Thursday, June 13, 2013

Around Observation 5.5

\(
\DeclareMathOperator{\pp}{pp}
\def\pcf{\rm{pcf}}
\DeclareMathOperator{\cov}{cov}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\DeclareMathOperator{\PP}{pp}
\DeclareMathOperator{\Sk}{Sk}
\)

I apologize for the lack of context for this post, but I'm just trying to get some ideas down.


Proposition: Assume \(\cf(\lambda)<\theta=\cf(\theta)<\lambda\).    Then

$(1)\quad\quad \pp(\lambda)=\pp_\theta(\lambda)\Longrightarrow\cov(\pp(\lambda),\lambda,\theta^+,\theta)=\pp(\lambda).$

Proof:   We know $\lambda<\pp(\lambda)$ so $\lambda<\cov(\pp(\lambda),\lambda,\theta^+,\theta)$. Thus, by the cov vs. pp theorem (Theorem 5.4 on page 87 of The Book) it suffices to prove the following:

If $\lambda\leq\mu\leq\pp(\lambda)$ and $\cf(\mu)=\theta$, then $\pp_{\Gamma(\theta)}(\mu)\leq\pp(\lambda)$.

Clearly for such a $\mu$ we have $\pp_{\Gamma(\theta)}(\mu)\leq\pp(\mu)$, so we must show

$(2)\quad\quad \lambda\leq\mu\leq\pp(\lambda)\wedge\cf(\mu)=\theta\Longrightarrow\pp(\mu)\leq\pp(\lambda)$

But  we know the following:

  • $\pp(\mu)=\pp_\theta(\mu)$ by definition.
  • $\pp(\lambda)=\pp_\theta(\lambda)$ by assumption, and
  • $\pp_\theta(\mu)\leq\pp_\theta(\lambda)$ by "inverse monotonicity" of $\pp_\theta$  (Conclusion 2.3(2) on page 57 of The Book)
Putting all these together gives us what we need.   Q.E.D.

As far as I know, it is still unknown if the hypothesis of the preceding proposition can fail.  We do know the following:

  • If $\lambda$ is a strong limit and $\cf(\lambda)\leq\theta=\cf(\theta)<\lambda$, then $\pp(\lambda)=\pp_\theta(\lambda)$.
  • Actually, instead of assuming $\lambda$ is a strong limit, it suffices to assume that $\pp_\theta(\mu)<\lambda$ for all sufficiently large $\mu<\lambda$ with $\cf(\mu)\leq\theta$.  This is part of Corollary 1.6 on page 321.
  • We could also assume $\pp(\lambda)<\lambda^{+(\cf(\lambda))^+}$ and obtain the same conclusion.
What is the point?  In the first place, the argument given above allows us to deduce Observation 5.5 on page 404.  In the second place...well, I'll write the next piece out tomorrow.   

Wednesday, June 12, 2013

Final Inequality Finale

\(
\DeclareMathOperator{\pp}{pp}
\def\pcf{\rm{pcf}}
\DeclareMathOperator{\cov}{cov}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\DeclareMathOperator{\PP}{pp}
\DeclareMathOperator{\Sk}{Sk}
\)

Continuing our last post, we need to prove that $X$ is a subset of $M\cap\lambda$.  Let

$N:=N_{\sup(I)}.$

Note that $X\subseteq N\cap\lambda$, so it suffices to prove the following:

$N\cap\lambda\subseteq M.$


Suppose by way of contradiction that this fails,  and define

$(1)\quad\quad \gamma(*)=\min(N\cap\lambda\setminus M).$

Since $\mu+1\subseteq M$, we know

$(2)\quad\quad\mu<\gamma(*)<\lambda.$

Next, define
$(3)\quad\quad\beta(*)=\min(M\cap N\setminus\gamma(*)).$

Claim 1: $\beta(*)<\lambda$

Proof:  Clearly $\lambda$ and $\cf(\lambda)$ are elements of $M\cap N$.  Since $\cf(\lambda)<\mu$, we know $\cf(\lambda)+1\subseteq M$.  Since $\cf(\lambda)<\theta$  and $N\cap\theta$ is an initial segment of $\theta$, it follows that $\cf(\lambda)+1\subseteq N$ as well.  From this we deduce that $M\cap N$ contains a cofinal subset of $\lambda$, and hence $\beta(*)<\lambda$.   Q.E.D.

