Wednesday, July 20, 2016

On a theorem of Gitik and Shelah II: Blanket Assumptions


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

Assumptions for the next few posts are:


  • $\kappa$ weakly compact
  • $\mu>2^\kappa$ is singular of cofinality $\kappa$
  • $A\subseteq\mu$ is a set of regular cardinals
  • $A$ is unbounded in $\mu$ of cardinality $\kappa$
  • $J$ is a $\kappa$-complete ideal on $A$ extending the bounded ideal
  • $\lambda=\tcf(\prod A/ J)$
Since $J$ extends the bounded ideal, we may as well assume

  • $(2^\kappa)^+<\min(A)$ 
Of course this implies  $|A|<\min(A)$, and so without loss of generality $A$ is progressive.

Let $D$ be an ultrafilter on $A$ disjoint to $J$.  Then $\cf(\prod A/D)=\lambda$ and hence $\lambda\in \pcf(A)$.  Furthermore, if $B_\lambda[A]$ is a pcf generator for $\lambda$ then $B_\lambda[A]$ must be in $D$ and therefore unbounded in $A$.

It follows that $\tcf(\prod B_\lambda[A]/J)=\lambda$ as well, and so we may assume (by passing to $B_\lambda[A]$ if necessary) that

  • $\lambda=\max\pcf(A)$

Our assumption that $2^\kappa<\min(A)$ tells us that

(1)  $\pcf(\pcf(A))=\pcf(A)$, and

(2)  $|\pcf(A)|\leq 2^\kappa<\min(A)=\min(\pcf(A))$


and so $\pcf(A)$ has a transitive set of generators, that is, a sequence $\langle B_\theta:\theta\in\pcf(A)\rangle$ such that

  • $B_\theta\subseteq \pcf(A)$
  • $B_\theta$ is a generator for $\theta$ in $\pcf(\pcf(A))=\pcf(A)$, and
  • $\tau\in B_\theta\Longrightarrow B_\tau\subseteq B_\theta$.




Friday, July 01, 2016

On a theorem of Gitik and Shelah I

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

We are going to be looking at Theorem 2.1 of [GiSh:1013]  "Applications of pcf for mild large cardinals to elementary embeddings" Annals of Pure and Applied Logic 164 (2013) 855--865:

Theorem (Gitik and Shelah)
Assume $\kappa$ is weakly compact and $\mu>2^\kappa$ is singular of cofinality $\kappa$. If $\lambda$ is a regular cardinal satisfying $\mu<\lambda<\pp^+_{\Gamma(\kappa)}(\mu)$, then there is an increasing sequence $\langle\lambda_i:i<\kappa\rangle$ of regular cardinals converging to $\mu$ such that $\lambda=\tcf(\prod_{i<\kappa}\lambda_i/J^{\bd}_\kappa)$.


In this post, we will simply unpack what the theorem says.


The assumption  $\mu<\lambda<\pp^+_{\Gamma(\kappa)}(\mu)$ means the following:

There is a set of regular cardinals $A\subseteq\mu$ and a $\kappa$-complete ideal $J$ on $A$ such that

  • $A$ is unbounded in $\mu$
  • $|A|=\kappa$
  • $J$ extends the ideal $J^{\bd}[A]$ of bounded subsets of $A$, and
  • $\lambda=\tcf(\prod A/ J)$.
Comments:

1. $A$ need not have order-type $\kappa$, and $J$ can potentially be much larger than the ideal of bounded subsets of $A$.

2. Recall a poset has true cofinality $\lambda$ if it has a totally ordered cofinal subset of cardinality $\lambda$.


So our assumptions tell us $\lambda$ can be represented as $\tcf(\prod A/ J)$ for a certain sort of $A$ and $J$.

The conclusion of the theorem is that in fact $\lambda$  has a very nice representation as a true cofinality:

There is a set of regular cardinals $D\subseteq \mu$ (in fact, $D$ will be a subset of $\pcf(A)\cap\mu$) such that

  • $D$ is unbounded in $\mu$ of order-type $\kappa$, and
  • $\lambda=\tcf(\prod D/ J^{\bd}[D])$


















Thursday, June 30, 2016

On Weak Compactness II [edited]


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

Lingering a little on the material from the last post just for curiosity's sake:

Theorem 

The following are equivalent for an inaccessible cardinal $\kappa$:

1.  $\kappa$ is weakly compact.

