Tag Archives: Choice calculus

Re: Paper and slides on indefiniteness of CH

Continuing the hopefully productive discussion of the Axiom of Choice, this time with maximality considerations brought in.

I have been discussing elemental ways to express the Axiom of Choice in a logical framework.

THEOREM 1. AxC, over ZF, can be expressed in the form such and such (purely universal) sentence of predicate calculus with equality has a model on any given infinite domain, and also in the form such and such (purely universal) sentence of predicate calculus with equality has a model on any given nonempty domain.

NOTE: I brought in the true \Pi^0_1 sentences into the previous discussion, to formulate the priveleged position of AxC over ZF in this logical realm. However, I should not have brought the \Pi^0_1 sentences in until I stated the above.

THEOREM 2. AxC cannot be expressed over ZF in the form such and such purely universal sentence of predicate calculus with equality in only relation symbols has a model on any given infinite domain, nor in the form such and such purely universal sentence of predicate calculus with equality in only relation symbols has a model on any given nonempty domain. There is a decision procedure for the true ones of these forms, and they all follow from ZF + “every set can be linearly ordered”.

We now want to discuss how we can express the Axiom of Choice as “maximality”, also in a logical framework.

QUESTION: Can AxC be expressed over ZF in the form such and such a purely universal sentence in relation symbols only, has a maximal model on every nonempty domain? Clearly, every such sentences are provable in ZFC.

In the past, I would never throw out a possible jewel like the above without putting a fair amount of effort into answering it. But I am 66 and trying to go into new fields like piano, math physics, mechanics, statistics, etc., to make good on a teenage ambition. So if by good fortune, the above Question turns out to be a jewel (maybe, maybe not), have fun.

THEOREM 3. AxC, over ZF, can be expressed in the following form. Every model of such and such purely universal sentence with relations only, has a maximal extension with the same underlying domain, satisfying the same purely universal sentence with relations only. Clearly, every such sentence is provable in ZFC.

Proof: Use R,S,P, where R,S are binary and P is unary. Assert that

i. R(x,y) iff not S(x,y). ii. P(x) and P(y) implies R(x,y).

QED

PROJECTS. We would like to get a general understanding of what sentences over ZF can be expressed in these various ways. Various test problems can be formulated, including decision procedure issues.

Consider countable choice, dependent choice, and “the reals are well ordered”. Can these be stated over ZF in any of the forms we have discussed? Or if we restrict the forms above by restricting the domains?

Harvey

Re: Paper and slides on indefiniteness of CH

Where we left off:

THE STRONGEST STATEMENT, OVER ZF PLUS THE TRUE \Pi^0_1 SENTENCES, OF THE FORM “SUCH AND SUCH (PURELY UNIVERESAL) SENTENCE IN PREDICATE CALCULUS WITH EQUALITY HAS A MODEL ON EVERY INFINITE DOMAIN” IS PROVABLY EQUIVALENT TO THE AXIOM OF CHOICE, OVER ZF PLUS THE TRUE \Pi^0_1 SENTENCES.

THE STRONGEST STATEMENT, OVER ZF PLUS THE TRUE \Pi^0_1 SENTENCES, OF THE FORM “SUCH AND SUCH (PURELY UNIVERESAL) SENTENCE IN PREDICATE CALCULUS WITH EQUALITY HAS A MODEL ON EVERY NONEMPTY DOMAIN” IS PROVABLY EQUIVALENT TO THE AXIOM OF CHOICE, OVER ZF PLUS THE TRUE \Pi^0_1 SENTENCES.

The first statement above is straightforward using Gödel’s Completeness Theorem.

I now rework the proof of the second statement, as there were some correctable flaws. It is quite easy to fall into the trap of assuming that you have a cardinal \lambda such that \lambda^+ is regular. The uniformity in condition 4 must be exploited properly.

We use the universal sentence \varphi expressing

  1. Equivalence relation E on a subset D' of the domain D.
  2. Define [x] only for x in D', where [x] is a subset of D'.
  3. Set S contained in D' which picks exactly one from each equivalence class of E on D'.
  4. Map which, given x in D', produces a bijection between [x] and S, depending only on [x].
  5. D\setminus D' is embeddable in D'\sqcup D'.

Note that \varphi has a model with domain any nonempty finite set.

Let D = B \cup \lambda^+, where \lambda is an infinite cardinal, and \lambda cannot be embedded into B. We prove that B is well ordered.

