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.

??? My question has nothing to do with ctm’s! It has nothing to do with the HP either (which I repeat can proceed perfectly well without discussing ctm’s anyway). I was referring to the many different forms of set-theoretic practice which disagree with each other on basic questions like CH. How do you assign a truth value to CH in light of this fact?

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).

Here come the irrelevant ctm’s again. But you do say that V = Ultimate L will “swamp all the others”, so perhaps that is your answer to my question. Now do you really believe that? You suggested that Forcing Axioms can somehow be “part of the picture” even under V = Ultimate L, but that surely doesn’t mean that Forcing Axioms are false and Ultimate L is true.

Pen and Peter, can you please help here? Pen hit me very hard for developing what could be regarded as “Sy’s personal theory of truth” and it seems to me that we now have “Hugh’s personal theory of truth”, i.e., when Hugh develops a powerful piece of set theory he wants to declare it as “true” and wants us all to believe that. This goes far beyond Thin Realism, it goes to what Hugh calls a “conception of V” which far exceeds what you can read off from set-theoretic practice in its many different forms. Another example of this is Hugh’s claim that large cardinal existence is justified by large cardinal consistency; what notion of “truth” is this, if not “Hugh’s personal theory of truth”?

Pen’s Thin Realism provides a grounding for Type 1 truth. Mathematical practice outside of set theory provides a grounding for Type 2 truth. Out intuitions about the maximality of V in height and width provide a grounding for Type 3 truth. How is Hugh’s personal theory of truth grounded?

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.

I want to know what you mean when you say “PD is true”. Is it true because you want it to be true? Is it true because ALL forms of good set theory imply PD? I have already challenged, in my view successfully, the claim that all sufficiently strong natural theories imply it; so what is the basis for saying that PD is true?

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.

I see no virtue in “back to square one” conjectures. In the HP the whole point is to put out maximality criteria and test them; it is foolish to make conjectures without doing the mathematics. Why should your programme be required to make “make or break” conjectures, and what is so attractive about that? As I understand the way Pen would put it, it all comes down to “good set theory” for your programme, and for that we need only see what comes out of your programme and not subject it to “death-defying” tests.

One more question at this point: Suppose that Jack had succeeded in proving in ZFC that does not exist. Would you infer from this that V = L is true? On what grounds? Your V = Ultimate L programme (apologies if I misunderstand it) sounds very much like saying that Ultimate L is provably close to V so we might as well just take V = Ultimate L to be true. If I haven’t misunderstood then I find this very dubious indeed. As Pen would say, axioms which restrict set-existence are never a good idea.

Best,

Sy