Re: Paper and slides on indefiniteness of CH

This message will try to say where we are in a generally understandable way.

1. Sol Feferman originally put out a request for comments on his paper and slides concerning the indefiniteness of CH. Sol maintains that CH = continuum hypothesis, is neither a mathematical NOR a logical problem. I think Sol is pretty definite that CH is never going to become a mathematical problem, and is definitely not currently a logical problem. And that there is some realistic possibility of it becoming a logical problem, provided some theory emerges of sufficiently widespread acceptance or interest in the set theory community, with the question of the status of CH within that theory.

NOTE: I have been wanting to turn to the Woodin program (Ultimate L) to see if this is clear, simple, and coherent enough to turn CH into a logical problem. I am planning to approach this from a high level generally understandable perspective shortly to see what shakes out. We have seen that the HP, at least in its present form, does not meet such standards.

2. My own position has a lot of overlap with Sol’s but differs in detail and emphasis. The bottom line is probably that both Sol and I agree that “the program of “settling CH” is not a relatively promising area of research in the foundations of mathematics”. In my foundational methodology, I never subscribe to philosophical views, but do think in terms of which side has the better arguments. I believe that arguments can always be strengthened, and attacks against arguments can also always be strengthened. I don’t come to a definite conclusion that “CH is not a mathematical problem”, even if prospects look poor right now for it being so. One thing is clear: it is not an important mathematical problem right now given the present research activities in mathematics today. It was at the time of Hilbert’s problem list. An interesting discussion is just why and how this changed, and whether this is simply due to Goedel/Cohen. I believe that the story goes well beyond Goedel/Cohen here, but this message is not about that. (Sol does deal with this, but I don’t think that it is the last word).

3. In particular, right now, the arguments against CH “having a definite truth value” are stronger than the arguments for CH “having a definite truth value”: but the argument against “having a definite truth value” being definite enough for most philosophically purposes, is, in my view, stronger than the argument for “having a definite truth value” being definite enough for most philosophical purposes. Incidentally, I was recently on the phone with a well known philosopher who strongly disagrees with me about this, and regards CH as (or equivalent to) a clearly stated problem in higher order logic, which he regards as automatically having a “definite truth value” in a “definite sense”.

4. There was little traffic in reaction to Sol’s original request for comments. Until Sy wrote forcefully and extensively about a “program” called HP = hyperuniverse program. Sy urged Sol to incorporate an account of HP in his paper(s). The headline statement of HP was very simple: study the countable transitive models of ZFC and their relationships, and this will reveal the “correct” or “best” axioms for set theory, including axioms that might well settle CH.

5. There were numerous attempts by Sy to justify the claim that such a study of countable transitive models of ZFC (dubbed the hyperuniverse) would provide the “correct” or “best” axioms for set theory, mostly under some form of “intrinsic maximality of the set theoretic universe”. Sy attempted to make this philosophically and foundationally coherent, but left a lot of objections by Pen and Peter unanswered (at least to the satisfaction of Pen and Peter and many others). One of his coworkers in HP answered Pen with answers in direct fundamental contradiction with those of Sy. No “reconciliation” between these very opposing views that Pen pointed out very clearly has been given.

6. On the philosophical side, Sy attempts to convince us that HP is a foundational program that responds to the “intrinsic maximality of the set theoretic universe”, without getting involved in the badly needed analysis of just what “intrinsic maximality of the set theoretic universe” means, or could mean, or should mean. Because of the lack of such a discussion – let alone creative ideas about it – it does not appear that anyone on this list, except a handful of HP coworkers, are being persuaded that HP is a legitimate foundational program.

7. In this connection, both Hugh and I believe that the HP is better viewed and better named as CTMP = countable transitive model program. There was an attempt by Sy to claim that ctms (countable transitive models) are of fundamental foundational importance for present purposes, based on the downward Skolem Lowenheim theorem. However, that is merely a technical point that comes after a framework for analyzing “intrinsic maximality of the set theoretic universe” is first accepted. In the absence of a careful and persuasive discussion of that framework, there is no relevant use of the Skolem Lowenheim theorem. Also, Hugh pointed out that in some of the recent HP proposals, the link between arbitrary sets and countable sets is broken. Another unanswered question of Hugh for Sy.