We write D'/E for the set of equivalence classes of E on D'. We write (D'/E)^* for the set of equivalence classes of E on K = D' \cap\lambda^+.

Case 1. |S| \geq lambda+. Then each [x] has at least \lambda^+ elements. Hence each [x], has at least one element from \lambda^+. Hence |S| = \lambda^+. For each x \in D', we associate first the unique element of S that is equivalent to x, and then the result of applying the bijection between [x] and S, given by 4, to x. Thus we have a one-one map from D' into S \times S. Hence D' is well ordered. By 5, D\setminus D' is also well ordered. Hence D is well ordered. In particular, B is well ordered.

Case 2. |S| \ngeq \lambda^+. Note that S has at most \lambda elements of \lambda^+. We have a bijection from D/E onto S. We have an embedding from (D/E)^* onto W contained in S. Clearly W is well ordered, and so must have at most lambda elements. Hence we have an embedding from (D/E)^* into \lambda. Using condition 4, we get a map that takes each u in (D/E)^* to a an embedding from u into S. Now every value of every such map can be indexed by an element of (D/E)^* and an element of K, using that (D/E)^* is well ordered. So we get a map that takes each u in (D/E)^* to an embedding from u into \lambda. This together with (D/E)^* having at most \lambda elements allows us to conclude that K has at most lambda elements. Therefore \lambda\setminus D' has lambda^+ elements. By 5, \lambda^+ is embeddable in D' \sqcup D'. Hence \lambda^+ is embeddable in D'. The preimage under this embedding of B must have at most \lambda elements. Hence \lambda^+ is embeddable in $latex K, which is a contradiction.

QED

What is the status (especially equivalence with AxC) and relative status in ZF of

There is a semigroup, group, abelian semigroup, abelian group, divisible abelian group, free group, ring, commutative ring, field, algebraically closed field, ordered field, ordered ring, discrete ordered ring, linear ordering, dense linear ordering, model of Presburger, on every infinite domain. And any other special cases that you get interested in.

Next time, I will start talking again about maximality again.

Harvey

Re: Paper and slides on indefiniteness of CH

THE STRONGEST STATEMENT, OVER ZF PLUS THE TRUE \Pi^0_1 SENTENCES, OF THE FORM “SUCH AND SUCH (PURELY UNIVERESAL) SENTENCE IN PREDICATE CALCULUS WITH EQUALITY HAS A MODEL ON EVERY INFINITE DOMAIN”, IS PROVABLY EQUIVALENT TO THE AXIOM OF CHOICE, OVER ZF PLUS THE TRUE \Pi^0_1 SENTENCES.

THE STRONGEST STATEMENT, OVER ZF PLUS THE TRUE \Pi^0_1 SENTENCES, OFTHE FORM “SUCH AND SUCH (PURELY UNIVERESAL) SENTENCE IN PREDICATE CALCULUS WITH EQUALITY HAS A MODEL ON EVERY NONEMPTY DOMAIN”., IS PROVABLY EQUIVALENT TO THE AXIOM OF CHOICE, OVER ZF PLUS THE TRUE \Pi^0_1 SENTENCES.

Such spinoffs very naturally arise in the course of using legitimate methods for conducting foundationally and philosophically motivated research. As you can see in the full email history here, I felt traction well before I had seen the above in any formulation, and saw the above in some formulation before I saw the above in its present formulation above. It is fairly clear that there is a rich new (I always worry about precisely how new anything is, of course) area surrounding the above observations. There is at least some new unifying theory of forms of the axiom of choice (some equivalent to the axiom of choice and others weaker), and probably much more.

My intention is to start dealing directing with “intrinsic maximality in set theory” in the next messages. Let’s see what I come up with.

COPY

Back to my persistent efforts to turn this mostly unproductive rarely generally understandable slogan ridden thread into something more.

In a previous posting, I indicated some important features of my general f.o.m. methodology. I have started to apply it to a notion that has been around for some time

*) intrinsic maximality of the set theoretic universe

as a way of generating or justifying axioms for set theory. It has clearly not been appropriately elucidated, and the notion is also under considerable attack these days.

In fact, there is folklore that it is a sound way of generating the axioms of ZFC. That specifically is being questioned by some even with regard to the AxC = axiom of choice.

Now the way I see it, informally “intrinsic maximality (of the set theoreitc universe)” means something like this:

**) the set theoretic universe is as large as possible or imaginable – consistent with the most elemental features of sets**

What elemental features of sets? Well, for this purpose, we take as a working idea, first and foremost, extensionality = two sets are equal if and only if they have the same elements. But what about foundation? Well, I just don’t know at this point what attitude we should take toward foundation for present purposes.

