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.