8. Another even more critical unanswered question of Hugh for Sy simply asks for clarity concerning what happens in CTMP (aka HP) after we see that the IMH contradicts the existence of inaccessible cardinals. IMH = inner model hypothesis, is the initial assertion coming out of the CTMP (aka HP). Hugh asked recently (and I think Hugh has been asking for months) for this clarity, and I have been expecting a response from Sy.

9. The issues of CTMP (aka HP) after IMH is critical. That IMH refutes the existence of an inaccessible cardinal should immediately make a thorough analysis of just what is meant by “intrinsic maximality of the set theoretic universe” urgent. Instead, Sy chose to reject IMH in favor of a number of “fixes”. Hugh has not gotten a satisfactory response as to what these “fixes” are. Furthermore, back channels affirm my suspicions that these “fixes” are ad hoc, taking the large cardinals as given, and then layering a kind of IMH on top of it. If this is the plan, then we really have, prima facie, a serious dose of philosophical incoherence.

10. With Hugh, Pen, Peter, Geoffrey, we see that Sy has to some extent lived up to his professional responsibility for interactive engagement (given that Sy has forcefully pushed the HP) – but only up to a point. There is still a significant degree of non responsiveness. If Sy would follow the principle of writing in generally understandable ways whenever practical, the non responsiveness and drawbacks would be apparent, and the discussion would be much more productive. However, with me, there is a complete refusal to engage. If he did engage in a professional manner, the prima facie emptiness of the HP would have gotten addressed months ago and either a new idea would have emerged from the interaction, or the HP would have simply morphed, rather pleasantly and uneventfully, into the not uninteresting CTMP.

11. Having been appalled at the utter waste of time for the overwhelming majority of people on this list, at least compared to what it could have been, I took the plunge and started a discussion of “intrinsic maximality in the set theoretic universe”. I was especially motivated by Sy saying that “intrinsic maximality of the set theoretic universe” doesn’t even generate AxC. This in the context of all sorts of pronouncements about what it does generate, of a comparatively technical nature. Sy’s coworker states unequivocally that “intrinsic maximality of the set theoretic unvierse” generates all of ZFC.

12. Already this has led to what appears to be an apparently new study of AxC and variants in the absolutely classically fundamental contexts of satisfiability of sentences in first order predicate calculus. But I am hoping to have a lot more to say about “intrinsic maximality” not only in the set theoretic universe, but much more generally. It appears that much of this thread is an important lesson in how NOT to do philosophy and foundations, and the fruits that come out if philosophy and foundations is done competently.