Prima facie, it would appear that AxC follows from **). Say, given an equivalence relation, we can certainly imagine the idea that we have picked one element from each equivalence class. But how do we systematize this?

I came up with the following more general idea. Instead of starting with an equivalence relation, we can instead start with an arbitrary set X. We can put “basic” conditions on a relation or function on X. We then consider the sentence

#) for all sets X there exists a relation or function satisfying a given condition.

Here are three of the simplest special cases.

For all X there exists a linear ordering on X. For all X there is a one-one function from X to X that is not onto. For all X there is a one-one function from X^2 into X.

Of course, the first is provable in ZFC. However, the other two are refutable in ZFC (even in ZF).

So this suggests the following.

##) for all infinite sets X there exists a relation or function satisfying a given condition.

Then consider these three examples.

For all infinite X there exists a linear ordering on X. For all infinite X there is a one-one function from X to X that is not onto. For all infinite X there is a one-one function from X^2 into X.

These are all provable in ZFC. The third is equivalent to AxC over a weak fragment of ZF. The conjunction of the first two does not imply AxC over ZF, and neither of the first two implies the other over ZF.

Thus it looks like we have stumbled upon a calculus that unifies a lot of important work concerning forms of the axiom of choice in set theory.

So now let’s try to get it all together.

DEFINITION 1. K(\text{infinite}) is the set of all sentences of set theory of the following form. For all infinite D there exists a model of \varphi with domain D. Here \varphi is a sentence in first order predicate calculus with equality. K(\text{nonempty}) is the set of all sentences of set theory of the following form. For all nonempty D there exists a model of \varphi with domain D.

But an important feature of the examples are that they are purely universal.

DEFINITION 2. K(\text{infinite},\pi) consists of “for all infinite D there exists a model of \varphi with domain D” where \varphi is purely universal. K(\text{nonempty},\pi) consists of “for all nonempty D there exists a model of \varphi with domain D” where phi is purely universal.

It appears that every element of the K’s, from the point of view of ZF, has two orthogonal components – its arithmetic part and its set theoretic part.

THEOREM 1. The following is provable in a weak fragment of ZFC. A sentence lies in K(\text{infinite}) if and only if it is satisfiable in some (every) infinite domain. A sentence lies in K(\text{nonempty}) if and only if it is satisfiable in every domain if and only if it is satisfiable in some (all) infinite domains and satisfiable in all nonempty finite domains. Thus the set of all true sentences in K(\text{infinite}) and K(\text{nonempty}) are complete and \Pi^0_1, respectively.

DEFINITION 3. Let ZFC* be ZFC together with the true \Pi^0_1 sentences.

THEOREM 2. Every element of K(\text{infinite}) and K(\text{nonempty}) is provable or refutable in ZFC*. In fact, every such element is either provable in a weak fragment of ZFC* or refutable in a weak fragment of ZF.

There are plenty of interesting special fragments of first order predicate calculus with equality that where validity and validity for infinite models are decidable – and demonstrably so in ZFC (even in a weak fragment of ZF). For K(\text{infinite}) and K(\text{nonempty}) based on such fragments, Theorem 2 will clearly hold with ZFC* replaced by ZFC. For these fragments of K(\text{infinite}) and K(\text{nonempty}), we should be able to get a particularly clear understanding of the status of the elements over ZF.

The program is to understand the status and relative status of the elements of K(\text{infinite}) and K(\text{nonempty}) over ZF*.

We have already seen that there is a variety of elements of K(\text{infinite},\pi) over ZF*, some of which are provably equivalent to AxC over a weak fragment of ZF*. However, what about elements of K(\text{nonempty}) and K(\text{nonempty},\pi)?

THEOREM 3. There is an element of $latex K(\text{infinite},\pi)$ and of K(\text{nonempty},\pi), respectively, that is provably equivalent to AxC over a weak fragment of ZF.

We have already seen that we can use “for every infinite D there is a one-to-one f:D^2 \to D“. But about about K(\text{nonempty},\pi)?

We now show that

*The axiom of choice can be expressed as the assertion that some given purely universal sentence is satisfiable in every nonempty domain. Same with “infinite domain”.

I looked into this more deeply than I did in posting #550. I think that a good way of proving this is as follows.

The sentence \varphi asserts the following.

