Peter has just written:

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”.

Pen has just written:

- What is the precise statement of ?
- Why should we think the study of countable models will shed light on V?

In response to 1, Sy has not been able to explain the point of the statement or even the content of the statement to anyone beyond a handful of specialists. So there is no way for philosophers or f.o.m. interested parties to evaluate the subtleties that may be raised by this — the exact nature of parameters, use of single sentences instead of theories, resulting inconsistencies and trivialities, levels of artificialness and non robustness, etcetera.

So the entire picture from the ground up, from first principles, of the parameter and theory-versus-single-sentence situation needs to be carefully examined, going all the way back to even way before the original IMH (inner model hypothesis), going back to earlier ideas of Jouko and others. The legitimate foundational programs generally have a great deal of robustness in this regard, and we need to look at the source of the nonrobustness here.

Generally speaking, when proposing some new statement, if one tries to justify it or explain it in generally understandable fundamental terms, then the defects and the merits come to the surface. When proposing any kind of legitimate foundational program, such moves are of course taken for granted.

The quicker one moves to talking to only a tiny number of people, the higher the suspicion level becomes — as to whether one is proposing a legitimate foundational program. I have done some telephoning with some of the most technically knowledgeable people on this list, and also philosophers, and they report that they don’t have any good idea as to what is actually being proposed here. The move here to talking to only a tiny number of people was almost instantaneous. A very bad sign.

The “response” to 2 is not really responsive. The core issue is that it is not realistic to even propose that relationships between countable transitive models of ZFC are going to shed any light on “maximality” or other genuinely foundational aspects of set theory. The use of countable transitive models may be an important technical tool in understanding principles formulated in much more promising foundational terms. The idea is that in establishing information about consistency or relatively provability or relative consistency or the like, about the actual statements of direct interest in a legitimate foundational program, one may be expected to use countable models in the proofs.

But then to cast a “foundational program” in terms of countable models is dubious. Any legitimate casting of an underlying legitimate foundational program (related to set theoretic truth, set theoretic axioms, set theoretic maximality, etc) is not going to be properly cast in terms of countable models.

So talking about “hyperuniverses” or “hyperuniverse programs”, etcetera, as some sort of legitimate foundational program does not appear, prima facie, to be justified or even reasonable. And the language is far too flowery to signify what it seems to actually be: a kind of detailed study of countable transitive models of ZFC.

I suggest that Sy rename his “program” as the ctm program. A detailed study of countable transitive models of set theory and their relationships. One particular family of relationships is arguably connected with some “maximality” ideas, that have not been sufficiently analyzed or even clarified. Maximality makes sense all through mathematics, and has different precise meanings depending on context. So since one is simply offering the study of ctm’s as a technical program, it is perfectly legitimate to talk of maximality properties just as any mathematician might do in core mathematics. E.g., maximal subgroups, maximal ideals, etcetera.

I have been hearing, down the grapevine, that some of the statements being offered after the original IMH (inner model hypothesis) cannot be phrased or phrased appropriately in terms of ctm’s. If that is the case, then this is yet another reason to avoid putting all of one’s eggs in the basket of some misnamed “hyperuniverse program” that is simply a study of countable transitive models of ZFC.

A crucial issue is the very real prospect that “maximality in set theory” – at least as we are talking and thinking about it – is a deeply flawed or deeply vague notion that has not led to – and may never lead to – any genuine foundational program. We know that all kinds of natural formulations lead to inconsistencies or trivialities. In fact, it appears to be remarkably resistant to robust formulation that provides us with any traction.

The mere fact that even trying to justify AxC in terms of some general understanding of “maximality in set theory” is already right now elusive (and controversial) is also a very bad sign.

Coming back to 1, the more time we spend in this thread without any kind of substantive discussion or even substantive explanation of just what these ideas amount to, in generally understandable terms, the more skeptical people are going to be that there really is a legitimate foundational program being proposed. It is of course quite OK for one to simply be enthusiastic about the detailed study of ctm’s.

Now for Peter’s comment:

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”.

This was in response to my earlier message:

The basic issue has been raised as to how the axiom of choice is to follow from “maximality”. This has been particularly well understood for a long time, e.g., in the following way.

THEOREM. In ZF, the following are equivalent. i. Every graph has a maximal clique. ii. The axiom of choice.

