Tag Archives: GCH

Re: Paper and slides on indefiniteness of CH

Dear Sy,

This concerns the (clarified) notion of being strongly unreachable from the outline on HP you circulated yesterday. (M is strongly unreachable if for all proper inner models N of M, for all sufficiently large M-cardinals \kappa, \kappa^+ as computed in N is strictly less than \kappa^+ as computed in M).

Suppose V is strongly unreachable (and just relative to \Sigma_2-definable classes from parameters to make this explicitly first order). Then there are no measurable cardinals and either

  1. V = HOD and in fact V = K, (so GCH holds, and much more); or
  2. global-PD holds.

(K refers to a natural generalization of the usual core model—the union of “lower-parts of structures”—and this could be L of course. This K must be very L-like because of having no measurable cardinals. Global-PD is the assertion that PD holds in all set-generic extensions).

These are not mutually exclusive possibilities. But I actually do not know if (2) is possible. This leads to some rather subtle questions about correctness, for example suppose that M is countable and M is the minimum correct model of ZFC+global-PD. Must M be strongly unreachable? It seems likely that the answer should be yes, but this looks quite difficult (to me anyway).

(“Correct” here means that the set-generic extensions of M are projectively correct)

Regards,
Hugh

Re: Paper and slides on indefiniteness of CH

If certain technical conjectures are proved, then “V=ultimate L” is an attractive axiom.

The paper “Gödel’s program“, available on my webpage, gives an argument for that. I would guess the technical conjectures are true; in any case, large cardinals should decide them.

“V = Ultimate L” implies CH.

I don’t think we should care how hard it is to understand the statement of  “V = Ultimate L”. But in fact, one semester of graduate set theory is all you need to understand it. The conjectures are just that it implies GCH, and is consistent with the existence of supercompacts. One graduate semester is enough to understand the conjectures.

Best,
John