Here are some questions and comments on the question of AC and “maximality”.
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.)
Harvey: The equivalence you mention between AC and the existence of maximal cliques is intriguing. You said that this topic (of how AC follows from “maximality”) has been well understood for a long time. What other results do you have in mind? I would be interested to hear whether you think that such results make a case for the claim that AC is indeed intrinsically justified on the basis of the “maximal” iterative conception of set? Since, like me, you put “maximality” in scare quotes I assume that the answer is “no”.
I share these doubts and that is one reason I have a weak grasp on the notion of “being intrinsically justified on the basis of the “maximal” iterative conception of set” and, consequently, cannot put much stock in it.
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.))
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.)
One way in which people try to argue for AC on the basis of “maximality” is that if one has a collection of non-empty sets then there must, by “maximality”, be a choice function since otherwise the universe of sets would be impoverished.
There are problems with this.
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 breath of the collection of choice functions. For AC to hold one needs the proper balance between (1) and (2). 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).
Second, (and relatedly), by parity of reasoning one could argue for AD on the grounds that by “maximality” there must be lots of winning strategies. But AD contradicts AC.
Let me now focus on one dimension of “maximality”, namely, that of interpretability power, and say something that’s a bit “far out”.
Some have maintained that what we are trying to maximize is interpretability power. Steel maintains this and I believe that Maddy maintains this. I am pretty certain that neither of them would argue for this on the basis of what is “intrinsically justified on the basis of the ‘maximal’ iterative conception of set” but that is because neither of them would put much stock in the notion of “being an intrinsic justification on the basis of the ‘maximal’ iterative conception of set”. But someone who does put stock in this notion, might argue along similar lines. So let us run with this idea.
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. 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.
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”. But raising skeptical concerns is all too easy and not very inspiring. So let us wait and see. Perhaps a unified notion will emerge and there will be convergence.