Putting things together, we have

$(4)\quad\quad\mu<\gamma(*)<\beta(*)<\lambda,$

and by definition of $\beta(*)$,

$(5)\quad\quad M\cap N\cap [\gamma(*),\beta(*))=\emptyset.$

We now analyze what sort of ordinal $\beta(*)$ might be.  In the first place, $\beta(*)$ must be a limit ordinal,  because otherwise its predecessor would violate (5).

Claim 2: $\beta(*)$ is a regular cardinal

Proof:  Suppose not, and let $\kappa(*)=\cf(\beta(*))$.  Since $\kappa(*)\in M\cap N$, it must be the case that

$(6)\quad\quad \kappa(*)<\gamma(*)<\beta(*).$

Let $f\in M\cap N$ be a strictly increasing map from $\kappa(*)$ onto a cofinal subset of $\beta(*)$, and let $\gamma<\kappa(*)$ be the least ordinal with  $\gamma(*)<f(\gamma)$.

We know $\gamma\in N$ (it is definable once we have $\gamma(*)$ available) but $\gamma$ cannot be an element of $M$ as otherwise we contradict (5).  This means that $\kappa(*)$ must be larger than $\mu$ as $N\cap\mu\subseteq M$.

Certainly $\kappa(*)$ is regular, so $\kappa(*)$ is a regular cardinal in $N\cap (\mu,\lambda)$.  Given our definition of $N$, we can find $i\in I$ such that $f$, $\gamma$, and $\kappa(*)$ are all in $N_i$.

Let $\mathfrak{a}=N_i\cap\lambda\cap\REG$.  We chose $\alpha_i$ and $\mu_i$ so that

$\kappa\in\mathfrak{a}\setminus\mu_i+1\Longrightarrow \sup(N_i\cap\kappa)<f_{\alpha_i}(\kappa).$

Since $i\in I$, it follows that $\mu_i\leq\mu$ and therefore

$(7)\quad\quad \gamma\leq\sup(N_i\cap\kappa(*))<f_{\alpha_i}(\kappa(*))<\kappa(*).$

Let $\delta(*)=f(f_{\alpha_i}(\kappa(*))$.  Since $f$ is strictly increasing, we know

$(8)\quad\quad \gamma(*)\leq f(\gamma)<\delta(*)<\beta(*).$

The ordinal $\delta(*)$ is in $N$ because everything needed to define it is available in $N_{i+1}$. However, since $i\in I$ it follows that the ordinal $\alpha_i$ is in $B$ and hence $f_{\alpha_i}\in M$.  Since $f$ and $\kappa(*)$ are also in $M$, we find $\delta(*)\in M$ as well.  But now we have contradicted (5).  Q.E.D.

Claim 3: $\beta(*)$ cannot be a regular cardinal

Proof:  This is almost the same argument as above.   Fix $i\in I$ with $\gamma(*)\in N_i$.  Then arguing as above we find

$(9)\quad\quad \gamma<\sup(N_i\cap\beta(*))<f_{\alpha_i}(\beta(*))<\beta(*)$

and   $f_{\alpha_i}(\beta(*))$ contradicts (5). Q.E.D.

Clearly Claim 2 and Claim 3 provide the needed contradiction and we have established what we need.

Final Inequality 2

\(
\DeclareMathOperator{\pp}{pp}
\def\pcf{\rm{pcf}}
\DeclareMathOperator{\cov}{cov}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\DeclareMathOperator{\PP}{pp}
\DeclareMathOperator{\Sk}{Sk}
\)

Picking up right where we left off, suppose $X\in [\lambda]^{<\theta}$.  We will find an acceptable model $M$ with $X\subseteq M\cap\lambda$.