  1. Equivalence relation E on D.
  2. Set D’ obtained by removing 0,1, or 2 elements from each equivalence class of E on D, as long as you leave at least one element after removal. Work with E on D’.
  3. Set S which picks exactly one from each equivalence class of E on D’.
  4. Map which, given x in D’, produces a bijection between [x] and S, depending only on [x].
  5. D\setminus D' is embeddable in D' \times D'.

Note that \varphi has a model with domain any nonempty finite set.

Let D = B \cup \lambda^+, where \lambda is an infinite cardinal, and \lambda cannot be embedded into B. We prove that B is well ordered.

Case 1. |S| \geq \lambda^+. Then each [x], x \in D', has at least \lambda^+ elements. Hence each [x], x in D', has at least one element of \lambda^+. Hence |S| = lambda^+. For each x \in D', we associate first the unique element of S that is equivalent to x, and then the result of the bijection between [x] and S given by 4. Thus we have a one-one map from D' into S \times S. Hence D' is well ordered. By 5, D\setminus D' is well ordered. Hence D is well ordered. In particular, B is well ordered.

Case 2. |S| \ngeq \lambda^+. Then no equivalence class has cardinality  \geq\lambda^+. Hence every equivalence class of E on D’ has fewer than \lambda^+ elements of \lambda^+. Hence every equivalence class of E on D has at most \lambda elements of lambda^+. Hence there are at least \lambda^+ equivalence classes of E on D. Hence there are at least \lambda^+ equivalence classes of E on D’. Hence every equivalence class of E on D’ has at least \lambda^+ elements. This is a contradiction.

QED

Next posting will start to engage with maximality.

Another way of saying this: we have characterized AxC as the strongest statement in any of K(\text{infinite}), K(\text{nonempty}), K(\text{infinite},\pi), K(\text{nonempty},\pi), over ZF plus the true \Pi^0_1 sentences.

Re: Paper and slides on indefiniteness of CH

I will now try to start setting up a legitimate foundational program surrounding “maximality in set theory”. I am not aware that we have one. If you know of any, please let us know.

I expect philosophers (and other interested parties) to weigh in raising all sorts of issues as I struggle to develop one, and then I should be able to play Ping Pong with further developments taking into account what they say.

I hope there is interest here in seeing such a real time development – and the quality of what comes out in terms of foundational programs of general intellectual interest.

A working environment for stating “maximality principles” or “principles that are inherent in maximality”, or whatever, is the following class of statements:

*) Let R be a binary relation on a set X. There exists f:X \to X such that some condition holds relating R,f.

We can take the conditions to be purely universal. Thus we have defined a countably infinite class of sentences of set theory.

EXAMPLE. Let R be a binary relation on X. There exists f:X \to X such that \forall x,y in X\ (R(x,y) implies R(x,f(x)).

We know that this example is provably equivalent to AxC.

Note that if we put a bound on the number of quantifiers over X that are allowed, and we don’t allow f to be iterated, then we have only finitely many instances, up to tautological equivalence.

Now it is “obvious” that the above Example is the simplest example leading to an equivalent of AxC. But

PROBLEM. State and prove rigorously that the above example is simplest.

Now for a crucial question. Is there an interesting criteria for determining whether an arbitrary sentence of set theory in this family *) of sentences of set theory, represents a legitimate maximality property?

Arguably, ANY sentence in *) that is “true” or “obvious” or “reasonable” represents an instance of maximality. It is saying that given any R,X, there is a certain kind of associated function from X into X.

I can hear the complaints already, but I am groping around here, as I feel a lot of traction.

CONJECTURE. All instances of *) without nesting of f are either a) provable in a weak fragment of ZF; b) refutable in a weak fragment of ZF; c) provably equivalent to AxC over a weak fragment of ZF. Furthermore, this decision is of low computational complexity.

A consequence of this Conjecture is that there are no CONFLICTS. I.e., we have the kind of ROBUSTNESS that we want for a legitimate foundational program.

MORE CONJECTURES. Allow successively broader and broader forms of *), starting with functions and not just relations.

Where is the threshold, where we can code too much and get pathology?

Normally I would just sequester this development until I had the time to fully see what is going on technically, but I am 66 and have much too many other things on my plate. Go have fun!

Philosophers – please complain so that I can make this more interesting. Hopefully we are just getting started.

Generally understandable?

Harvey

Re: Paper and slides on indefiniteness of CH

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:

  1. What is the precise statement of \textsf{SIMH}^\#(\omega_1)?
  2. 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