It is most convenient to define a graph as a pair (V,E), where E is an irreflexive symmetric binary relation on V. A clique is a set where any two distinct elements are related by E. Maximal means inclusion maximal.

Alternatively, one can use digraphs in the sense of paris (V,E), where E is a binary relation on V. A clique is a set where any two elements are related by E. (You can also use: any two distinct elements are related by E).

I am following my usual modus operandi – I don’t take philosophical positions, but rather develop foundational programs based on a mixture of these considerations: 1. Motivation from philosophical considerations. 2. Mathematical traction. When I offer some foundational program, the philosophical/foundational story is presented in generally understandable terms, and that story may be primitive and highly attackable. In fact, initially it is usually primitive and highly attackable. But then, there are further developments that make it more responsive to philosophical/foundational considerations – often discovered in the process of being attacked – and the stories get better over time, and the process gets repeated and repeated. A real foundational thinker puts forward the matters in very generally understandable terms, and engages openly with the attacks. And then makes key further developments – again openly and in completely generally understandable terms – also subject to modified attacks, and the process repeats itself. At a Princeton visit, I recently coined the term “ping pong” in this connection. And when the developments even threaten to be not generally understandable, of general intellectual interest, that raises a major red flag. I better come up with better ideas for further development, or I had better reconsider the validity of the emerging foundational program. For every ongoing serious foundational program, there are dozens of failed attempts that descend into perhaps not uninteresting technical programs. But there is no substitute for the real thing, and the real methodology.

In particular, I was actually surprised to see that a “foundational program” is being proposed that is generated by “maximality”, yet the underlying notion of “maximality” is not sufficiently clear as to even determine that AxC is itself generated by “maximality”!

In fact, I automatically assumed that AxC was considered by those active on the list as the almost paradigm case of “maximality” that is so “obvious” that it needs no explanation. In fact, I thought it was considered “conventional wisdom” that does not even have to be said, that all of ZFC is generated by “maximality”.

Now I am always sensitive to weaknesses, and as you have seen, I consider even the concept of “arbitrary permutation of {1,…,1000}”, including quantification over such, as already worthy of very serious foundational programs. So I never reflexively bought into the idea that it is completely obvious that ZFC is generated by “maximality”. Nevertheless, at least I thought that people generally thought they had a clear enough understanding of “maximality” to be totally convinced that ZFC fits under this. I am sort of shocked, but of course I will play.

I now see what Peter means in his question to me. He is essentially inquiring as to whether I have something in mind other than the “obvious” — which from reasonable points of view, is not “obvious” at all.

In fact, this interchange with Peter is yet another indication that the way “maximality” is being used in this thread needs to be carefully analyzed before one can even think of proposing a genuine foundational program.

The naive discussion goes like this. We can certainly imagine picking exactly one from each equivalence class. So if the set theoretic universe doesn’t have such a set picking exactly one from each equivalence class (and nothing outside the domain of the equivalence relation), then the set theoretic universe is missing something.

So if this is questioned — and I am quite happy questioning it — then we need to go deeply into just what kind of “maximality” of the set theoretic universe we are talking about. In fact, this now appears all of the more imperative to address this.

On the other hand, it is interesting to look for deeper arguments that the AxC is responsive to “maximality”. I have been playing around with this for a little bit, and I see how elusive even this enterprise is — filled with trivialities and inconsistencies and so forth.

So let me pose this as a challenge: clarify the notion of “maximality in set theory” in order to formulate a fundamental principle that implies AxC.

On a productive note, I am starting to formulate a calculus of “simple choice principles” of which “maximal cliques exist” is a special case. The main challenge in the calculus is to give a decision procedure for determining the status and relative status of the instances. This is a developing example of my modus operandi I referred to above. It is fully attackable, but appears to have serious mathematical traction.

However, timing is important, and as long as Hugh, Pen, Peter and others want to discuss some purported legitimate foundational program, no matter how problematic, I will refrain from proposing this one.

But I welcome private email indicating interest in hearing about “choice calculus”. It’s not going to “solve” philosophical problems, but I find it at least suggested by the questioning of just what can be meant by “maximality principles” in general. In fact, I have other projects undeveloped over the years that can also be viewed with this lens, also leading to calculi. In fact, there seems to be emerging a genuine umbrella project called “set theoretic calculi”.

Harvey