Start by fixing a continuous increasing $\in$-chain  $\langle N_i:i<\theta\rangle$  of elementary submodels of $\mathfrak{A}$ satisfying:

  • $\langle N_j:j\leq i\rangle\in N_{i+1}$,
  • $|N_i|<\theta$
  • $N_i\cap\theta$ is an initial segment of $\theta$
  • $\{X,p\}\in N_0$

This is certainly possible as $\theta$ is a regular cardinal.   Note that our requirement that $N_i\cap\theta$ is an initial segment of $\theta$ has a lot of power:  for example, it implies that $X\subseteq N_0$, and that $N_j\subseteq N_i$ whenever $j<i<\theta$.


Given $i<\theta$, let us consider $\mathfrak{a}:= N_i\cap\lambda\cap\REG$.  We know the following facts:
  • $|\mathfrak{a}|=|N_i|<\theta$, and
  • $\mathfrak{a}$ is cofinal in $\lambda$ (as $\cf(\lambda)<\theta$).
This means that $\prod\mathfrak{a}$ is "covered" by our family $\mathcal{F}$:  if $g\in\prod\mathfrak{a}$ then there is an $f\in\mathcal{F}$ and $\mu<\lambda$ such that $g(\kappa)<f(\kappa)$ whenever $\kappa\in\mathfrak{a}\cap (\mu,\lambda)$.

We apply the observation in the previous paragraph to the characteristic function of $N_i$ on $\mathfrak{a}$. This gives us an ordinals $\alpha_i<\lambda(1)$ and $\mu_i<\lambda$ such that
$$\kappa\in\mathfrak{a}\cap (\mu_i,\lambda)\Longrightarrow \sup(N_i\cap \kappa)<f_{\alpha_i}(\kappa).$$
Moreover, we can choose $\alpha_i$ and $\mu_i$ in $N_{i+1}$, as the above argument can be done in the model $N_{i+1}$.  This is important, because it guarantees that the function $i\mapsto\alpha_i$ is one-to-one.

Since $\cf(\lambda)<\theta=\cf(\theta)<\lambda$, there is a $\mu<\lambda$ such that 
$$|\{i<\theta:\mu_i\leq\mu\}|=\theta.$$
Fix such a $\mu$ (without loss of generality greater than $\cf(\lambda)$), and define 
$$I^*:=\{i<\theta:\mu_i\leq\mu\}$$.

Now look at the set $A^*=\{\alpha_i:i\in I^*\}$.   This set is in $[\lambda(1)]^\theta$ because we made sure that the function $i\mapsto\alpha_i$ is one-to-one.  By the definition of $\mathcal{P}$, there is a set $B^*\in\mathcal{P}$ such that $A^*\cap B^*$ is infinite.


Let $\alpha$ be the supremum of the first $\omega$ elements of $A^*\cap B^*$, and find $I\subseteq I^*$
of order-type $\omega$ so that

  • $\{\alpha_i:i\in I\}$ is strictly increasing,
  • each $\alpha_i$ is in $A^*\cap B^*$, and
  • $\sup\{\alpha_i:i\in I\}=\alpha$
Now let $B=B^*\cap \alpha$, and define

$M=\Sk_\mathfrak{A}(B\cup p\cup\mu+1)$

Clearly $M$ is an acceptable model, so we need to prove that $X\subseteq M$.

To be continued...




Tuesday, June 11, 2013

Final Inequality 1

\(
\DeclareMathOperator{\pp}{pp}
\def\pcf{\rm{pcf}}
\DeclareMathOperator{\cov}{cov}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\DeclareMathOperator{\PP}{pp}
\DeclareMathOperator{\Sk}{Sk}
\)

Recall
  • $\cf(\lambda)<\theta=\cf(\theta)<\lambda$, 
  • $\lambda(1):=\cf_{<\theta}(\prod(\lambda\cap\REG), <_{J^\bd_\lambda})$, and
  • $\lambda(2)$ is the minimum cardinality of a family $\mathcal{P}\subseteq[\lambda(1)]^{<\lambda}$ such that for any $B\in[\lambda(1)]^\theta$ there is an $A\in\mathcal{P}$ with $A\cap B$ infinite.

