Monday, July 28, 2014

Towards simultaneous resolutions Part II

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

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

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


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 $Ch_N$

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


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)

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

[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

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

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

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

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.

Tuesday, July 01, 2014

Club obedient sequences I

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

We will follow Abraham-Magidor for a little bit.

Assume the following:

  • $A$ is a progressive set of regular cardinals
  • $\kappa$ is a regular cardinal with $|A|<\kappa<\min(A)$,
  • $\langle f_\xi:\xi<\delta\rangle$ is a sequence of functions in $\prod A$ with $\cf(\delta)=\kappa$.

Given a club $E\subseteq\delta$ of order-type $\kappa$, we define

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


Note the following:
  • $h_E$ is in $\prod A$ as $\kappa<\min(A)$,
  • if $C\subseteq E$ are clubs in $\delta$, then $h_C\leq h_E$ in $\prod A$, and
  • there is a club $C\subseteq \delta$ of order-type $\kappa$ such that $h_C\leq h_E$ for any club $E\subseteq \delta$.
The last point follows by an easy proof by contradiction. If there is no such club $C$, then we can build a decreasing sequence $\langle E_\alpha:\alpha<|A|^+\rangle$ of clubs in $\delta$ such that
  • $E_\alpha=\bigcap_{\beta<\alpha}E_\beta$ if $\alpha$ is a limit ordinal, and
  • $h_{E_{\alpha}}\nleq h_{E_{\alpha+1}}$
For each $\alpha<|A|^+$, we can find $a_\alpha\in A$ such that

$h_{E_{\alpha+1}}(a_\alpha)< h_{E_\alpha}(a_\alpha)$

and then stabilize to find a single $a\in A$ such that 

$h_{E_{\alpha+1}}(a)< h_{E_{\alpha}}(a)$

for unboundedly many indices $\alpha$.

Since the sequence of clubs $E_\alpha$ is decreasing, we get an infinite decreasing sequence of ordinals. Contradiction.


Now suppose  $J$ is an ideal over $A$, and the sequence $\langle f_\xi:\xi<\delta\rangle$ is $<_J$-increasing.  In this case, if $C$ is the "minimal" club as above, then 

$\xi\in C\Longrightarrow f_\xi\leq f_C$,

and hence $f_C$ is a $<_J$-upper bound for the sequence $\langle f_\xi:\xi<\delta\rangle$.

The function $f_C$ is called the minimal club-obedient bound of $\langle f_\xi:\xi<\delta\rangle$





Characteristic functions

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

Definition 1
Let $N$ be an elementary substructure of $H(\chi)$ for some sufficiently large regular $\chi$.
The characteristic function of $N$ is defined by

$\ch_N(\theta)=\sup(N\cap\theta)$

whenever $\theta$ is a regular cardinal.

We've seen in previous blog posts how important the characteristic functions of models are when connecting pcf theory to classical cardinal arithmetic.

We're going to be looking at obtaining refinements of the following cornerstone result of pcf theory:



Suppose $A$ is a progressive set of regular cardinals and let $\lambda=\max\pcf(A)$.   If

$$|A|<\kappa=\cf(\kappa)<\min(A)$$

then there is a family $F\subseteq\prod A$ of size $\lambda$ such that  $\ch_N\restr A\in F$ whenever $N$ is a $\kappa$-presentable elementary substructure of $H(\chi)$.


[This is essentially Lemma 3.4 on page 63 of Cardinal Arithmetic, but it is also an immediate consequence of Corollary 5.9 of Abraham-Magidor]


Wednesday, June 11, 2014

$\kappa$-presentable structures

Our topic in this post is Definition 5.1 in the Abraham-Magidor Handbook Article [AM]

Assume $\chi$ is some sufficiently large regular cardinal, and $\langle H(\chi),\in, <_\chi\rangle$ is as usual.

An elementary substructure $M$ of $H(\chi)$ is $\kappa$-presentable for a regular cardinal $\kappa$ if $M=\bigcup_{i<\kappa} M_i$ for some sequence $\langle M_i:i<\lambda\rangle$ where
  1. Each $M_i$ is an elementary substructure of $H(\chi)$,
  2. $i<j<\lambda\Longrightarrow M_i\subseteq M_j$, 
  3. if $\delta<\lambda$ is a limit, then $M_\delta=\bigcup_{i<\delta} M_i$,
  4. $M$ has cardinality $\kappa$,
  5. $\kappa+1\subseteq M$, and
  6. $M_i\in M_{i+1}$ for each $i<\kappa$ (so $M_i\in M_j$ whenever $i<j<\kappa$)
No assumption is made on the cardinality of $M_i$ for $i<\kappa$, so each $M_i$ could have cardinality $\kappa$, or cardinality less than $\kappa$.

This is clearly related to the notion of a structure being internally approachable of length $\kappa$; this concept suffices for the [AM] presentation, but I'm not certain it suffices for what Shelah is doing in [Sh:371].  This is one of the details that I'd like to clear up.


Back to the grind: [Sh:371] revisited

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

Summer is here again, and once again I've got some time to write about pcf theory.

I've got a backlog of things I want to present, so what I will do is pick up with a project from 2012, in which I was working to redo some of the work in [Sh:371] in light of Abraham and Magidor's excellent Handbook Article (denoted  [AM] in what follows).

The first thing that needs to be done is to clarify the relationship between the Corollary 5.9 in [AM] and Claims1.3 and 1.4 on page 316 of Cardinal Arithmetic [CA].

In particular, is the work leading to the proof of Corollary 5.9 in [AM] enough to push through the proof of Claim 1.4?  Shelah works with "minimally club-obedient sequences", while Abraham and Magidor assume the existence of generators and work with something weaker than minimally club-obedient sequences.

Our next few posts will lay out the relevant definitions and details surrounding the above (vague) discussion.