Harvey

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 On Sun, Oct 19, 2014 at 5:06 AM, Radek Honzik wrote: Dear all, I will attempt to answer briefly the questions posted by Harvey. My view on HP is different from Sy’s, but I see HP as a legitimate foundational program. The most accomplished contributors to this list seem to be doubtful. Thanks for your replying. Your reply doesn’t contain a mish mash of hidden assumptions, not so hidden assumptions, question begging, ignoring criticisms, missed opportunities for joining issues, evasions, crude bragging about important busy activities, undefined terms, total lack of respect for the audience who does not keep up with specialist jargon, and a long list of other sins that make this thread an example of how not to do or even talk about foundations and philosophy. I appreciate that you have for the most part avoided these sins. 0] At a fundamental level, what does “(intrinsic) maximality in set theory” mean in the first place? Let me write IMST instead of “(intrinsic) maximality in set theory” for the sake of brevity. I doubt IMST can mean more than “viewing sets as big as possible, without the use of considerations based on practice of set theory as the main incentive”. I think you mean “viewing the set theoretic universe as inclusive as possible”? Your sentence has an indirect construction that really is not necessary and slows down the reader. “Intrinsic” is thus temporarily reduced to “non-extrinsic”; in view of the heavy philosophical discussions around this notion, I prefer to give it this more restrictive meaning. Note that “extrinsic”, unlike “intrinsic”, has a well-defined inter-subjective meaning. This leaves us with the word “big”; I guess that this is the primitive term, which cannot be defined by anything more simple — at least on the level of general discussion. Again, I don’t think you want to use “big” here – I am suggesting something very similar – the set theoretic universe is as inclusive as possible. I regard this as only an informal starting point and the job is to systematically explore analyses of it. Admittedly, this definition is far from informative. For me, HP is a way of explicating this definition in a mathematical framework. Making its meaning more precise, and by the same token, less general. A discussion should be if other approaches — which set out to get real mathematical results — retain more of the general meaning of the term IMST. No approach can retain all the meaning of ISMT because it is by definition vague and subjective; thus HP should not be expected to do that. But there is the real possibility of saying something generally understandable, surprising, and robust. I haven’t seen anything like that in CTMP (aka HP). 1] Why doesn’t HP carry the obvious name CTMP = countable transitive model program. Because the program was formulated by Sy with the aim of having wider application than the study of ctm’s. Since the name “hyperuniverse” specifically refers to the countable transitive models of ZFC, period, it amount to nothing more than a propogandistic slogan designed to lure the listener into thinking that there is something profound going on having to do with the foundations of set theory. But since nothing yet has come out of this special study of ctms for foundations of set theory, even propogandistic slogans about ctms are premature. 2] What does the choice of a countable transitive model have to do with “(intrinsic) maximality in set theory”? Countable models are a way of explicating IMST. It is a technical convenience which allows us to use model-theoretic techniques, not available for higher cardinalities. What you have here is an unanalyzed idea of “intrinsic maximality in set theory”, and before that is analyzed to any depth, you have the blanket assumption that countable transitive models are going to be the way you can formulate what is going to become the analysis of “intrrinsic maximality in set theory”. The real agenda is a creative or novel analysis of “intrinsic maximality in set theory”, and BEFORE that is accomplished any “proof” that ctms will do by some sort of Lowenheim Skolem argument is bogus. Of course, you can set up some idiosyncratic framework, pretend that you going to make this your analysis of “intrinsic maximality in set theory”, and then cite Lowenheim Skolem. But that is bogus. After you make some creative or novel analysis, and work through the problematic issues (inconsistencies and other non robustness arising out of parameters, sets of sentences, etcetera), and have a framework that is credible and well argued, you can cite the Lowenheim Skolem theorem – if it really does apply correctly – to say that any claims of a certain form are equivalent to claims of the form with ctms. That would make some sense, but I still would not advise it since the proper framework, if there is any, is not going to be based on ctms. Ctms would only be a convenience. This is the serious conceptual error being made in endless emails by Sy trying to justify the use of ctms. An unjustified framework for treating “intrinsic maximality in set theory” is alluded to, and to the extent that it is precise, one quotes Skolem Lowenheim to argue that one can wlog work with ctms. This is a very serious question begging sin. The issue at hand is first to have a novel or creative and well argued and thought out framework for treating “intrinsic maximality in set theory”. AFTER THAT, one can talk about the convenience – but NOT the fundamental nature of – using ctms. This reversal of proper order of ideas – putting the cart before