# 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

Hugh wrote:

Suppose $M$ is a ctm and $M \vDash \textsf{ZFC}$. Must $M$ have an outer model of $\textsf{ZFC}$ which is cardinal preserving and not a set forcing extension? Number Theory is full of problems which seem completely out of reach. And one can easily generate such problems in a foundational guise.

I have a couple of questions about Hugh’s question.

1. The question as formulated involves both forcing extensions and general (outer) extensions. Are there appropriate formulations that

a. Do not refer to forcing at all. b. Given any ctm $M$, refer only to forcing extensions.

2. What are the known natural conditions on $M$ that are known to be sufficient? Also for variants in 1 above.

With regard to Hugh’s second statement, I think he is proposing an informal family of questions that are at present all intractable. I and I think others would like to get a better sense of just what this informal family of questions is or is like.

Concerning Hugh’s questions involving big numbers. I remember raising on the FOM email list this kind of question many years ago:

Is the $n$-th digit in the binary expansion of $\sqrt2$ zero?

Here $n$ say is some ridiculously large Ackerman number, or whatever. This gives us intractable $\Delta_0$ sentences which can be completely solved with a computer of absurd size. I conjectured that there is no proof or refutation of this question in ZFC + large cardinals in any remotely reasonable sized proof.

Continuing from my earlier messages that try to get an understanding of maybe how AxC follows from some foundational important sense of “set theoretic maximality”.

I need to present more ideas to make further progress, but before moving on to this, let’s pause and see what we have discovered already.

It appears that we have stumbled on a very fundamental class K of set theoretic statements with the following properties.

PRESENTATION OF K. K consists of all sentences of set theory of the form “a given sentence in first order predicate calculus with equality has a model on any infinite set domain”.

THREE EXAMPLES

1. Every set can be linearly ordered.
2. Every infinite set $A$ has a one-to-one function from $A^2$ into $A$.
3. For every infinite A, there is a one-to-one function from $A x \{0\} \cup A x \{1\}$ into $A$.

It is well known that 2 is equivalent to AxC over ZF. (1 and 3) does not imply 2 over ZF. 1 implies 3 and 3 implies 1 are not provable in ZF.

Also notice that these examples are in K for UNIVERSAL SENTENCES. Also note that 1 is in K’ for UNIVERSAL SENTENCES WITH NO FUNCTION SYMBOLS. Thus we have two important fragments of K here.

We have the following main results for the original full K.

1. In a weak fragment of ZFC we can prove that the set of true elements of K is arithmetical – in fact, complete $\Pi^0_1$.
2. Some elements of K are provably equivalent, in a weak fragment of ZF, to AxC.
3. Some elements of K are provable in a weak fragment of ZFC, not provable in ZF, and do not imply AxC over ZF.
4. In fact, in a weak fragment of ZFC, every element of K is provably equivalent to a Pi01 sentence.

But note that satisfiability in infinite models of universal sentences with no function symbols in infinite models is decidable (due to Ramsey). So for this fragment, we get the sharper

1′. Every sentence in the fragment of K is provable or refutable in a weak fragment of ZFC. 2. Same. 3. Same. 4′. Same as 1′.

A foundational program is to gain a complete understanding of K and its interesting fragments. What does this entail?

It would appear that there are two orthogonal components to truth of elements of K. One is the purely”combinatorial” component. The other is the purely “set theoretic” component. For this reason, it is probably best to work over ZF* = ZF + the true $\Pi^0_1$ sentences. Of course, we can define important fragments of K, where every element is provable or refutable in a weak fragment of ZFC – in which case the purely “combinatorial” component is trivial. But generally, we want to work over ZF*. We seek

i. Interesting necessary or sufficient conditions on elements of K to be provable in ZF*.ii. Interesting necessary or sufficient conditions on elements of K to imply AxC over ZF*. iii. Interesting necessary or sufficient conditions for determining whether one element of K implies another over ZF*.

For “innocent” enough fragments of K, we can realistically hope to get precise necessary and sufficient conditions, and associated decision procedures. Also, in general, robustness would be very good to have – that for present purposes, ZF behaves exactly like weak fragments of ZF.

There is an important variant of K which we call K’.

PRESENTATION OF K’. K’ consists of all sentences of set theory of the form “a given sentence in first order predicate calculus with equality has a model on any nonempty set domain”.

We have the same 1-4 above, and also 1′-4′. BUT, there is a problem with Examples 2,3. IN FACT, is there an instance of K’ which is provably equivalent to AxC over ZF?

It appears that the answer is yes.

*) Every nonempty set D is in one-one correspondence with $A^2$ disjoint union B disjoint union C, where B,C are subsets of A.

*) is equivalent to AxC, and lies in the purely universal part of K’.

QUESTIONS. Are the elements of K,K’ provably equivalent over ZF? Are the purely universal elements of K,K’ provably equivalent over ZF?

In the next message I hope to bring maximality back into the picture, while maintaining the fundamental character of the investigation.

Harvey

# Re: Paper and slides on indefiniteness of CH

Dear Peter,

Suppose it should turn out that the “choiceless” large cardinals are consistent. (This is a hierarchy of large cardinals that extends beyond $\textsf{I}0$. 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.

Thanks for bringing up the choiceless cardinals, Peter.  As a strictly amateur kibitzer, my hunch has been that their consistency alone wouldn’t be enough to unseat AC, that they’d have to generate something mathematically attractive (analogous to ordinary LCs generating the #’s, say).  In any case, if you (or one of your co-workers) were willing, I suspect more than a few of us would be very interested to hear about the state of play on these cardinals.

All best,
Pen

# Fwd: Paper and slides on indefiniteness of CH

There is a nice solution to an even more attractive formulation. Look
at all sentences of set theory given by

*) Let $X$ be an infinite set. There exist constants, relations, and
functions obeying a given first order sentence in predicate calculus
with equality.

EXAMPLE. Let $X$ be an infinite set. There exists one-to-one $f:X^2 \to X$. An extremely simple sentence in predicate calculus.

CONJECTURE. The above Example is the simplest example under *) that is
provably equivalent to AxC over ZF.

Tarski proved that this example is provably equivalent to AzC over a
weak fragment of ZF.

THEOREM. It is provable in a weak fragment of ZFC that the set of true
instances of *) is complete co-r.e. Every instance of *) is either
refutable in a tiny fragment of ZF, or provable in ZFC together with
the true Pi01 sentences.

CONJECTURE. Every reasonably simple instance of *) is either refutable
in a weak fragment of ZF or provable in ZFC. For reasonably simple
instances of *), you can determine which implies which over ZF.

Coming back to “set theoretic maximality”, there is the general idea
that I have been playing with on this list. Namely, perhaps there is a
good robust notion of “imaginable property of the set theoretic
universe”, and we want to say that “any imaginable property of the set
theoretic universe is in some sense actualized”. I know this is
fraught with all kinds of non robustness, inconsistencies,
trivialities, and the like. More tractable might be “any imaginable
kind of set that can be added to the set theoretic universe is in some
sense already present in some form”.

But for statements with enough simplicity, my feeling is that there
may be some criteria whereby we can accept them or reject them as
exhibiting “maximality”.

I’m not ready to be able to put this all together into a legitimate
foundational program targeting “set theoretic maximality” — but
hopefully moving in that direction..

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