2.  For every sequence of functions $\langle f_\alpha:\alpha<\kappa\rangle$ with $f_\alpha:\alpha\rightarrow\alpha$, there is an $f:\kappa\rightarrow\kappa$ such that for every $\alpha<\kappa$ there is a $\beta\geq\alpha$ with $f_\beta\restr\alpha= f\restr\alpha$.

3. For every sequence of sets $\langle A_\alpha:\alpha<\kappa\rangle$ with $A_\alpha\subseteq\alpha$, there is a set $A\subseteq\kappa$ such that for every $\alpha<\kappa$ there is a $\beta\geq\alpha$ with $A_\beta\cap\alpha = A\cap\alpha$.

4. For every $C$-sequence $\langle C_\alpha:\alpha<\kappa\rangle$ there is a closed unbounded $C\subseteq\kappa$ such that for all $\alpha<\kappa$ there is a $\beta\geq\alpha$ with $C_\beta\cap\alpha= C\cap\alpha$.


We'll work this out next time to make sure, but (1) implies (2) by Shelah's result used last time,  and (4) implies (1) by a result of Todorcevic  (Theorem 6.3.5 of his book "Walks on Ordinals and their Characteristics").


One thing to note is that subtle cardinals, weakly ineffable cardinals, and ineffable cardinals all are defined using something that looks like condition (3).  It might be worthwhile to see if the corresponding modifications of (2) and (4) give equivalent characterizations.

See  Cantor's Attic for more information.


--------------

Edit:  (2) follows from (3) immediately, as we just set $f_\alpha$ to be the characteristic function of $A_\alpha$, and then $f$ is the characteristic function of the $A$ we need.

(4) follows from (3) immediately, as it is easily seen that if we apply (3) to a $C$-sequence $\langle C_\alpha:\alpha<\kappa\rangle$ then the resulting $A$ must be closed.










On weak compactness


\(
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\def\cf{\rm{cf}}
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\def\restr{\upharpoonright}
\def\bd{\rm{bd}}
\def\subs{\subseteq}
\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\newcommand{\restr}{\upharpoonright}
\DeclareMathOperator{\ch}{Ch}
\DeclareMathOperator{\PP}{pp}
\DeclareMathOperator{\Sk}{Sk}
\)

We're firing the blog up again after a long hiatus.  Our immediate plan is to give a simplified proof of one of the main results from [GiSh:1013]  (Gitik-Shelah:  Applications of pcf for mild large cardinals to elementary embeddings).

We start with a lemma on weakly compact cardinals. I haven't seen this exact formulation before but certainly there aren't any new ideas involved.

Theorem

The following two conditions are equivalent for an uncountable cardinal $\kappa$:

1.  $\kappa$ is weakly compact.

2. If $c:[\kappa]^2\rightarrow\sigma$ with $\sigma<\kappa$, there is a function $f:\kappa\rightarrow\sigma$ and a set $H\subseteq\kappa$ of cardinality $\kappa$ such that for each $\alpha<\kappa$ we have $c(\alpha,\beta)=f(\alpha)$ for all sufficiently large $\beta\in H$.

Proof:

(2) implies (1)

Suppose $c:[\kappa]^2\rightarrow \{0, 1\}$, and let $f$ and $H$ be as in (2).   Define a function $g:\kappa\rightarrow\kappa$ by letting $g(\alpha)$ equal the least $\gamma<\kappa$ such that
\begin{equation}
\beta\in H \wedge \gamma\leq\beta\Longrightarrow c(\alpha,\beta)= f(\alpha).
\end{equation}

Thin out $H$ to $H_0$ of cardinality $\kappa$ so that
\begin{equation}
\alpha<\beta\text{ in }H\Longrightarrow g(\alpha)\leq\beta.
\end{equation}

This means that for $\alpha<\beta$ in $H_0$, $c(\alpha,\beta)=f(\alpha)$.  Now choose $\epsilon\in\{0,1\}$ so that $\{\alpha\in H_0:f(\alpha)=\epsilon\}$ has cardinality $\kappa$.  This set is homogeneous for the coloring $c$, and we conclude that $\kappa$ is weakly compact.



