Tag Archives: Projective sets under not-PD

Re: Paper and slides on indefiniteness of CH

Dear Sy,

My point is that the non-rigidity of HOD is a natural extrapolation of ZFC large cardinals into a new realm of strength.  I only reject it now because of the Ultimate-L Conjecture and its implication of the HOD Conjecture. It would be interesting to have an independent line which argues for the non-rigidity of HOD. This is the only reason I ask.

Please don’t confuse two things: I conjectured the rigidity of the Stable Core for purely mathematical reasons. I don’t see it as part of the HP. Indeed, I don’t see a clear argument that the nonrigidity of inner models follows from some form of maximality.

It would be nice to see one such reason (other than then non V-constructible one).

You seem to feel strongly that maximality entails some form of V is far from HOD. It would seem a natural corollary of this to conjecture that the HOD Conjecture is false, unless there is a compelling reason otherwise. If the HOD Conjecture is false then the most natural explanation would be the non-rigidity of HOD but of course there could be any number of other reasons.

In brief: HP considerations would seem to predict/suggest the failure of the HOD Conjecture. But you do not take this step. This is mysterious to me.

I am eager to see a well grounded argument for the HOD Conjecture which is independent of the Ultimate-L scenario.

Why am I so eager?  It would “break the symmetry” and for me anyway argue more strongly for the HOD Conjecture.

But I did answer your question by stating how I see things developing, what my conception of V would be, and the tests that need to be passed. You were not happy with the answer. I guess I have nothing else to add at this point since I am focused on a rather specific scenario.

That doesn’t answer the question: If you assert that we will know the truth value of CH, how do you account for the fact that we have many different forms of set-theoretic practice? Do you really think that one form (Ultimate-L perhaps) will “have virtues that will swamp all the others”, as Pen suggested?

Look, as I have stated repeatedly I see the subject of the model theory of ctm’s as separate from the study of V (but this is not to say that theorems in the mathematical study of ctm’s cannot have significant consequences for the study of V). I see nothing wrong with this view or the view that the practice you cite is really in the subject of ctm’s, however it is presented.

For your second question, If the tests are passed, then yes I do think that V = Ulitmate L will “swamp all the others” but only in regard to a conception of V, not with regard to the mathematics of ctm’s. There are a number of conjectures already which I think would argue for this. But we shall see (hopefully sooner rather than later).

Look: There is a rich theory about the projective sets in the context of not-PD (you yourself have proved difficult theorems in this area). There are a number of questions which remain open about the projective sets in the context of not-PD which seem very interesting and extremely difficult. But this does not argue against PD. PD is true.

Sample current open question: Suppose every projective set is Lebesgue measurable and has the property of Baire. Suppose every light-face projective set has a light-face projective uniformization. Does this imply PD? (Drop light-face and the implication is false by theorems of mine and Steel, replace projective by hyper projective and the implication holds even without the light-face restriction,  by a theorem of mine).

If the Ultimate L Conjecture is false then for me it is “back to square one” and I have no idea about an resolution to CH.