# Re: Paper and slides on indefiniteness of CH

Dear Harvey (and all other who follow the thread),

I will try to respond to your comments, hopefully providing some more clarification.

1] You say that it is crucial to answer the following:

Do we or do we not want to take the structure of ctms as somehow reflecting on the structure of the actual set theoretic universe?” It is important regarding the legitimacy of the use of ctm’s in HP.

Answer: For me, “the actual set-theoretical universe” (= V) is a meaningful term only if it refers to syntactical consequences of ZFC plus some other, explicitly assumed axioms. I do have an intuition about sets which might go beyond this, but it is just that – intuition, not expressible in words with any reasonable degree of accuracy. Starting with this modest assumption about V, it feels natural to work with models of ZFC and look at their properties (the usual “double” role of set theory — metatheory, and theory). The basic idea of HP, which I like, is that perhaps we learn more about our intuition by working with these models, providing we ask the right questions. Since V is either a set of provable sentences, or a vague subjective notion, the question how ctm’s correspond to V is off the target – ctm’s form a reasonably large collection of models, rich enough to provide a field for answering our questions (we decide at the beginning that ill-founded models and large models do not add more significant benefits; we choose transitive models = standard models, to have the standard numbers, formulas, etc).

This evidently does not answer your worries because you do think, as you wrote, that “But there is the real possibility of saying something generally understandable, surprising, and robust [about intrinsic maximality of sets], and therefore probably about V as well. I guess I am less optimistic, and therefore acknowledge that there will always be — at the beginning of the analysis — some “technical convenience”, it is just the question which convenience you prefer.

2] The question of whether AC follows from our intuition about sets. You asked,

Then what is all this talk on the traffic doubting whether AxC is supported by “intrinsic maximality of the set theoretic universe?

Answer: It seems to me that there is so much discussion regarding AC because people hope there is some “hidden proof” of AC from IMST (intrinsic maximality of set-theoretic universe), or some such similar notion. I do not share this hope myself – for the reason that the assumptions of IMST are too subjective to give rise to a widely acceptable argument (while I consider it probable there is some hidden clever proof of Fermat’s theorem, for instance — because here we have objective assumptions).

Aside. I confess i do not quite understand the meaning of “maximal iterative concept of sets = MIC”, either (MIC is sometimes used to argue for axioms of ZF+AC). Or rather, I understand the term MIC if it means an application of a transfinite recursion theorem as provable in ZF, in some maximal sense; I do not see how it can be used to argue for the axioms of ZF+AC (ordinals were defined by Cantor in set theory precisely to make proper sense of (transfinite) iteration, not conversely).

3] Finally, there is the name HP (and related vocabulary), to which you strongly object.

Answer: To me, the name should indicate family-resemblance to “multiverse” (which is open to similar discussion regarding its appropriateness). But let us for a moment forget about the name: would the project seem more convincing with a different name? If yes, I suggest the discussion continues while ignoring the current name; if no, the same applies. Let us not be distracted by the choice of vocabulary.

Best regards to all,
Radek

# 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.

# 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

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

# Re: Paper and slides on indefiniteness of CH

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., $\textsf{IMH}^\#$ 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 $\textsf{I}0$. 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.

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 “$\aleph_1$ 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

# Re: Paper and slides on indefiniteness of CH

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., $\textsf{IMH}^\#$ 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”.

COMMENTS:

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 $\textsf{I}0$. 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.

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.

Best,
Peter

# Re: Paper and slides on indefiniteness of CH

Dear Sy,

1.  What is the precise statement of SIMH#(omega_1)?

I am surprised that anybody cares about this! For what it’s worth, it’s just the SIMH# but restricted to the parameter omega_1 (i.e. #-generation plus if a sentence with parameter omega_1 holds in a cardinal-preserving, #-generated outer model then it holds in an inner model). I thought that I had an argument for its compatibility with arbitrary LCs, but I was wrong, and was tricked by the fact that the IMH# (without the “S”) is indeed so compatible.

Thank you for this clarification.  Your exchange with Hugh went like this:

I think it is worth pointing out to everyone that $\textsf{SIMH}^\#$, and even the weaker $\textsf{SIMH}^\#(\omega_1)$ which we know to be consistent, implies that there is a real x such that $x^\#$ does not exist.

No, that is not true. The $\textsf{IMH}^\#$ is compatible with all large cardinals. So is the $\textsf{SIMH}^\#(\omega_1)$.

I wasn’t aware that you had changed your mind.  My impression is that this exchange is what initially generated interest in the exact statement of $\textsf{SIMH}^\#(\omega_1)$.

2.  Why should we think the study of countable models will shed light on V?

Pen, have you also missed the e-mail that I sent to Geoffrey on 24.September?

Thank you for reminding me of your letter to Geoffrey. I confess I have some trouble digesting this (which is probably why I forgot about it), but perhaps others would care to comment.

PS: I am excited to hear the answer to my AC question:
Is the prevalent view among philosophers of set theory that AC is not derivable from the MIC?

I can only speak for myself.  My own view is that AC is at least compatible with the MIC, and that MIC took over from an ill-determined conglomeration of different thoughts about ‘collections’ partly because the extrinsic evidence for Choice (from set theory and from the rest of mathematics, as documented in Moore’s history) forced out any ‘determined by a rule’ or ‘extension of a predicate’-type notion.

All best,
Pen

# Re: Paper and slides on indefiniteness of CH

Dear Hugh,

In ZF if there is an extendible cardinal (or just proper class of supercompact cardinals formulated properly in ZF) then there is a class forcing extension in which AC holds and one preserves all supercompact cardinals (and much more).

Thanks for pointing this out. It suggests that for the choiceless-HP, AC might be compatible with maximality if the existence of supercompacts is (still unclear), and also that any “good set theory” compatible with supercompacts has a chance of being at least “simulated” in models of useful axioms compatible with AC.

Of course I’d prefer just to always assume choice but do want to stay open-minded about that as we don’t know what the set theory of the 22nd century might look like.

Best,
Sy

# Re: Paper and slides on indefiniteness of CH

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).

Harvey