(1) implies (2)

We use a result due to Shelah (Theorem 1 of  [Sh:94] "Weakly compact cardinals: a combinatorial proof") which states that an inaccessible cardinal $\kappa$ is weakly compact if and only if for every family of functions $f_\alpha:\alpha\rightarrow\alpha$ (for $\alpha<\kappa$) there is a function $f:\kappa\rightarrow\kappa$ such that
$$(*) \qquad\qquad (\forall\alpha<\kappa)(\exists\beta<\kappa)[\alpha\leq\beta\wedge f_\beta\upharpoonright\alpha= f\restr\alpha].$$

Suppose $c:[\kappa]^2\rightarrow\sigma$ for some $\sigma<\kappa$.

 For $\alpha<\sigma$ we let $f_\alpha:\alpha\rightarrow\alpha$ be identically 0 (these $\alpha$ will be irrelevant), and for $\sigma\leq\alpha<\kappa$ we define $f_\alpha:\alpha\rightarrow\alpha$ by

$$f_\alpha(\gamma)=c(\gamma,\alpha).$$

Now fix a function $f:\kappa\rightarrow\kappa$ as in Shelah's theorem. We define an increasing sequence $\langle \beta_\xi:\xi<\kappa\rangle$ by the following recursion:


First, we set $\beta_0 =\sigma$.

Given $\langle \beta_\zeta:\zeta<\xi\rangle$, we define

$$\alpha_\xi=\sup\{\beta_\zeta:\zeta<\xi\}+1$$

and then use (*) to choose $\beta_\xi$ such that

$$\alpha_\xi\leq\beta_\xi\text{ and }f_{\beta_\xi}\restr\alpha_\xi = f\restr\alpha_\xi.$$


The preceding generates an increasing sequence $\langle\beta_\xi:\xi<\kappa\rangle$, and we note our construction guarantees that $c(\alpha,\beta_\xi)=f(\alpha)$ whenever $\alpha\leq\beta_\zeta$ and $\zeta<\xi<\kappa$.



In particular, if $\alpha<\kappa$ then $c(\alpha,\beta_\xi)= f(\alpha)$ for all sufficiently large $\xi<\kappa$, and $H:=\{\beta_\xi:\xi<\kappa\}$ is as required.









Monday, July 28, 2014

Towards simultaneous resolutions Part II

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

Suppose now that $A$ is $N$-admissible. Since

\(N\subseteq\bigcup_{\alpha<\kappa} M_\alpha\),

we know there is an $\alpha\in C$ with $A\in M_\alpha$.

The closed unbounded set $C[A]$ from our Technical Lemma is therefore also a member of $M_\alpha$
(it is definable from parameters available in $M_\alpha$), and since $\alpha = M_\alpha\cap\kappa$,
it follows that

$\alpha = M_\alpha\cap\kappa\in C[A]$

All we needed here was for $A$ to be a member of $M_\alpha$, so this holds for all sufficiently large $\alpha\in C$.

In particular, if $\alpha<\beta$ in $C$ and $A\in M_\alpha$ is $N$-admissible, then $b[A,\alpha,\beta]$
is a generator for $\lambda(A)$  as in our previous posts.  Also note that $b[A,\alpha,\beta]$ is definable from $A$ (which gives us $\lambda(A)$ and $\bar{f}^A$) together with $N_\alpha$ and $N_\beta$, so we have the following:

Assume $\alpha<\beta<\gamma$ in $C$, and $A\in M_\alpha$ is $N$-admissible.  Then

(1) \(b[A,\alpha,\beta]\) is a generator for $\lambda(A)$ in $\pcf(A)$,

(2) $a\in b[A,\alpha,\beta]\Longrightarrow \sup(N\cap a) = f^A_{\sup(N\cap\lambda(A))}(a)$, and

(3)  $b[A,\alpha,\beta]\in M_\gamma$.


As a transition to our next post, note that $A\setminus b[A,\alpha,\beta]$ is also $N$-admissible (it's going to be an element of $N_{\beta+1}$), and $A\setminus b[A,\alpha,\beta]$ is in $M_\gamma$, so we are set up to iterate things.


Towards simultaneous resolutions Part I

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


Background


Recall that in our discussions, $\kappa$ is a regular cardinal, and  $N$ is a $\kappa$-presentable elementary submodel of $H(\chi)$ for some sufficiently large regular $\chi$.

