Dear Peter,

Thanks for the very interesting mail. Some comments below.

On Mon, 13 Oct 2014, Koellner, Peter wrote:

Dear All:

Here are some questions and comments on the question of AC and “maximality”.

QUESTIONS

Sy: In response to the result Hugh mentioned — which bears on Choiceless-HP — you wrote that the existence of supercompacts was “still unclear”. In your letter of August 8 to Pen you wrote: “I remain faithful to the extrinsically-confirmed fact that large cardinal axioms are consistent.” From that I assume that you take it to be extrinsically confirmed that supercompact cardinals are consistent. But here you say that the question of their existence is unclear. I would be very interested to hear what you have to say about what it would take to achieve the leap from consistency to existence. Would it be to show that such principles are intrinsically justified on the basis of the “maximal” iterative conception of set? If so do you have in mind any candidates for doing that? (I am aware of the fact that while, e.g., is consistent with all large cardinals it does not imply them.)

One can get the existence of small large cardinals from “maximality in height” (an extension of reflection). At the moment the criteria that combine “maximality in height” with “maximality in width” either contradict large large cardinals or are just consistent with them, without implying their existence. For a while, Radek and I thought that “omniscience” would engender Ramsey cardinals, but there were 2 problems with that: It is not clear if “omniscience” is derivable from the MIC and it turned out that it doesn’t need Ramseys after all! The other possibility regarded stronger forms of reflection, such as what Victoria Marshall was talking about (and which I at one point called “Magidor” or “embedding” reflection). But I ended up seeing serious problems with that. So in answer to your question: I don’t have a good suggestion at this point for obtaining the existence of measurable cardinals from the MIC!

Considerations of maximality have certainly served as a useful heuristic that has led to some wonderful mathematics, and, in some cases, to a unified program, as in the case of forcing axioms (which can be construed as generalizations of the Baire Category Theorem). But it seems (to me at least) that the notion of “maximality” is a rather vague notion, one that has many dimensions, depending on what exactly it is that one is trying to maximize (for example, whether it is generalizations of the Baire Category Theorem, or the sort of inter-relations between candidate “V”s that Sy is investigating, or interpretability power). Moreover, there are widely conflicting intuitions as to when something follows from “maximality”. For example, some claim (and I believe Magidor is an example) that Vopenka’s Principle is so justified, while Sy would claim that it is not. (I say this with some hesitancy since although I have discussed the matter with both Magidor and Sy I am not sure that they attach the same significance to the term “intrinsically justified”. (Philosophy is one of those areas that can seem tedious and frustrating at times since instead of proceeding freely in a shared language and simply saying things about the world, one must, at times turn inward and discuss the language itself and the various senses that are attached to terms.))

In the HP I am only talking about one specific meaning of maximality: the height and width maximality of the universe of sets. I see the HP analysis as a new source of candidates for set-theoretic truth, based on the concept of set. Now the word “intrinsic” is very strong, and as you pointed out can lead to “foot-stamping”. I like to identify “intrinsic” with “derivable from the MIC” but that is not necessary to preserve the idea of the programme:

I am happy to take up Pen’s suggestion of taking the maximality of V (in height and width) as a “heuristic” for generating new axioms, with one proviso. Whereas she doesn’t regard the source of a new axiom as being of any importance whatsoever (it all comes down to good set theory and mathematics) I regard it as important to try to understand the maximality of V (in height and width) and therefore not to erase the source of an axiom derivable from it. Further, as I said, I think there is a good chance of arriving at an optimal HP criterion, which means that one can have a consistent theory of Type 3 (HP) truth, which I regard as extremely unlikely for Type 1 truth (what’s good for set theory as a branch of math) and unexplored for Type 2 truth (what’s good for math outside of set theory).

Shifting focus to the question of AC more specifically, some have claimed that AC is intrinsically justified on the basis of the “maximal” iterative conception of set. (I believe that Ramsey argued this and that Tait defends the claim since he argues for something much stronger, namely, that AC follows from the meaning of higher-order quantifiers.)

I see! So it’s not out of the question!

First, the question of whether AC holds depends on two things — (1) the breath of the collections of non-empty sets that one has to select from and (2) the breadth of the collection of choice functions. For AC to hold one needs the proper balance between (1) and (2).

Perfectly said, I couldn’t agree more.

It is not straightforward that “maximality” implies AC because in addition to giving us lots of choice functions it also makes matters harder by giving us lots of collections of non-empty sets to choose from. What one gets out of “maximality” depends on where one puts the emphasis — on (1) or (2). For example, if one puts the emphasis on (2) then one can make a case for AC but if one puts the emphasis on (1) then one can make a case for things like Reinhardt cardinals (which provide us with so many sets that it is hard to find choice functions for them).

But I don’t follow this argument (more on Reinhardt cardinals below).

Suppose it should turn out that the “choiceless” large cardinals are consistent. (This is a hierarchy of large cardinals that extends beyond . The first major marker is a Reinhardt cardinal. After that one has Super Reinhardt cardinals and then the hierarchy of Berkeley cardinals. (This is something that Woodin, Bagaria, and myself have been recently investigating.)) Suppose that the principles in this hierarchy are consistent. Then if we are to follow the principle of “maximality” — in the sense of maximizing interpretability power — these principles will lead us upward to theories that violate AC.

Do you have a proof that “ZFC + There is an inner model with a Reinhardt cardinal” is inconsistent? I.e., is there a “Morris-style” phenomenon at work here, where no model with a Reinhardt cardinal has an outer model with choice? If not, then it seems to me that if Reinhardt cardinals make you feel uncomfortable because they contradict AC then a nice alternative is “ZFC + There is an inner model with a Reinhardt cardinal”. The same goes with any of the stronger large cardinals axioms that contradict choice.

So I repeat my comment to John: There are many theories of equal interpretative power. For example, “there exists a huge cardinal” and “ is huge in an inner model.” There is some arbitrariness in the choice of which theory you pick within a given equivalence class of interpretability power.

On this picture, AC would be viewed like V = L, as a limiting principle, a principle that holds up to a certain point in the interpretability hierarchy (while one is following a “natural” path) and then gets turned off past a certain stage.

Not exactly, if all you have to do is use inner models instead of V. But again, maybe you have a proof that no model of choice has an inner model with a Reinhardt cardinal, in which case my suggestion doesn’t work.

I really hope that these “choiceless large cardinals” are not consistent (and that is something we are trying to show). But my point is that if they are and one runs wild with “maximality” considerations then one can put together a case for the negation of AC.

In summary, it seems that there is not enough unity and convergence in this enterprise to inspire confidence in the notion of “being intrinsically justified on the basis of the “maximal” iterative conception of set”.

Now you have lost me. You switched from the MIC to interpretability power. Why is the latter subsumed under the former? We can’t even get to measurable cardinals using the maximal iterative conception.

As I said, I don’t insist on “intrinsic justification” anymore as I see how heavily-laden that phrase is, but it is clear to me that there is something important about the maximality of V worthy of investigation, whether you call it “intrinsic justification” or just an “intrinsic source for new axioms”. Indeed without a lot of work the HP will generate contradictory new axioms (just like set-theoretic practice) so “intrinsic justification” is at best something that one can consider after the successful completion of the programme.

Best, Sy