On Mon, 15 Sep 2014, Koellner, Peter wrote:
You proposed “strong unreachability” as intrinsically justified on the
basis of the maximal iterative conception of set, writing: “It is
compelling that _unreachability (and strong unreachability) with
reflection is faithful to maximality but these criteria have not yet
been systematically investigated”.
Nothing “intrinsic” comes out of this programme until the spectrum of mathematical criteria that in some way mirror maximality have been studied, compared, unified and shown consistent. When I say “compellingly faithful to maximality” I do not mean “intrinsically justified”! I just mean that they should be regarded as criteria worth exploring when studying the different ways that maximality can be formulated mathematically, that is all.
At the end of your mail you raise the valid “bifurcation” issue. I cannot say with confidence that this won’t happen. It is too early to make a judgment on that.
This concerns the (clarified) notion of being strongly unreachable from the outline on HP you circulated yesterday.
Thanks for thinking about this,
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 as computed in N is strictly less than as computed in M).
Suppose V is strongly unreachable (and just relative to -definable classes from parameters to make this explicitly first order). Then there are no measurable cardinals and either
- V = HOD and in fact V = K, (so GCH holds, and much more); or
- 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)