Previously we have fixed some progressive set of regular cardinals $A$ with $A\in N$ and $|A|<\kappa<\min(A)$.  In this post, we want to let $A$ vary while $N$ remains fixed.


Let us agree to call a set $A$ of regular cardinals $N$-admissible if it satisfies the following:

  • $A\in N$, and
  • $|A|<\kappa<\min(A)$.

Given a progressive set of regular cardinals $A$, we let  $\lambda(A)$ denote $\max\pcf(A)$, and let $\bar{f}^A$ be the $<_\chi$-least  minimally obedient universal sequence $\bar{f}^{A}$ for $\lambda(A)$. (Here $<_\chi$ is the well-ordering of $H(\chi)$ that is in the background whenever we talk about ``elementary submodels of $H(\chi)$''.)

Our technical lemma tells us the following:

Whenever $A$ is $N$-admissible, there is a club $C[A]\subseteq\kappa$ with the property that for any $\alpha<\beta$ in $C[A]$, the set

$b[A,\alpha,\beta]=\{a\in A: \sup(N_\alpha\cap a)< f^A_{sup(N\cap \lambda(A))}(a)\}$

sastifies the following:

(1) $b[A,\alpha,\beta]$ is a generator for $\lambda(A)$ in $\pcf(A)$, and

(2) $a\in b[A,\alpha,\beta]\Longrightarrow \sup(N\cap a) = f^A_{\sup(N\cap\lambda(A))}(a)$.


Moving forward, let $\langle M_\alpha:\alpha<\kappa\rangle$ be an $\in$-increasing and continuous
chain of elementary submodels of $H(\chi)$  with $N$ (and $\langle N_\xi:\xi<\kappa\rangle$) in $M_0$, such that for each $\alpha<\kappa$:

  • $|M_\alpha|<\kappa$
  • $M_\alpha\cap\kappa$ is an initial segment of $\kappa$, and 
  • $\langle M_\beta:\beta<\alpha\rangle\in M_{\alpha+1}$.
Now define

$C:=\{\delta<\kappa:\delta=M_\delta\cap \kappa\}.$

We know $C$ is closed and unbounded in $\kappa$.  In our next post, we'll look at some properties of this club $C$.

[Warning:  the requirements on the $M_\alpha$ may be edited as I move through the proof in the next couple of posts.]















Monday, July 07, 2014

Resolutions of $Ch_N$

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\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{\ch}{Ch}
\DeclareMathOperator{\PP}{pp}
\DeclareMathOperator{\Sk}{Sk}
\)


Assume the following:


  •  $A$ is a set of regular cardinals, 
  • $|A|<\kappa=\cf(\kappa)<\min(A)$,
  • $N$ is $\kappa$-presentable as witnessed by $\langle N_\alpha:\alpha<\kappa\rangle$ with $A\in N_0$,
  • for each $\lambda\in\pcf(A)$, $\bar{f}^\lambda=\langle f_\xi^\lambda:\xi<\lambda\rangle$ is a minimally obedient universal sequence for $\lambda$,
  • $\bar{F}=\{\bar{f}^\lambda:\lambda\in \pcf(A)\}\in N_0$.


Let us declare

$A_0:= A$,

$\lambda_0 := \max\pcf(A)$, and

$\gamma_0:=\sup(N\cap \lambda_0)$

Our technical lemma implies (among other things) that there is a set $B_{\lambda_0}\in N$ such that


  • $B_{\lambda_0}$ is a generator for $\lambda_0$ in $\pcf(A)$, and
  • $\ch_N\restr B_{\lambda_0} = f^{\lambda_0}_{\gamma_0}\restr B_{\lambda_0}$

Since $B_{\lambda_0}\in N$ [THIS IS CRITICAL!], we know the set 

$A_1:= A_0\setminus B_{\lambda_0}$

is in $N$ as well.  Furthermore, since $\lambda_0$ is $\max\pcf(A_0)$ and $B_{\lambda_0}$ is a generator for $\lambda_0$, we know

$\lambda_1:=\max\pcf(A_1)<\lambda_0$

If we set

$\gamma_1 : = \sup(N\cap\lambda_1)$

then we are essentially back in the same situation described at the start of this post (we may have to lose an initial segment of the sequence $\langle N_\alpha:\alpha<\kappa\rangle$ to guarantee $A_1\in N_0$, but this is trivial.)