the horse – is a major error in work in foundations and philosophy. IN ADDITION, on the mathematical level, I quote from Hugh. This indicates that even in frameworks proposed beyond IMH, there is no Lowenheim Skolem argument, and one is compelled to make the move that ctms are fundamental, rather than just a convenience. Here is the exchange: Sy wrote: More details: Take the IMH as an example. It is expressible in V-logic. And V-logic is first-order over the least admissible (Goedel-) lengthening of V (i.e. we go far enough in the L-hierarchy built over V until we get a model > of KP). We apply LS to this admissible lengthening, that’s all. Hugh wrote This is of course fine for IMH. But this does not work at all for $\textsf{SIMH}^\#$. One really seems to need the hyperuniverse for that. Details: $\textsf{SIMH}^\#$ is not in general a first order property of $M$ in $L(M)$ or even in $L(N,U)$ where $(N,U)$ witnesses that $M$ is #-generated. MY COMMENT: So we may be already seeing that in some of these approaches being offered, one must buy into the fundamental appropriateness of ctms in the philosophy, and not just an automatic freebie from the Lowenheim Skolem theorem. FURTHERMORE, IMH, where reduction to ctm makes sense through Skolem Lowenheim, has not even been seriously analyzed as an “intrinsic maximality in set theory” by serious foundational and philosophical standards. There is a large array of issues, including inconsistencies and non robustness involving parameters and sets of sentences, and so forth. Aside: I do not quite understand why the discussion rests so heavily on this issue: everyone seems to accept it readily when we talk about forcing (I know it can be eleminated in forcing, but the intuition — see Cohen’s book — comes from countable models). Would it make a difference if the models had cardinality omega_1, or omega and omega_1, or should they be proper classes etc? Larger cardinalities would introduce technical problems which are inessential for the aims of HP. The crucial issue can be raised as follows. 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? I am interested in seeing what happens under both answers. What is totally unacceptable is to make the hidden assumption of “yes we do” while pretending “no we are not because of the Löwenheim Skolem theorem”. That is just bad foundations and philosophy. I am going to explore what happens when we UNAPOLOGETICALLY say “we do”. No bogus Löwenheim Skolem. 3] Which axioms of ZFC are motivated or associated with “(intrinsic) maximality in set theory”? And why? Which ones aren’t and why? IMST by historical consensus includes at this moment ZFC. “Historical consensus” for me means that many people decided that the vague meaning of IMST extends to ZFC. I do think that this depends on time (take the example of AC). HP is a way to raise some new first-order sentences as candidates for this extension. Then what is all this talk on the traffic doubting whether AxC is supported by “intrinsic maximality of the set theoretic universe?” 4] What is your preferred precise formulation of IMH? E.g., is it in terms of countable models? Yes. 5] What do you make of the fact that the IMH is inconsistent with even an inaccessible (if I remember correctly)? If this means that IMH needs to be redesigned, how does this reflect on whether CTMP = HP is really being properly motivated by “(intrinsic) maximality in set theory”? I view the process of obtaining results in HP like an experiment in explicating the vague meaning of IMST. It is to be expected that some of the results will be surprising, and will require interpretation. This is a good attitude. However, there is still not much that has come out, and it is still unclear whether this will change. So declaring it a “program” without having the right kind of ideas in hand, and coining a jargon name, is way premature. 6] What is the simplest essence of the ideas surrounding “fixing or redesigning IMH”? Please, in generally understandable terms here, so that people can get to the essence of the matter, and not have it clouded by technicalities. It is a creative process: explicate IMST by principle $P_1$ — after some mathematical work, it outputs varphi (such as $P_$1 = IMH, $\varphi$ = no inaccessible). Then try $P_2$, etc ($P_2$ can be a “redesigned”, or “modified” version of $P_1$). Of course, one hopes that his/her understanding of set theory will be helpful in identifying P’s which have potential to output nice (good, deep) mathematics. It is essential that the principles $P$‘s should be as practice-independent as possible (= intrinsic, in my reading); that is what makes the program foundational (again, in my more narrow sense). Taking into account what you are saying, and the difficulties that Hugh has been pointing out about the post IMH proposals, this does not have enough of the features of a legitimate foundational program at this stage. It has the features of a legitimate exploratory project without a flowery name and pretentious philosophy. We don’t know if it is going to develop into a legitimate foundational program, which would justify flowery names and pretentious philosophy. 