List of PCF Hypotheses

November 17, 2011 at 11:09 | Posted in Uncategorized | 2 Comments

This post represents a short detour. I want to take a look at the last section of the paper [Sh:420]: “Advances in Cardinal Arithmetic”. The published version of this paper is a bit hard to track down, but Shelah’s archive contains an approximation.

Anyway, the last section commences with a list of hypotheses:

  1. {{\rm pp}(\lambda)=\lambda^+} for every singular {\lambda}. (The “Shelah Strong Hypothesis” or (SSH).)
  2. If {\mathfrak{a}} is a progressive set of regular cardinals, then {|{\rm pcf}(\mathfrak{a})|\leq|\mathfrak{a}|}.
  3. If {\mathfrak{a}} is a progressive set of regular cardinals, then {{\rm pcf}(\mathfrak{a})} does not have a weakly inaccessible accumulation point.
  4. For every {\lambda}, {\{\mu<\lambda:\mu\text{ singular and }{\rm pp}(\mu)\geq\lambda\}} is countable. (The “Shelah Weak Hypothesis” or (SWH).)
  5. For every {\lambda}, {\{\mu<\lambda:\mu\text{ singular of countable cofinality and }{\rm pp}(\mu)\geq\lambda\}} is countable.
  6. For every {\lambda}, {\{\mu<\lambda:\mu\text{ singular of uncountable cofinality and }{\rm pp}_{\Gamma({\rm cf}\mu)}(\mu)\geq\lambda\}} is finite.

Are the above hypotheses true? Well, the first of these is the only one whose negation is known to be consistent (relative to large cardinals), so potentially any of the others could be a theorem of ZFC.

Edit: See James’s comment for news on (2).

How are they related? Shelah points out the following:

  • (1) implies (2) implies (3)
  • (1) implies (4) implies (5)
  • (1) implies (6)
  • (5) and (6) together imply (4)
  • (4) implies (2)

I think I’d like to take a few posts to map out the proofs of the above, and maybe comment on what I know about the strength of the various hypotheses.

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2 Comments »

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  1. I was at a meeting in Singapore during the summer and I heard
    Moti Gitik give a talk “On a certain PCF configuration” with a proof that (2) is
    consistently false (from quite weak hypotheses, at the level of strong cardinals). I
    saw him again in Bristol back in September and the proof was still standing.

  2. Excellent, James! Thanks for the info!


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