Any iteration of this procedure must stop after finitely many steps (i.e., eventually $A_i=\emptyset$) as the sequence of $\lambda_i$ is decreasing.

Thus, we end up with a sequence $\lambda_0>\lambda_1>\dots>\lambda_n$ of elements of $\pcf(A)$, and
corresponding generators $\{B_{\lambda_i}:i\leq n\}$ from $N$ such that

(*)   $A=\bigcup_{i\leq n}B_{\lambda_i}$

and

(**) $\ch_N\restr B_{\lambda_i} = f^{\lambda_i}_{\gamma_i}\restr B_{\lambda_i}$ for each $i\leq n$.

By the obedience of $\bar{F}$, we know

$f_{\gamma_i}^{\lambda_i}\leq \ch_N\restr A$ for each $i\leq n$,

and therefore

(***)  $\ch_N\restr A = \max\{f^{\lambda_0}_{\gamma_0},\dots, f^{\lambda_n}_{\gamma_n}\}.$


We call the sequence $\langle b_{\lambda_i}:i\leq n\rangle$ an $\bar{F}$-resolution of $\ch_N\restr A$.  More formally, 
an $\bar{F}$-resolution of $\ch_N\restr A$ is a sequence $\langle B_i:i\leq n\rangle$ so that, letting $\lambda_i = \max\pcf(B_i)$, we have
  • the sets $B_i$ are pairwise disjoint,
  • each $B_i\in N$
  • $B_i$ is a generator for $\lambda_i$ in $\pcf(A)$,
  • $\lambda _{i+1}<\lambda_i$ for $i<n$, and
  • $\ch_{N}\restr B_i  =  f^{\lambda_i}_{\sup(N\cap\lambda_i)}\restr B_i$ for each $i\leq n$.
Our technical lemma implies that these resolutions exist; this is the conclusion of Corollary 5.9 of Abraham-Magidor.  

Why the extra bells and whistles in presentation?

Note that any resolution of $\ch_N$ is actually an element of $N$, as it is a finite sequence of elements of $N$.  What if instead of a single $A\in N$ we were given a collection $\langle A_\alpha:\alpha<\theta\rangle$ where each $A_\alpha\in N$ is a set of regular cardinals satisfying

$|A_\alpha|<\kappa<\min(A)$.

Certainly we can build a resolution for each $A_\alpha$ using the above procedure, and each of these resolutions is in $N$.  However,  we would like to have some way of guaranteeing that the entire sequence of resolutions is an element of $N$.  We will see that this can be done, assuming something a little stronger than $\kappa$-presentability for $N$.






Proof of Lemma, Part (B)

\(
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\def\cof{\rm{cof}}
\def\ran{\rm{ran}}
\DeclareMathOperator{\ch}{Ch}
\DeclareMathOperator{\PP}{pp}
\DeclareMathOperator{\Sk}{Sk}
\)

Let's take a stab at proving our technical lemma.

Recall $N$ is $\kappa$-presentable as witnessed by $\langle N_\alpha:\alpha<\kappa\rangle$, and let us define
  • $\gamma=\sup(N\cap\lambda)$
  • $\gamma_\alpha= \sup(N_\alpha\cap\lambda)$, and
  • for $\alpha<\beta<\kappa$, $b(\alpha,\beta)=\{a\in A:\sup(N_\alpha\cap a)\leq f_{\gamma_\beta}(a)\}$.
We are trying to produce a club $C\subseteq\kappa$ such that for any $\alpha<\beta$ in $C$ we have
  • $b(\alpha,\beta)$ is a generator for $\lambda$ in $\pcf(A)$, and
  • $\ch_N\restr b(\alpha,\beta)= f_\gamma\restr b(\alpha,\beta)$.
For the last statement to hold, we need only show

$b(\alpha,\beta)\subseteq \{a\in A: \sup(N\cap a)\leq f_\gamma(a)\}$.

Recall that in our last post we dealt with the set

$B^*=\{a\in A: \sup(N\cap a)\leq f_\gamma(a)\}$,

so the last requirement is equivalent to demanding $b(\alpha,\beta)\subseteq B^*$.