We wish to prove $\cov(\lambda,\lambda,\theta,2)\leq\lambda(2)$.  This requires us to produce a family
$\mathcal{P}^*\subseteq[\lambda]^{<\lambda}$ such that
  • $|\mathcal{P}^*|\leq\lambda(2)$, and
  • for any $B\in [\lambda]^{<\theta}$ there is an $A\in\mathcal{P}^*$ with $B\subseteq A$.
We start by fixing a family $\mathcal{F}\subseteq \prod(\lambda\cap\REG)$ witnessing the definition of $\lambda(1)$, say $\mathcal{F}=\{f_\alpha:\alpha<\lambda(1)\},$ and let $\mathcal{P}\subseteq[\lambda]^{<\lambda}$ be as in the definition of $\lambda(2)$.





To find $\mathcal{P}^*$, let us assume $\chi$ is a regular cardinal much larger than any of the cardinals considered above, and let (as usual) $\mathfrak{A}$ denote the structure $\langle H(\chi),\in, <_\chi)$,where $<_\chi$ is some well-ordering of $H(\chi)$.  We will use $\Sk_\mathfrak{A}(B)$ to denote the Skolem hull of $B$ in the structure $\mathfrak{A}$.


Ad hoc definition:
Let us say that an elementary submodel $M$ of $\mathfrak{A}$ is acceptable if  
$M=\Sk_{\mathfrak{A}}(B\cup p\cup\mu+1)$
where
  • $B$ is an initial segment of some member of $\mathcal{P}$, 
  • $p=\{\theta,\lambda,\lambda(1),\lambda(2), \mathcal{F}, \mathcal{P},...\}$, and
  • $\mu<\lambda$
(We have been sloppy with $p$.  The intent is that $p$ encodes the parameters associated with the preceding discussion.)


If $M$ is an acceptable model, then $|M|<\lambda$.  Also note that there are at most $\lambda(2)=|\mathcal{P}|$ acceptable models.

We define $\mathcal{P}^*\subseteq[\lambda]^{<\lambda}$ to be all sets of the form $M\cap\lambda$ where $M$ is an acceptable model.

It remains to show that $\mathcal{P}^*$ has the required covering property.  We will do this in the next couple of posts.





Wednesday, June 05, 2013

Continuing

\(
\DeclareMathOperator{\pp}{pp}
\def\pcf{\rm{pcf}}
\DeclareMathOperator{\cov}{cov}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\DeclareMathOperator{\PP}{pp}
\)

Continuing our series of posts, let us recall


  1. $\cf(\lambda)<\theta=\cf(\theta)<\lambda$, 
  2. $\lambda(1):=\cf_{<\theta}(\prod(\lambda\cap\REG), <_{J^\bd_\lambda})$, and
  3. $\lambda(2)$ is the minimum cardinality of a family $\mathcal{P}\subseteq[\lambda(1)]^{<\lambda}$ such that for any $B\in[\lambda(1)]^\theta$ there is an $A\in\mathcal{P}$ with $A\cap B$ infinite.
I still owe you a proof that
$$\cov(\lambda,\lambda,\theta,2)\leq\lambda(2)\leq\cov(\lambda(1),\lambda,\theta^+,\theta).$$

The first of these inequalities requires a bit of machinery, so I'll save that for later and instead give the trivial proof of the second one and then make some comments on the relationship between $\lambda(1)$ and $\lambda(2)$.

Proposition: $\lambda(2)\leq\cov(\lambda(1),\lambda,\theta^+,\theta)$.

Let $\mathcal{P}\subseteq [\lambda(1)]^{<\lambda}$ have the property that any $B\in [\lambda(1)]^{\theta}$ is covered by a union of fewer than $\theta$ sets from $\mathcal{P}$.  Given such a $B$, it follows immediately that there is an $A\in \mathcal{P}$ for which $A\cap B$ is infinite.


Moving on to something a little more interesting, what can we say about the relationship between $\lambda(1)$ and $\lambda(2)$?  

