# Re: Paper and slides on indefiniteness of CH

Dear Harvey,

Ok, I will add some comments to my response. Below is simply how I currently see things. It is obvious based on this account that an inconsistency in PD would render this picture completely vacuous and so I for one would have to start all over in trying to understand V. But that road would be much harder given the lessons of the collapse caused by the inconsistency of PD. How could one (i.e. me) be at all convinced that the intuitions behind ZFC are not similarly flawed?

I want to emphasize that what I describe below is just my admittedly very optimistic view. I am not advocating a program of discovery or anything like that. I am also not arguing for this view here. I am just describing how I see things now. (But that noted, there are rather specific conjectures which if proved, I think would argue strongly for this view. And if these conjectures are false then I will have to alter my view.)

This view is based on a substantial number of key theorems which have been proved (and not just by me) over the last several decades.

Starting with the conception of V as given by the ZFC axioms, there is a natural expansion of the conception along the following lines.

The Jensen Covering Lemma argues for $0^\#$ and introduces a horizontal maximality notion. This is the first line and gives sharps for all sets. This in turn activates a second line, determinacy principles.

The core model induction now gets under way and one is quickly led to PD and $\text{AD}^{L(\mathbb R)}$, and reaches the stage where one has iterable inner models with a proper class of Woodin cardinals. This is all driven by the horizontal maximality principle (roughly, if there is no iterable inner model with a proper class of Woodin cardinals then there is a generalization of L relative to which V is close at all large enough cardinals and with no sharp etc.).

Adding the hypothesis that there is a proper class of Woodin cardinals, one can now directly define the maximum extension of the projective sets and develop the basic theory of these sets. This is the collection of universally Baire sets (which has an elementary definition). The important point here is that unlike the definitions of the projective sets, this collection is not defined from below. (There is a much more technical definition one can give without assuming the existence of a proper class of Woodin cardinals).

Continuing, one is led to degrees of supercompactness (the details here are now based on quite a number of conjectures, but let’s ignore that).

Also a third line is activated now. This is the generalization of determinacy from $L(\mathbb R) = L(P(\omega))$ to the level of $L(P(\lambda))$ for suitable $\lambda > \omega$. These $\lambda$ are where the Axiom I0 holds. This axiom is among the strongest large cardinal axioms we currently know of which are relatively consistent with the Axiom of Choice. There are many examples of rather remarkable parallels between $L(\mathbb R)$ in the context that AD holds in $L(\mathbb R)$, and $L(P(\lambda))$ in the context that the Axiom I0 holds at $\lambda$.

Now things start accelerating. One is quickly led to the theorem that the existence of the generalization of L to the level of exactly one supercompact cardinal is where the expansion driven by the horizontal maximality principles stops. This inner model cannot have sharp and is provably close to V (if it exists in the form of a weak extender model for supercompactness). So the line (based on horizontal maximality) necessarily stops (if this inner model exists) and one is left with vertical maximality and the third line (based on I0-like axioms).

One is also led by consideration of the universally Baire sets to the formulation of the axiom that V = Ultimate L and the Ultimate L Conjecture. The latter conjecture if true confirms that the line driven by horizontal maximality principles ceases. Let’s assume the Ultimate L Conjecture is true.

Now comes (really extreme) sheer speculation. The vertical expansion continues, driven by the consequences for Ultimate L of the existence of large cardinals within Ultimate L.

By the universality theorem, there must exist $\lambda$ where the Axiom I0 holds in Ultimate L. Consider for example the least such cardinal in Ultimate L. The corresponding $L(P(\lambda))$ must have a canonical theory where of course I am referring to the $L(P(\lambda))$ of Ultimat L.

It has been known for quite some time that if the Axiom I0 holds at a given $\lambda$ then the detailed structure theory of $L(P(\lambda)) = L(V_{\lambda+1})$ above $\lambda$ can be severely affected by forcing with partial orders of size less than $\lambda$. But these extensions must preserve that Axiom I0 holds at $\lambda$. So there are natural features of $L(P(\lambda))$ above $\lambda$ which are quite fragile relative to forcing.

Thus unlike the case of $L(\mathbb R)$ where AD gives “complete information”, for $L(P(\lambda))$ one seems to need two things: First the relevant generalization of AD which arguably is provided by Axiom I0 and second, the correct theory of $V_\lambda$. The speculation is that V = Ultimate L provides the latter.

The key question will be: Does the global structure theory of $L(P(\lambda))$, as given in the context of the Axiom I0 and V = Ultimate L, imply that V = Ultimate L must hold in $V_\lambda$?

If this convergence happens at $\lambda$ and the structure theory is at all “natural” then at least for me this would absolutely confirm that V = Ultimate L.

Aside: This is not an entirely unreasonable possibility. The are quite a number of theorems now which show that $\text{AD}^{L(\mathbb R)}$ follows from its most basic consequences.

For example it follows from just all sets are Lebesgue measurable, have the property of Baire, and uniformization (by functions in $L(\mathbb R))$ for the sets $A \subset \mathbb R \times \mathbb R$ which are $\Sigma_1$-definable in $L(\mathbb R)$ from parameter $\mathbb R$. This is essentially the maximum amount of uniformization which can hold in $L(\mathbb R)$ without yielding the Axiom of Choice.

Thus for $L(\mathbb R)$, the entire global structure theory, i.e. that given by $\text{AD}^{L(\mathbb R)}$, is implied by a small number of its fundamental consequences.

Regards,
Hugh

# Re: Paper and slides on indefiniteness of CH

I see the importance you are attaching to your Ultimate L Conjecture – particularly getting a proof “by the current scenarios of course”. Care to make rough probabilistic predictions on when you will prove it “by the current scenarios of course”? Until then, your point of view seems to be that statements like Con(HUGE) are wide open, and you currently are not willing to declare any confidence in them.

Fascinating as this is, I think people here might be even more interested in the implications Con(EFA) arrows Con(PA) arrows Con(Z) arrows Con(ZFC) arrows Con(ZFC + measurable) arrows Con(ZFC + PD). Maybe you can comment on at least one of these arrows? — or maybe Peter Koellner?

Based on all my experience to date I have a conception of V in which PD holds. Based on that conception it is impossible for PD to be inconsistent. But that conception may be a false conception.

If it is a false conception then I do not have a conception of V to fall back on except for the naive conception I had when I first was exposed to set theory. This is why for me, if ZFC+PD is inconsistent I think that ZFC is suspect. This is not to say that I cannot or will not rebuild my conception to that of V which satisfies ZFC etc. But I would need to understand how all the intuitions etc. that led me to a conception V with PD went so wrong.

My question to those who feel V is just the integers (and maybe just a bit more) is: How do they assess that Con ZFC+PD is relevant to their conception? The conception of the set theoretic V with or without PD are very different conceptions with deep structural differences. I just do not see that happening yet to anywhere near the same degree in the case where the conception V  is just the integers. But as your work suggests this could well change.

But even so, somehow a structural divergence alone does not seem enough (to declare that Con ZFC+PD is an indispensable part of that conception).  Who knows, maybe there is an arithmetically based strongly motivated hierarchy of “large cardinals” and Con PD matches something there.

If one’s conception of V is the integers and one is never compelled to declare Con ZFC+PD as true based on whatever methodology one is using to refine that conception, then it seems to me that the only plausible conjecture one can make is that ZFC+PD is inconsistent. This was the cryptic point behind item (2) in my message which inspired your press releases.

To those still reading and to those who have participated, I would like to express my appreciation. But I really feel it is time to conclude my participation in this email thread. Classes are about to begin and I shall have to return to Cambridge shortly.

Regards,
Hugh