Tag Archives: Sharps

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.