What I would like to prove here is that if $\lambda(1)$ is strictly less than $\lambda(2)$, then there must be a cardinal of cofinality $\theta$ in the interval $(\lambda,\lambda(1)]$.  Said another way, we show
$$\lambda(1)<\lambda^{+\theta}\Longrightarrow \lambda(1)=\lambda(2).$$

There's a cheap proof of this that is easy only by way of quoting  Shelah's cov vs. pp theorem (Theorem 5.4 on page 87 of Cardinal Arithmetic).  This proof works by noting that 
$$\lambda(1)<\lambda(2)\Longrightarrow \lambda(1)<\cov(\lambda(1),\lambda,\theta^+,\theta),$$
and by the cov vs. pp theorem, this latter cardinal is equal to

$\lambda(1)+\sup\{\pp_{\Gamma(\theta^+,\theta)}(\lambda^*):\lambda\leq\lambda^*\leq\lambda(1)\wedge\cf(\lambda^*)=\theta\}.$

Since we know $\lambda(1)<\cov(\lambda(1),\lambda,\theta^+,\theta)$, we conclude there must be a singular cardinal of cofinality $\theta$ in the interval $[\lambda,\lambda(1)]$. Since $\cf(\lambda)<\theta$, the only way this can happen is if 
$$\lambda^{+\theta}\leq\lambda(1).$$

The other proof I have in mind is also easy, and it doesn't require any fancy pcf theory:

For a cardinal $\kappa$, let $P(\kappa)$ be the statement that there is a family $\mathcal{P}\subseteq[\kappa]^<\lambda$ of size $\kappa$ with the property that for any $B\in [\kappa]^\theta$, there is an $A\in\mathcal{P}$ with $A\cap B$ infinite.

Clearly $P(\lambda)$ holds by way of "initial segments" as $\cf(\lambda)<\theta$, and an easy induction shows that $P(\kappa)$ holds if $\lambda\leq\kappa<\lambda^{+\theta}$.  Thus, if $P(\lambda(1))$ fails it must be the case that $\lambda^{+\theta}\leq\lambda(1)$.

Why should we care about the above?  My interest here lies in the question of whether or not the cardinals $\lambda(1)$ and $\cov(\lambda,\lambda,\theta^+,2)$ are actually equal.  What we've shown above is that if they are not, then there's a sizable gap between $\lambda$ and $\cov(\lambda,\lambda,\theta, 2)$.

Cutting down the case where $\lambda$ has size cofinality $\omega$, the above shows:
$$\cf_{\aleph_0}(\prod(\lambda\cap\REG), <_{J^\bd_\lambda})<\cov(\lambda,\lambda,\aleph_1, 2)\Longrightarrow \lambda^{+\omega_1}\leq\cov(\lambda,\lambda,\aleph_1, 2).$$


Thursday, May 30, 2013

Easy Parts

\(
\DeclareMathOperator{\pp}{pp}
\def\pcf{\rm{pcf}}
\DeclareMathOperator{\cov}{cov}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\DeclareMathOperator{\PP}{pp}
\)

This is a direct continuation of the previous post.  I want to refresh your memory about some definitions, and then proceed to a proof of the easy parts of the chain of inequalities we were investigating.

We have assumed $\cf(\lambda)<\theta=\cf(\theta)<\lambda$.

Definition
  1. $\pp_{<\theta}(\lambda)$ is defined to be the supremum of all cardinals of the form $\cof(\prod\mathfrak{a}/\mathcal{U})$, where
    • $\mathfrak{a}$ is cofinal in $\lambda\cap\REG$,
    • $|\mathfrak{a}|<\theta$, and
    • $\mathcal{U}$ is an ultrafilter on $\mathfrak{a}$ disjoint to the ideal of bounded subsets of $\mathfrak{a}$.
  2. $\cf_{<\theta}(\prod(\lambda\cap\REG), <_{J^\bd_\lambda})$ is the minimum cardinality of a family of functions $\mathcal{F}\subseteq\prod(\lambda\cap\REG)$ such that for any cofinal $\mathfrak{a}\subseteq\lambda\cap\REG$ of cardinality $\lambda$ and any $g\in\prod\mathfrak{a}$, there is an $f\in \mathcal{F}$ such that $g<f\restr\mathfrak{a}$ modulo the bounded ideal on $\mathfrak{a}$
  3. $\cov(\lambda,\lambda,\theta,2)$ is the minimum cardinality of a family $\mathcal{P}\subseteq [\lambda]^{<\lambda}$ such that for any $B\in [\lambda]^{<\theta}$ there is a $A\in\mathcal{P}$ with $B\subseteq A$

