# Re: Paper and slides on indefiniteness of CH

Dear Sy,

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.

Of course you do not see any virtue in “back to square one” conjectures. Fine, we have different views (yet again),

Why should your programme be required to make “make or break” conjectures, and what is so attractive about that?

I find it quite interesting if philosophical considerations lead to specific “make or break” conjectures. Especially if there is no obvious purely mathematical basis on which to make the conjecture. The HOD Conjecture is a good example. There is no purely mathematical reason (that I know of) to make that conjecture (that the HOD Hypothesis is provable from say ZFC + extendible).  It is a prediction from the Ultimate L scenario (just as is the (provability) of the $\Omega$ Conjecture).

Of course there is another reason for identifying such conjectures. They provide test questions for future progress. If one can refute from large cardinals that the $\Omega$ Conjecture holds then one refutes the Ultimate L Conjecture and moreover shows that there is a failure of inner model theory based on sequences of extenders.

One more question at this point: Suppose that Jack had succeeded in proving in ZFC that $0^\#$ does not exist. Would you infer from this that V = L is true? On what grounds?

Not necessarily. But I certainly would no longer declare as evident that V is not L. The question of V versus L would for me,  reopen.

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.

If the Ultimate L Conjecture is true (provable in ZFC + extendible) then V = Ultimate L becomes a serious possibility which (to me anyway) cannot just be dismissed as is now the possibility that V = L.

For me, the “validation” of V = Ultimate L will have to come from the insights V = Ultimate L gives for the hierarchy of large cardinals beyond supercompact. (These are the “other tests which will have to be passed”).

If that does not happen or if the genuine insights come from outer models of V = Ultimate L, or even from something entirely unrelated to Ultimate L, then for me the case for V = Ultimate L will weaken, possibly significantly.

On the other hand, if in the setting of V = Ultimate L, a whole new hierarchy of large cardinals is revealed, otherwise invisible, then things get interesting. Here it might be the Axiom $\textsf{I}0$, in but the context of V = Ultimate L, which could be key.

You will respond that is sheer speculation without foundation or solid evidence. It is sheer speculation. We shall see about the evidence.

Maybe it is time to try once again to simply agree that we disagree and wait for future mathematical developments before continuing this debate.

Regards,
Hugh