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 Sy wrote: In other words, we can discuss lengthenings and shortenings of V without declaring ourselves to be multiversers. Similarly we can discuss “thickenings” in quotes. No multiverse yet. But then via a Loewenheim-Skolem argument we realise that it suffices to work with a countable little-V, where it is natural and mathematically extremely useful to regard lengthenings and “thickenings” as additional universes. Thus the reduction of the study of Maximality of V to the study of mathematical criteria for the selection of preferred “pictures of V” inside the Hyperuniverse, The Hyperuniverse is of course entirely dependent on V; if we accept a new axiom about V then this will affect the Hyperuniverse. For example if we accept a little more than first-order reflection then a consequence is that the Hyperuniverse is nonempty. If Sy would slow down and carefully explain in universally understandable terms just what he is talking about, we would all probably recognize that the use of the “Löwenheim-Skolem argument” is bogus. I’m not sure that Sy is aware that there are some standards for doing philosophy and foundations of set theory (or anything else). Perhaps Sy believes that with enough energetic offerings of slogans, and enough seeking of soundbites from philosophers (which he has found are not all that easy to get), you can avoid having to come up with real foundational/philosophical ideas that work. I am not aware of a single person on this email list who is inclined to believe that CTMP (aka HP) constitutes any kind of legitimate foundational program for set theory – at least on the basis of anything offered up here. (CTMP appears to be a not uninteresting technical study, but even as a technical study, it currently suffers from a lack of systemization – at least judging by what is being offered up here). If there is a single person on this email list who thinks that CTMP (aka HP) constitutes any kind of legitimate foundational program for set theory, I think that we would all very much appreciate that they come forward and say why they think so, and start offering up some clear, deliberate, and generally understandable answers to the questions I raised a short time ago. I copy them below. Now I am not primarily here to tear down silly propoganda. Enough of this has already been done by me and others. I am making efforts to steer this discussion into productive channels that meet that great standard: being generally understandable to everybody, with no attempt to mask flawed ideas — or seemingly unsound ideas — in a mixture of technicalities, slogans, and propoganda I invited Sy to engage in a productive discussion that would meet at least minimal standards for how foundations and philosophy can be discussed, and he has refused to engage dozens of times. So again, if there is anybody here who thinks that CTMP (aka HP) is a legitimate foundational program for set theory, please say so, and engage in the following questions I posted recently: In the meantime, I am finishing up a wholly positive message that I hope you are interested in. QUESTIONS – lightly edited from the original list Why doesn’t HP carry the obvious name CTMP = countable transitive model program. That is my suggestion and has been supported by Hugh. What does the choice of a countable transitive model have to do with “(intrinsic) maximality in set theory”? Avoid quoting complicated technicalities, meaningless slogans, or idiosyncratic jargon and adhere to generally understandable considerations. At a fundamental level, what does “(intrinsic) maximality in set theory” mean in the first place? Which axioms of ZFC are motivated or associated with “(intrinsic) maximality in set theory”? And why? Which ones aren’t and why? What is your preferred precise formulation of IMH? E.g., is it in terms of countable models? What do you make of the fact that the IMH is inconsistent with even an inaccessible (if I remember correctly)? If this means that IMH needs to be redesigned, how does this reflect on whether CTMP = HP is really being properly motivated by “(intrinsic) maximality in set theory”? What is the simplest essence of the ideas surrounding “fixing or redesigning IMH”? Please, in generally understandable terms here, so that people can get to the essence of the matter, and not have it clouded by technicalities. Overall, it would be particularly useful to avoid quoting complicated technicalities or idiosyncratic jargon and adhere to generally understandable considerations. After all, CTMP = HP is being offered as some sort of truly foundational program. Legitimate foundational programs lend themselves to generally understandable explanations with overwhelmingly attractive features. Harvey Re: Paper and slides on indefiniteness of CH Pen wrote I realize this is frustrating for you, but what you’ve said so far about the ‘reduction to ctms’ hasn’t yet produced understanding (for me) or conviction (for Hugh or Harvey). I’m still stuck, as above, on figuring out where the ctms live, what makes everything ‘ideal’, and so on. They have other concerns. There is no issue if the CTMP (aka HP) is taken to be a detailed study of countable transitive models – a not uninteresting subject totally within set theory. The dubious step is to make a claim that CTMP (aka HP) is a foundational program aimed at “set theoretic maximality” or “set theoretic truth” or whatever. Even bolder would be that “there is a prior fundamental notion of a ctm”, but I have not seen even the slightest indication that that is where proponents of CTMP (aka HP) want to go. Once it is accepted that this is really CTMP, then one can freely go about trying to build an attractive systematic subject of CTMP, unfettered with the problem of trying to assign “foundational or philosophical significance” to the developments. Legitimately assigning “foundational or philosophical” significance to CTMP is a rather delicate business requiring — if it can be done at all — major new foundational or philosophical