Tag Archives: Inner models and consistency

Re: Paper and slides on indefiniteness of CH

Dear Harvey,

I think it is a bit presumptuous of me to answer these questions in an email thread with such a large cc. I will however answer (1) since that is quite closely related to issues already discussed.

The short answer to (1) is actually no. Let me explain. If the Ultimate L Conjecture is true than I would certainly move supercompact to the safe zone on equal footing with PD. I do not currently place supercompact there.

What about huge cardinals? Ultimate L will be constructed as the inner model for exactly one supercompact cardinal. But unlike all other inner model constructions, this inner model will be universal for all large cardinal axioms we know of (large cardinals such as huge will be there if they occur in the parent universe within which Ultimate L is constructed but they play no role in the actual construction).

Therefore if the Ultimate L Conjecture is true (and the proof is by the current scenarios of course), inner model theory can no longer provide direct evidence for consistency since the large cardinals past supercompact play no role in the construction of Ultimate L. This is just as for L, the large cardinals compatible with L play no role in the construction of L.

So if the Ultimate L Conjecture is true then there is serious challenge. How does one justify the large cardinals beyond supercompact? My guess is that their justification will involve how they affect the structure of Ultimate L. For example, consider the following conjecture.

Conjecture: Suppose V = Ultimate L. Suppose \lambda is an uncountable cardinal such that the Axiom of Choice fails in L(P(\lambda)). Then there is a non-trivial elementary embedding j:V_{\lambda+1} \to V_{\lambda+1}.

If conjectures such as this are true then it seems very likely that for large cardinals beyond supercompact, their true natures etc., are really only revealed within the setting of V = Ultimate L and it is only in that unveiling that one is able to make the case for consistency. In this scenario, V = Ultimate L is not a limiting axiom at all, it is the axiom by which the true nature of the large cardinal hierarchy is finally uncovered.

But it is important to keep in mind, the Ultimate L Conjecture could be false. Indeed one could reasonably conjecture that it is the conception of a weak extender model which is not correct once one reaches the level of supercompact even though it seems to be correct below that level.

However at present and for me, the Ultimate L Conjecture is the keystone in a very tempting and compelling picture.