----------------------------------------
Observation 1:  For all sufficiently large $\alpha<\kappa$ we have

$f_\gamma(a)<\sup(N\cap a)$ if and only if $f_\gamma(a)<\sup(N_\alpha\cap a)$, and

therefore for all sufficiently large $\alpha<\kappa$ the set of $a\in A$ for which $\sup(N_\alpha\cap a)$ is less than or equal to $f_\gamma(a)$ is just the set $B^*$:

$(\forall^*\alpha<\kappa)[\{a\in A: \sup(N_\alpha\cap a)\leq f_\gamma(a)\}] = B^*$.
----------------------------------------

Next, let us fix a club $E\subseteq \gamma$ for which

$f_\gamma = \sup\{f_\xi:\xi\in E\}$.

By minimality, we may assume that $E\subseteq \{\gamma_\alpha:\alpha<\kappa\}$, and that the least $\alpha$ for which $\gamma_\alpha\in E$ is ``sufficiently large'' in the sense of Observation 1.

Now let $C\subseteq\kappa$ be the club such that $E = \{\gamma_\alpha:\alpha\in C\}$, so for each $a\in A$, we have

$f_\gamma(a) = \sup\{f_{\gamma_\alpha}(a):\alpha\in C\}$.

We will show that $C$ is as required, so let us fix $\alpha<\beta$ in $C$




--------------------------------------------------
Observation 2:  $b(\alpha,\beta)\subseteq B^*$
--------------------------------------------------

Proof:

By definition,

$b(\alpha,\beta)=\{a\in A: \sup(N_\alpha\cap a)\leq f_{\gamma_\beta}(a)\}$

Since $\beta\in C$, we know $\gamma_\beta\in E$ and $f_{\gamma_\beta}(a)\leq f_\gamma(a)$,  so

$b(\alpha,\beta)\subseteq\{a\in A:\sup(N_\alpha\cap a)\leq f_\gamma(a)\}$.

But $\alpha\in C$ is ``sufficiently large'', so the set on the right is just $B^*$, and we are done.

---------------------------------------------------
Observation 3: $b(\alpha,\beta)$ is a generator for $\lambda$.
---------------------------------------------------

Since $b(\alpha,\beta)\subseteq B^*$, it suffices to prove

$B^*\setminus b(\alpha,\beta) \in J_{<\lambda}$.

Let $B\in N_0$ be a pcf generator for $\lambda$.  As in the previous post, we know that the sequence $\langle f_\xi\restr B:\xi<\lambda\rangle$ is cofinal in $\prod B$ modulo $J_{<\lambda}$.

The function $\ch_{N_\alpha}\restr B$ is in $N_{\alpha + 1}\cap\prod B$, and so there is a $\xi\in N_{\alpha+1}\cap\lambda$ with

$\ch_{N_\alpha}\restr B \leq_{J_{<\lambda}} f_\xi\restr B$

Since $\alpha+1\leq\beta$, it follows that  $\xi\in N_{\beta}\cap\lambda$ hence $\xi<\gamma_\beta$ and

$\ch_{N_\alpha}\restr B \leq_{J_{<\lambda}} f_{\gamma_\beta}\restr B$

that is,

$\{a\in B:  f_{\gamma_\beta}(a)< \sup(N_\alpha\cap a)\}\in J_{<\lambda}$.

Since $B^*$ is also a generator for $\lambda$, we know $B^*\setminus B\in J_{<\lambda}$, and so

$\{a\in B^*: f_{\gamma_\beta}(a)< \sup(N_\alpha\cap a)\}\in J_{<\lambda}$.

But the above set is precisely $B^*\setminus b(\alpha,\beta)$, so we are done.



This completes the proof of our Technical Lemma.


Thursday, July 03, 2014

Proof of Lemma, Part (A)

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

[WARNING:  Edits may follow as I notice typos or add clarifications]

Let $N$, $A$, $\lambda$, and $\bar{f}$ be as in the last post, and define

$B^*=\{a\in A: \sup(N\cap a) = f_{\sup(N\cap\lambda)}(a)\}$.

Our first step in proving our technical lemma is the following:
_________________________________________________________________________
Goal 1:  $B^*$ is a generator for $\lambda$.
_________________________________________________________________________

Let $\gamma$ denote $\sup(N\cap\lambda)$, and let $B\in N$ be a generator for $\lambda$. (Such a $B$ exists in $N$ as $\lambda$ and $A$ are in $N$).  We will show that $B$ and $B^*$ are equal modulo the ideal $J_{<\lambda}$, and this will suffice.
-----------------------------------------------------
Stage 1: Some easy observations
-----------------------------------------------------
Note that $\gamma<\lambda$ as $N$ has cardinality $\kappa<\min(A)$, and if we define

$\gamma_\alpha:=\sup(N_\alpha\cap\lambda)$

then the sequence $\langle \gamma_\alpha:\alpha<\lambda\rangle$ enumerates a club of $\gamma$ of order-type $\kappa$.  Note as well that each $N_\alpha$ is in $N$, and this means each $\gamma_\alpha$ is as well.