insights. We just haven’t been seeing anything like that on this thread in connection with CTMP (aka HP). My plan was to engage proponents of CTMP (aka HP) in a careful and deliberate attempt to discover the needed major new foundational or philosophical insights that might just lift CTMP to a legitimate foundational program in the present sense. However, proponents of CTMP were unwilling to play. Therefore, I started my own thread trying to do just that. Starting, very slowly and deliberately, with trying to derive AxC from “intrinsic maximality in set theory”, with plans to address “there exists nonconstructible real”, and CH. Obviously I haven’t succeeded at this point, but I am personally thrilled with the spinoff subjects being created. Of course, it is not CTMP. Harvey Re: Paper and slides on indefiniteness of CH Claudio Ternullo wrote The HP is about the collection of all c.t.m. of ZFC (aka the “hyperuniverse” [H]). A “preferred” member of H is one of these c.t.m. satisfying some H-axiom (e.g., IMH). Your coauthor has not explained why HP doesn’t carry the name CTMP = countable transitive model program. That is my suggestion and has been supported by Hugh. Why not? What does the choice of a countable transitive model have to do with “(intrinsic) maximality in set theory”? At a fundamental level, what does “(intrinsic) maximality in set theory” mean in the first place? Which axioms of ZFC are motivated or associated with “(intrinsic) maximality in set theory”? And why? Which ones aren’t and why? What is your preferred precise formulation of IMH? E.g., is it in terms of countable models? What do you make of the fact that the IMH is inconsistent with even an inaccessible (if I remember correctly)? If this means that IMH needs to be redesigned, how does this reflect on whether CTMP = HP is really being properly motivated by “(intrinsic) maximality in set theory”? What is the simplest essence of the ideas surrounding “fixing or redesigning IMH”? Please, in generally understandable terms here, so that people can get to the essence of the matter, and not have it clouded by technicalities. Overall, it would be particularly useful to avoid quoting complicated technicalities and adhere to generally understandable considerations. After all, CTMP = HP is being offered as some sort of truly foundational program. Legitimate foundational programs lend themselves to generally understandable explanations with overwhelmingly attractive features. I have not been able to engage your coauthor in this way, so perhaps this is going to fall on you. Sorry about that (smile). Harvey Re: Paper and slides on indefiniteness of CH I now have a much clearer idea of just why this discussion can have this much traffic, yet actually have such limited productivity. Of course, I can simply continue to put up real time foundations, which seems to be working pretty well, and generate little interaction here. I was expecting a lot of interaction, and I am only beginning to understand why this is not likely to be the case. Actually, the diagnosis of this is itself interesting. Fundamentally a foundationalist like me operates under a very different methodology than philosophers or mathematicians, or for that matter, mathematical logicians, including those with foundational or philosophical pretensions. There is also the great silent majority on this email list who write nothing. So from time to time, I go offline and try to get some feedback on how I am doing. I get some responses, and all of them quite encouraging for me to continue. This leaves the majority of people from which I have no feedback yet. So right now, in the absence of negative feedback, I will continue. I also have had plans to push the traffic back to Sol’s views as expressed in his manuscript. That CH is neither a mathematical nor a logical problem. This goes to the heart of the mathematical and logical status of higher set theory, something which has not been seriously discussed here. It is generally speaking an awkward topic since many people on this list, and in particular Hugh, Pen, Peter, Sy have made a living mostly under the premiss that higher set theory is a legitimate area of research mathematically and philosophically. In varying senses, neither Sol nor I accept this prima facie. Speaking for myself, I regard the status of higher set theory as a genuinely open issue, with supporting and non supporting arguments. I definitely work on both sides of the fence concerning varying forms of legitimacy of higher set theory. Or put differently, I definitely work on many sides of the fence on this. A lot of the traffic concerns, in one way or another, “maximality in set theory”, and in particular a “program” being pushed, going under the flowery name of HP = hyperuniverse program. This is being put forth as some sort of foundational program based on some idea of “intrinsic maximality of the set theoretic universe”. The rationale of this program was questioned repeatedly – to varying degrees – by Hugh, Pen, Peter, and me. One bottom line is that I recommended on the traffic that the name of the “program” be changed to CTMP = countable transitive models program. Then Hugh later endorsed my recommendation on the traffic. The obvious reasons for this name change is that in fact the “program”, such as it is, is presented as a study of countable transitive models of ZFC. So it would simply be called CTMP if one is compelled to call it a program at all. But there is a foundational pretension being put forward, that it somehow supports a notion of “intrinsic maximality of the set theoretic universe”. As i have said repeatedly on the traffic, this simply cannot be the case. Countable models may be used as a major tool in establishing facts about some perhaps coherent formulation of “intrinsic maximality of the set theoretic universe”. But facts about countable models are not going to be the direct source of coherent treatments of anything like “intrinsic maximality of the set theoretic universe”. With regard to “intrinsic maximality of the set theoretic universe”, the traffic is interesting. There are some unexpected features. 