We will show

$\pp_{<\theta}(\lambda)\leq\cf_{<\theta}(\prod(\lambda\cap\REG), <_{J^\bd_\lambda})\leq\cov(\lambda,\lambda,\theta,2).$

This gives us the first two inequalities in our chain, and both are easy.


For the first, suppose $\mathcal{F}$ is as described, and let $\mathfrak{a}$ and $\mathcal{U}$ be as in the definition of $\pp_{<\theta}(\lambda)$

The set $$\{\mathcal{f}\restr\mathfrak{a}:f\in\mathcal{F}\}$$
(or rather the set of equivalence classes of these functions modulo $\mathcal{U}$) is cofinal in $\prod\mathfrak{a}/\mathcal{U}$ because $U$ is disjoint to the bounded ideal on $\mathfrak{a}$ and the inequality follows immediately.


For the second, suppose $\mathcal{P}\subseteq[\lambda]^{<\lambda}$ is as described above.
For $A\in\mathcal{P}$, let
us define a function $f_A\in\prod(\lambda\cap \REG$) by
$$f_A(\kappa)=
\begin{cases}
\sup(A\cap \kappa)+1 & \text{if this is less than $\kappa$,}\\
0 &\text{otherwise}.
\end{cases}
$$

Does this collection of functions work? Suppose $B$ is a cofinal subset of $\lambda\cap\REG$ of cardinality $<\theta$, and let $g\in\prod B$. Choose $A\in\mathcal{P}$ such that $\ran(g)\subseteq A$.

If $\kappa\in B$ is sufficiently large (say $|A|<\kappa$), then
$$g(\kappa)\leq\sup(A\cap\kappa)<f_A(\kappa)<\kappa,$$
and we are done.

Tuesday, May 28, 2013

A chain of inequalities

\(
\def\pp{\rm{pp}}
\def\pcf{\rm{pcf}}
\def\cov{\rm{cov}}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\)

I spent the weekend reading the very last section of Shelah's Cardinal Arithmetic, and I want to try my hand at writing a reasonable exposition of it.

The main concern he has is the following problem::

Question:
Suppose $\lambda$ is a singular cardinal.  Is $\pp(\lambda)=\cov(\lambda,\lambda,(\cf(\lambda))^+,2)$?

He discusses this question quite a bit in the Analytical Guide for Cardinal Arithmetic.  The "interesting" case is when $\lambda$ is a fixed point of cofinality $\aleph_0$.

In this post, I want to gather together a few inequalities culled from the book and try to build a coherent picture of what's going on.  I'm going to leave some things undefined for now just to get the statement out.
Over my next few posts I'll try to define everything and provide proofs for each of the inequalities.

__________
Theorem
Suppose $\cf(\lambda)<\theta=\cf(\theta)<\lambda$.

Define
$$\lambda(1)=\cf_{<\theta}(\prod(\lambda\cap\REG),<_{J^\bd_{\lambda}})$$

and let $\lambda(2)$ be the minimum cardinality of a family $\mathcal{P}\subseteq[\lambda(1)]^{<\lambda}$ such that for any infinite $B\in[\lambda(1)]^{\theta}$, there is an $A\in \mathcal{P}$ with $A\cap B$ infinite.

