ \ \ \ \ (2)

where

  • {\mathfrak{a}} is a progressive set of regular cardinals cofinal in {\mu},
  • {|\mathfrak{a}|<\theta},
  • {I} is a {\sigma}-complete ideal on {\mathfrak{a}} containing the bounded subsets of {\mathfrak{a}}, and
  • {{\rm tcf}\left(\prod\mathfrak{a}, <_I\right)} exists.

Proposition 1 With notation as above, we have

\displaystyle  {\rm PP}^0_{\Gamma(\theta,\sigma)}(\mu)={\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu), \ \ \ \ \ (3)

and so

\displaystyle  {\rm pp}_{\Gamma(\theta,\sigma)}(\mu)=\sup{\rm PP}^1_{\Gamma(\theta,\sigma)}(\mu). \ \ \ \ \ (4)

We will tend to use the above result as a definition in the future.

Second Project: Introduction

April 8, 2011 at 12:42 | Posted in Uncategorized | Leave a comment

One of the goals I set for myself is to use this blog to work through some of the “cov vs. pp” material in Shelah’s oeuvre, so I want next to take a look at the so-called “cov vs. pp theorem”: Theorem 5.4 on page 87 of Cardinal Arithmetic. This proof seems to date from the era before the discovery of pcf generators, so the theorem cries out for a new proof. The argument form Lemma 3.5 on page 381 of Cardinal Arithmetic looks like it might do the job, so that’s what my second project will be — I will attempt to give a reasonable proof of the following:

Theorem 1 Suppose {\aleph_0<\sigma\leq{\rm cf}(\mu)<\theta<\mu} where {\sigma} and {\theta} are regular. Then

\displaystyle  {\rm cov}(\mu,\mu,\theta,\sigma)=^+{\rm pp}_{\Gamma(\theta,\sigma)}(\mu). \ \ \ \ \ (1)

There’s a lot of notation packed into the statement of this theorem, so we’ll use the rest of this post to define everything. Roughly, the theorem pins down how cardinals determined by pcf theory correspond to some natural measures of the powerset of a singular cardinal {\mu}. This connection is well-known in the case where {\mu} is not a fixed point; the above theorem holds even if {\mu} is a fixed point, but at the cost of assuming the cofinality of {\mu} is uncountable.

Definition 2 Let {\sigma}, {\theta}, and {\mu} be as above.

  1. {{\rm cov}(\mu,\mu,\theta,\sigma)} is the minimum cardinality of a set {\mathcal{P}\subseteq[\mu]^{<\mu}} such that any element of {[\mu]^{<\theta}} can be covered by a union of fewer than {\sigma} sets from {\mathcal{P}}.
  2. {{\rm pp}_{\Gamma(\theta,\sigma)}(\mu)} is defined to be the supremum of all cardinals {\kappa} for which there is a cardinal {\tau<\theta}, a {\sigma}-complete ideal {I} on {\tau}, and a sequence {\langle \mu_i:i<\tau\rangle} of regular cardinals below {\mu} such that
    • {\{i:\mu_i<\epsilon\}\in I} for each {\epsilon<\mu}, and
    • {\kappa={\rm tcf}(\prod_{i<\tau}\mu_i, <_I)}
  3. Notice that the pp number is the supremum of a set of cardinals, while the cov number is defined to be the cardinality of a certain collection of sets. When we say these two cardinals are {=^+}, we mean that if the covering number is regular, then it is realized as the true cofinality of some product as in the definition of the pp number.

We noted earlier that the theorem assumes the singular cardinal in question has uncountable cofinality. The question of whether this hypothesis is necessary is still open; eventually, we will look at this problem in more detail.

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