1. My interest was piqued by the use of “intrinsic maximality”. I have long thought that intrinsic justifications, generally, are extremely important – even essential. And there are all sorts of great opportunities and challenges to uncover new intrinsic justifications in a wide variety of foundational arenas – not just maximality. 2. I was surprised to learn that Pen and Peter both are so highly skeptical of “intrinsic justifications” specifically with regard to set theoretic axioms, and perhaps more generally as well. I am not sure of Hugh’s position on this. Why was I surprised? I knew that they like to emphasize “extrinsic justifications”, but I had thought that they still endorsed “intrinsic justifications” to a fair extent. Another reason I was surprised, was implicit. I believe that the usual discussion of “extrinsic reasons for set theory” is deeply flawed and represents a lack of acknowledgement of many key features in mathematics generally, and many key attitudes of mathematicians. Specifically, there is a kind of RESTRICT! or DON’T MAXIMIZE! that is going on pervasively – at some important level – all through mathematics and with mathematicians generally. The “extrinsic/maximize” proponents (including Pen and Peter) surely have a defense against this loud attack. They can try to draw a distinction between what mathematics and mathematicians want to RESTRICT! and what would amount to restriction in set theory such as restricting to L. And then the debate goes on, with deeper issues as to the very point of what mathematics is and what higher set theory is, taking front and center. I am up for this debate, but I have as of yet no indication that Pen and Peter and others are up for this debate. 3. Looking at the major gap between something like the original IMH (inner model hypothesis) and any kind of truly coherent presentation of “intrinsic maximality”, I continually tried to get a truly coherent presentation of just what “intrinsic maximality” we are talking about, in truly fundamental terms. I was completely ignored perhaps a dozen times in one way or another. Now I see why I was ignored by Peter and Pen on this – because they do not advocate any relevant kind of “intrinsic maximality of the set theoretic universe” at all. But I directly addressed Sy on this, as he is strongly advocating “intrinsic maximality of the set theoretic universe”, and he is not even acknowledging my requests for a deliberate and careful discussion of just what “intrinsic maximality of the set theoretic universe” is supposed to mean, or why it does not deserve interactive discussion on this thread, is highly unprofessional. 4. Since people do not normally openly act in highly unprofessional ways in front of dozens of people, there has to be an explanation. The explanation is, of course, that Sy is no danger of being viewed as highly unprofessional, because “everybody will know that there must be something personal going on”. Actually, there is nothing personal going on, as far as I am concerned. I am simply treating Sy and always have as a legitimate member of the profession. His ideas are as good as many people’s, and his ideas are as bad as many people’s. I don’t discriminate. 5. Since all of the traffic, except Sol and me, took ZFC “for granted”, I automatically assumed that the prevailing view here was that ZFC was intrinsically justified – and specifically, intrinsically justified by “maximality of the set theoretic universe”. I have long been interested and have foundational programs concerning trying to “justify ZFC intrinsically” – including as a transfer from the finite (Transfer Program). So now I see that probably Pen and Peter do not subscribe to an “intrinsic story for ZFC”. Even more startling was Sy’s being dubious about AxC being “intrinsically justified by set theoretic maximality”. This from somebody pushing a misnamed “program” HP put forth as being responsive to “intrinsic set theoretic maximality”! 6. So I got interested in trying to give an “intrinsic maximality” justification for AxC. I have just got started with this, but already there has been a very attractive spinoff subject that I recently wrote about. I am hoping for a lot more spinoff subjects to come, even if I don’t succeed in getting an intrinsic maximality justification for AxC. In fact, more broadly, the topic of giving intrinsic justifications for ZFC is extremely attractive. This includes the possibility of proving that there are no intrinsic justifications for ZFC of certain kinds. At some point soon, I will leverage off of ZFC, and look at intrinsic justifications for “there exists a nonconstructible real”, especially through “intrinsic maximality”. Again also looking for negative results indicating that this cannot be intrinsically justified. Then I plan to start talking hopefully systematically and fundamentally about CH. 7. In my continuation, I will start by reviewing the spinoff subject already generated concerning AxC that I have previously discussed. Harvey 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