Then

$$\pp_{<\theta}(\lambda)\leq\lambda(1)\leq\cov(\lambda,\lambda,\theta, 2)\leq\lambda(2)\leq\cov(\lambda(1),\lambda,\theta^+,\theta).$$
__________

The main case of interest is when $\theta=\cf(\lambda)=\aleph_0$.  In this situation, we get:
$$\lambda(1)=\cf_{\aleph_0}(\prod(\lambda,\cap\REG, <_{J^\bd_{\lambda}}),$$

and 

$$\lambda(2)=\min\{|\mathcal{P}|:\mathcal{P}\subseteq[\lambda(1)]^{<\lambda}\wedge(\forall B\in [\lambda(1)]^{\aleph_1})(\exists A\in\mathcal{P})[|A\cap B|\geq\aleph_0]\},$$

while the conclusion is

$$\pp(\lambda)\leq\lambda(1)\leq\cov(\lambda,\lambda,\aleph_1, 2)\leq\lambda(2)\leq\cov(\lambda(1),\lambda,\aleph_2,\aleph_1).$$




Friday, May 24, 2013

Note on [Sh:371] Claim 3.6

\(
\def\pp{\rm{pp}}
\def\pcf{\rm{pcf}}
\def\cov{\rm{cov}}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\)

I wanted to write out a proof of Claim 3.6 on page 340 of Cardinal Arithmetic for my own benefit.  Be warned that the proof will be clunky as I haven't tried to smooth it out yet.

Theorem (Shelah)
Assume the following:
  • $\mu$ is a singular cardinal.
  • $\cf(\mu)\leq\theta<\mu$, and
  • $\pp_\theta(\mu)<\mu^{+\theta^+}$.
Then $\pp_{<\mu}(\mu)=\pp_\theta(\mu)$.

Proof:

It suffices to prove that whenever $\mathfrak{a}$ is an unbounded subset of $\mu\cap\REG$ of cardinality less than $\mu$, then there is a $\sigma<\mu$ such that
$$\max\pcf(\mathfrak{a}\setminus\sigma)\leq\pp_\theta(\mu).$$
Note that if $|\mathfrak{a}|<\sigma<\mu$, then $\mathfrak{a}\setminus\sigma$ is progressive. This means that $\max\pcf(\mathfrak{a}\setminus\sigma)$ is defined for all sufficiently large $\sigma<\mu$, and furthermore we need only verify the above for progressive $\mathfrak{a}$.

Let $\mathfrak{a}$ be a progressive cofinal subset of $\mu\cap\REG$, and assume by way of contradiction that

$$\sigma<\mu\Longrightarrow \pp_\theta(\mu)<\max\pcf(\mathfrak{a}\setminus\sigma).$$

Let $\langle \mathfrak{b}_\lambda:\lambda\in\pcf(\mathfrak{a})\rangle$ be a generating sequence for $\pcf(\mathfrak{a})$, and let us define
$$X:=\pcf(\mathfrak{a})\cap (\mu, \pp_\theta(\mu)].$$

Given a finite subset $x$ of $X$ and $\sigma<\mu$, we know
$$(\mathfrak{a}\setminus\sigma)\setminus\bigcup_{\lambda\in x}\mathfrak{b}_\lambda\neq\emptyset$$
as otherwise we would have
$$\max\pcf(\mathfrak{a}\setminus\sigma)\leq\max\{\lambda:\lambda\in x\}\leq\pp_\theta(\mu),$$
contradicting our assumption.

Since we assume $\pp_\theta(\mu)<\mu^{+\theta^+}$, we know $|X|\leq\theta$, so let $\langle x_i:i<\theta\rangle$ enumerate all finite subsets of $X$ (possibly with repetition).  Also fix an increasing
sequence $\langle \mu_j:j<\cf(\mu)\rangle$ of cardinals cofinal in $\mu$ with $\theta<\mu_0$.

