Re: Paper and slides on indefiniteness of CH

Dear Sy,

Let me respond to your questions about “choiceless large cardinal axioms”.

(1) First, let me first recall the context of your first comment: I was outlining a way in which “maximality” (understood in one way) could arguably lead to a violation of AC. The version of “maximality” that I was using was based on maximizing interpretability power. You pointed out that this form of maximality had to do with theories while the general form you considered had to do with sets. That is true. But (and this may not have been clear) I was imagining someone who used “maximality considerations” to parallel what people say about maximizing interpretability power but instead to make the claim that such theories were “intrinsically justified on the “maximal” iterative concept of set” (something that John would not maintain). I was imagining someone like Magidor who thought that Vopenka was so justified (in some sense at least) and went on further to make the claim that Reinhardt cardinals and Super Reinhardt cardinals and Berkeley cardinals and so on … were so justified. I can imagine someone making such a case, especially in light of the vagueness surrounding “maximality”. Anyway, that was the set-up of my rather fanciful scenario.

As you know I don’t even see how to get measurables out of the MIC. One could try to embarrass someone who thinks they can by suggesting that such a claim would lead them into inferring not-AC from Maximality via Reinhardt cardinals. But I don’t think that works, because one has to draw the line when ZFC speaks up: I like the axiom “not every real is generic over HOD” as a natural form of Maximality, but the reality is that ZFC won’t tolerate this. I don’t consider this to be a flaw in the idea that Maximality entails V not-equal HOD, but only a reality check, a mathematical constraint on adopting something that is conceptually appealing (a frequent occurrence in the HP).

Yes, I am aware of that. I was not saying that on your conception of “the “maximal” iterative conception of set” you should be able to get measurables. And I was not endorsing the view that “maximality” leads to Reinhardt cardinals and hence a failure of AC. Remember, I had just said that I don’t buy the arguments that “maximality” implies AC or that “maximality” implies not-AC. Indeed, I had said that I don’t even have a grip on “maximality”. It seems to me to a useful heuristic and nothing more than that. And given the vagueness of the notion and the different ways different people employ it, with little convergence (except along specific dimensions — as with forcing axioms), it seems that “anything goes”. I then went on to outline a fanciful story about choiceless large cardinals.

I confess: I was sort of trying to change the subject, simply because I find this topic fascinating.

(B) One could keep V = L by appealing to \Sigma^1_2 absoluteness and instead of asserting the existence of the large cardinal axiom assert that there is a countable transitive model of ZFC that satisfies the large cardinal axioms.

I take both of these to be examples of unnatural theories. In each case the new theory refers to the old theory it in such a way that it is not taken at face value. It is a case of “kicking away the ladder after one has climbed it”. Or, to change the metaphor, the new theory is parasitic on the old and if we kill the host then the new theory doesn’t have a chance.

Isn’t \text{AD}^{L(\mathbb R)} parasitic on AD?

No, \text{AD}^{L(\mathbb R)} is not parasitic on AD in the sense of “parasitic” as I was using it. Of course, \text{AD}^{L(\mathbb R)} involves the concept of determinacy. So does the statement “all open sets are determined” but that statement is not parasitic on AD. For every concept C we are almost always interested in figuring out whether C holds of a class of objects that falls short of everything (of which it makes sense to ask “does C hold of X?”). One exception (not a very interesting one) is when C is “is identical to itself”.

In the case of constructibility, when Goedel introduced L he was quite clear in saying that the construction of L (as he was using it) made sense only in the context where one presupposed all of the ordinals, as given by ZFC. If one took the predicativist considerations seriously one would not arrive at L (satisfying ZFC) but rather something like L_{\Gamma_0}. And when Jensen showed that various things (like \square) hold in L he was simply restricting his attention to a certain inner model and not presupposing that those things (like \square) really held.

Moreover, in the case where C is the concept of determinacy (which is at issue here) no one to my knowledge has ever maintained that all sets of reals are determined (any more than when dealing with the concept of Lebesgue measurability anyone has maintained that all sets of reals are Lebesgue measurable).

In the very paper — Mycielski and Steinhaus (1962) — in which AD was introduced the authors wrote:

“Our axiom can be considered as a restriction of the classical notion of a set leading to a smaller universum, say of determined sets, which reflect some physical intuitions which are not fulfilled by the classical sets (e.g. paradoxical decompositions of the sphere are eliminated by [AD]). Our axiom could be considered as an axiom added to the classical set theory claiming the existence of a class of sets satisfying [AD] and the classical axioms without the axiom of choice.”

Shortly thereafter, in 1964, proposed that the axiom held in an inner model: “a subclass of the class of all sets with the same membership relation. It would be still more pleasant if such a submodel contains all the real numbers.” Solovay and Takeuti pointed out that the obvious inner model is L(\mathbb R). And, in the late 1960s Solovay conjectured that under suitable large cardinal assumptions — in particular, the existence of a supercomact — that AD actually holds in L(\mathbb R).

As Kanamori puts it: “ZF + AD was never widely entertained as a serious alternative to ZFC, and increasingly from the early 1970’s onward consequences of ZF + AD were regarded as what holds in L(\mathbb R) assuming AD^{L(\mathbb R)}.”

The case of Reinhardt cardinals is entirely different. This axiom was proposed.

The statement “I am such and such a class-size forcing extension (satisfying ZFC) of an inner model of (say) a Super Reinhardt cardinal” (which is based on a fairly deep theorem) is parasitic on the inner model in a way that “all open sets are determined” (or “all projective sets are determined” or “all sets of reals in L(\mathbb R) are determined”) is NOT parasitic on the statement “all sets of reals are determined”.

(3) You asked about whether (from a mathematical point of view) such a move was even possible in the case of choiceless large cardinal axioms. The situation is delicate.

An old result of Woodin is that if there is a Super Reinhardt cardinal then one can force AC via a class-size forcing. A modification of this result shows that if any choiceless large cardinal axiom is consistent with an extendible cardinal then one can force AC over an model of the choiceless large cardinal and an extendible.

The question then is whether choiceless large cardinals are consistent with an extendible. In general, if the HOD Conjecture holds then choiceless large cardinals are not consistent with an extendible cardinal. For example, this is true of Berkeley cardinals. If the HOD Conjecture fails then a whole new world — a very bizarre world — opens up. That’s what we are investigating.

As a journal editor I’ve had to deal with papers that analyse some family of new large cardinal notions for their own sake and the question comes up each time: What is the point of this if the new notions are not known to be useful for something else in set theory?

Are these choiceless LC axioms useful for set theory with AC? (Pen asked a related question.) If not, then the most generous interpretation of this study is that it prepares mathematical tools for the day when such applications are found.

Let me first describe (in my letter to Pen) why I am interested in choiceless large cardinal axioms. To anticipate that let me say here that, yes, I am interested in an “application” of these axioms: The goal is to show that they imply 0 = 1.

Best,
Peter

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