Wednesday, June 11, 2014

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.





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