For each $i<\theta$ and $j<\cf(\mu)$, choose 

$$\theta(i, j)\in (\mathfrak{a}\setminus\mu_j)\setminus\bigcup_{\lambda\in x_i}\mathfrak{b}_\lambda$$

and define
$$\mathfrak{b}=\{\theta(i,j):i<\theta, j<\cf(\mu)\}.$$

Clearly $\mathfrak{b}$ satisfies the following:
  • $\mathfrak{b}\subseteq\mathfrak{a}$ (hence $\mathfrak{b}$ is a progressive set of regular cardinals)
  • $\mathfrak{b}$ is cofinal in $\mu$, and
  • $|\mathfrak{b}|\leq\theta$.
----------------------------
Claim 1: There is a $\sigma<\mu$ and a finite $x\subseteq X$ such that $\mathfrak{b}\setminus\sigma\subseteq\bigcup\{\mathfrak{b}_\lambda:\lambda\in x\}.$

Proof. Given the definition of $\pp_\theta(\mu)$, we know there is a $\sigma_0<\mu$ such that
$$\max\pcf(\mathfrak{b}\setminus\sigma_0)\leq\pp_\theta(\mu),$$
hence
$$\mathfrak{b}\setminus\sigma_0\in J_{\leq\pp_\theta(\mu)}[\mathfrak{a}].$$

This means we can find $\mathfrak{c}\in J_{<\mu}[\mathfrak{a}]$ and finite $x\subseteq X$ such that
$$\mathfrak{b}\setminus\sigma_0\subseteq \mathfrak{c}\cup\bigcup\{\mathfrak{b}_\lambda:\lambda\in X\}.$$

Since sets in $J_{<\mu}[\mathfrak{a}]$ are bounded below $\mu$, the result follows immediately.
--------------------------

We finish by noting that our construction of $\mathfrak{b}$ guarantees that
$\mathfrak{b}\setminus\bigcup\{\mathfrak{b}_\lambda:\lambda\in x\}$ is unbounded in $\mu$, and this contradicts the conclusion of Claim 1.








Tuesday, May 21, 2013

Let's try this again

This is a test post to see how well I can get Mathjax to work here.

When I started my blog on wordpress  a couple of years ago, I used it as an experiment in exposition.  The first project I worked on was an explication of the first theorem in Shelah's paper [Sh:430], which states

\(
\def\pp{\rm{pp}}
\def\cov{\rm{cov}}
\def\cf{\rm{cf}}
\def\REG{\sf {REG}}
\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\)

Theorem
If  $\mu$ is singular of uncountable cofinality and $\pp(\mu)=\mu^+$, then $\cov(\mu,\mu,\cf(\mu)^+,2)=\mu^+$.

(Here $\cov(\mu,\mu,\cf(\mu)^+,2)$ is the minimum cardinality of a family $\mathcal{F}\subseteq[\mu]^{<\mu}$ with the property that for any $B\in [\mu]^{\cf(\mu)}$ there is an $A\in\mathcal{F}$ such that $B\subseteq A$. This is potentially smaller than the cofinality of $[\mu]^{\cf(\mu)}$.)


I've been interested in seeing what occurs if we try to remove the "uncountable cofinality" assumption.  What I can show is that the statement

$$\cov(\mu,\mu,\cf(\mu)^+, 2)=\mu^+$$

is equivalent to the existence of a family $\langle f_\alpha:\alpha<\mu^+\rangle$  in $\prod(\mu\cap\REG)$ such that

  • $\alpha<\beta\Longrightarrow f_\alpha<f_\beta$ modulo the bounded ideal on $\mu\cap\REG$, and
  • whenever $\mathfrak{a}$ is a cofinal  subset of $\mu\cap\REG$ of cardinality $\cf(\mu)$, the restrictions $\langle f_\alpha\restr\mathfrak{a}:\alpha<\mu^+\rangle$ are cofinal in $\prod\mathfrak{a}/J^\bd[\mathfrak{a}]$
The family $\langle f_\alpha:\alpha<\mu^+\rangle$ could rightly be called a "master scale" for $\mu$:
the assumption $\pp(\mu)=\mu^+$ is equivalent to "every cofinal $\mathfrak{a}\subseteq\mu\cap\REG$ of cardinality $\cf(\mu)$ carries a scale'', while the ostensibly stronger assumption about $\cov$ gives us a single "master scale'' that works uniformly for all such $\mathfrak{a}$.

This concludes my test post.  This was certainly much much easier than at wordpress because of the ability to define macros!