Since $\bar{f}$ is minimally obedient at cofinality $\kappa$, there is a club $E\subseteq\gamma$ of
order-type $\kappa$ such that

$f_\gamma(a) = \sup\{f_\xi(a):\xi\in E\}$ for each $a\in A$.

This remains true for any club $D\subseteq E$ (by minimality), so there is a club $C\subseteq \kappa$ such that

$f_\gamma(a)=\sup\{f_{\gamma_\alpha}(a):\alpha\in C\}$

Notice that $A\subseteq N$ (as $A\in N$, $|A|<\kappa$, and $\kappa+1\subseteq N$). Since we know each $\gamma_\alpha$ is in $N$ as well, it follows that for each $a\in A$,  $f_\gamma(a)$ is the supremum of a set of ordinals in $N\cap a$, and therefore

$f_\gamma(a)\leq \sup (N\cap a)$ for each $a\in A$, that is

$f_\gamma \leq \ch_N$.

Thus, $B^*$ may be defined equivalently by:

$B^*=\{a\in A: \sup(N\cap a)\leq f_{\gamma}(a)\}$.


-----------------------------------------------------
Stage 2:  $B^*\setminus B\in J_{<\lambda}$
-----------------------------------------------------

Note that $A\setminus B$ is in $N$.  Since $B$ is a generator for $\lambda$, we know

$\lambda\notin\pcf(A\setminus B)$

and therefore $\prod(A\setminus B)/J_{<\lambda}$ is $\lambda^+$-directed.

This means that the sequence $\langle f_\xi\restr(A\setminus B):\xi<\lambda\rangle$ is bounded mod $J_{<\lambda}$, and so there is a function $h\in \prod A$ such that for each $\xi<\lambda$,

$\{a\in (A\setminus B): h(a)\leq f_\xi(a)\}\in J_{<\lambda}$.

In particular,

$\{a\in (A\setminus B): h(a)\leq f_\gamma(a)\}\in J_{<\lambda}$.

Now comes the important point:  we can assume that the function $h$ is in $N$ because $N$ sees everything needed to describe such an $h$.

Once we have such an $h\in N\cap\prod A$, it follows that $h(a)\leq \sup(N\cap a)$ for all $a\in A$,
and so

$\{a\in (A\setminus B): \sup(N\cap a)\leq f_\gamma(a)\}\in J_{<\lambda}$


But this says exactly that $B^*\setminus B\in J_{<\lambda}$, as required.

-----------------------------------------------------
Stage 3: $B\setminus B^*\in J_{<\lambda}$
-----------------------------------------------------

Note that $a\in B\setminus B^*$ means $a\in B$ and $f_\gamma(a)<\sup(N\cap a)$.  Our discussion in
Stage 1 tells us that there is an $\alpha(a)<\kappa$ such that $f_\gamma(a)<\sup(N_{\alpha(a)}\cap a)$.
Since $|A|<\kappa$, if follows that there is a single $\alpha<\kappa$ such that

$B\setminus B^* = \{a\in B: f_\gamma(a)<\sup(N_\alpha\cap a)\}$.

We will show that the set on the right is in $J_{<\lambda}$.

To do this, we need to use the fact that $\bar{f}$ is a universal sequence for $\lambda$, as this implies (see Theorem 4.13 of Abraham-Magidor) that $\langle f_\xi\restr B:\xi<\lambda\rangle$ is cofinal in $\prod B/J_{<\lambda}$.

Since $\ch_{N_\alpha}\restr B\in N$, this implies that there is a $\xi\in N\cap\lambda$ such that

$\ch_{N_\alpha}\restr B <_{J_{<\lambda}} f_\xi\restr B$.

Since $\xi<\gamma$, we know $f_\xi <_{J_{<\lambda}} f_\gamma$, and therefore

$\ch_{N_\alpha}\restr B <_{J_{<\lambda}} f_\gamma\restr B$.

But this means

$\{a\in B:  f_\gamma(a)\leq \sup (N_\alpha\cap a)\}\in J_{<\lambda}$

Since $B\setminus B^*\subseteq\{a\in B: f_\gamma(a)\leq \sup(N_\alpha\cap a)\}$, we are done.















Wednesday, July 02, 2014

Technical Lemma

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

In this post, I want to fulfill a promise by stating a lemma that is both a more technical version of Lemma 5.8 of Abraham-Magidor and a simplified version of something from Cardinal Arithmetic.   We are modifying the very nice work of Abraham and Magidor so that we can push through the result of Shelah that we will need.   The proof of the lemma will occupy subsequent posts.

Main Technical Lemma
Assume the following:

  •  $A$ is a progressive set of regular cardinals satisfying $|A|^+<\min(A)$
  •   $\kappa$ is a regular cardinal satisfying $|A|<\kappa<\min(A)$
  • $N$ is a $\kappa$-presentable elementary submodel of $H(\chi)$ for some sufficiently large regular $\chi$
  • $\langle N_\alpha:\alpha<\kappa\rangle$ witnesses the $\kappa$-presentability of $N$
  • $\lambda=\max\pcf(A)$
  • $\bar{f}=\langle f_\xi:\xi<\lambda\rangle$ is a minimally obedient universal sequence for $\lambda$
  • $\{\lambda, \bar{f}, A\}\in N_0$
Furthermore, for $\alpha<\beta<\kappa$ let us define

$b(\alpha,\beta)=\{a\in A: \sup(N_\alpha\cap a)\leq f_{\sup(N_\beta\cap\lambda)}(a)\}.$

Then there is a club $C\subseteq\kappa$ such that for any $\alpha<\beta$ in $C$,

  1. $b(\alpha,\beta)$ is a pcf-generator for $\lambda$, and
  2. $\ch_N\restr b(\alpha,\beta)=f_{\sup(N\cap\lambda)}\restr b(\alpha,\beta)$


This is a souped up version of Lemma 5.8 from Abraham-Magidor, and it is also a close relative of Claim 1.2 on page 315 of Cardinal Arithmetic.

Note for future reference that each set $b(\alpha,\beta)$ is in $N$, as they are definable from parameters available in $N$.


Club Obedient Sequences II

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

Continuing with some preliminary material from Abraham-Magidor:

Definition
Suppose that $\lambda\in\pcf(A)$.  A $<_{J_<\lambda}$-increasing sequence $\bar{f}=\langle f_\xi:\xi<\lambda\rangle$ in $\prod A$ is a universal sequence for $\lambda$ if $\bar{f}$ is cofinal in $\prod A/D$ for every ultrafilter $D$ on $A$ satisfying $\lambda = \cf(\prod A/D)$.


We discussed universal sequences on our old blog a bit during the series of posts on existence of pcf generators.  The important fact is that they exist whenever $A$ is a progressive set of regular cardinals and $\lambda\in\pcf(A)$.  [Theorem 4.2 on page 1180 of Abraham-Magidor.]

Definition
Suppose that $\lambda\in\pcf(A)$ and $\bar{f}=\langle f_\xi:\xi<\lambda\rangle$ is a universal sequence for $\lambda$.  Suppose that $\kappa$ is a regular cardinal with $|A|<\kappa<\min(A)$ (so we need $|A|^+<\min(A)$).  We say that $f$ is minimally obedient (at cofinality $\kappa$) if for every $\delta<\kappa$ of cofinality $\kappa$, $f_\delta$ is the minimal club-obedient bound of $\langle f_\xi:\xi<\delta\rangle$.
We say $\bar{f}$ is minimally obedient if $|A|^+<\min(A)$ and $\bar{f}$ is minimally obedient at cofinality $\kappa$ for every regular $\kappa$ satisfying $|A|<\kappa<\min(A)$.


If $|A|^+<\min(A)$ and $\lambda\in\pcf(A)$, then minimally obedient universal sequences exist, as one need only modify a given universal sequence in a straightforward way.  The details are spelled out on page 1192 of Abraham-Magidor.

One should view minimal obedience as a form of continuity for the sequence $\langle f_\xi:\xi:<\lambda\rangle$.  In fact, Shelah's name for this condition is $^b$-continuity.