Using general purpose LLMs to help with set theory questions

Are general purpose LLMs useful to figuring things out in set theory? Here is a story about two experiences I recently had. Don’t worry about the mathematical details.

Last week I wanted to know whether one can prove a certain strengthened version of Cantor’s Theorem without using the Axiom of Choice. I asked Gemini. The results were striking. It looked like a proof, but at crucial stages degenerated into weirdness. It started the proof as a reductio, and then correctly proved a bunch of things, and then claimed that this leads to a contradiction. It then said a bunch of stuff that didn’t yield a contradiction, and then said the proof was complete. Then it said a bunch more stuff that sounded like it was kind of seeing that there was no contradiction.

The “proof” also had a step that needed more explanation and it offered to give an explanation. When I accepted its offer it said something that sounded right, but it implicitly used the Axiom of Choice, which I expressly told it in the initial problem it wasn’t supposed to. When I called it on this, it admitted it, but defended itself by saying it was using a widely-accepted weaker version of Choice (true, but irrelevant).

ChatGPT screwed up in a different way. Both LLMs produced something that at the local level looked like a proof, but wasn’t. I ended up asking MathOverflow and getting a correct answer.

Today, I was thinking about Martin’s Axiom which is something that I am very unfamiliar with. Along the way, I wanted to know if:

1. There is an upper bound on the cardinality of a compact Hausdorff topological space that satisfies the countable chain condition (ccc).

Don’t worry about what the terms mean. Gemini told me this was a “classic” question and the answer was positive. It said that the answer depended on a “deep” result of Shapirovskii from 1974 that implied that:

2. Every compact Hausdorff topological space satisfying the ccc is separable.

A warning bell that I failed to heed sufficiently was that Gemini’s exposition of Shapirovskii included the phrase “the cc(X) = cc(X) implies d(X) = cc(X)”, which is not only ungrammatical (“the”?!) but has a trivial antecedent.

I had trouble finding an etext of the Shapirovskii paper (which from the title is on a relevant topic), so I asked ChatGPT whether (2) is true. Its short answer was: “Not provable in ZFC.” It then said that the existence of a counterexample is independent of the ZFC axioms. Well, I Googled a bit more, and found that the falsity of (2) follows from the ZFC axioms given the highest-ranked answer here as combined with the (very basic) Tychonoff theorem (I am not just relying on authority here: I can see that the example in the answer works). Thus, the “Not provable” claim was just false. I suspect that ChatGPT got its wrong answer by reading too much into a low-ranked answer on the same page (the low ranked answer gave a counterexample that is independent of the ZFC axioms, but did not claim that all counterexamples are so independent).

A tiny bit of thought about the counterexample to (2) made it clear to me that the answer to (1) was negative.

I then asked Gemini in a new session directly about (2). It gave essentially the same incorrect answer as ChatGPT, but with a bit more detail. Amusingly, this contradicts what Gemini said to my initial question.

Finally, just as I was writing this up, I asked ChatGPT directly about (1). It correctly stated that the answer to (1) is negative. However, parts of its argument were incorrect—it gave an inequality (which I haven’t checked the correctness of) but then its argument relied on the opposite inequality.

So, here’s the upshot. On my first set theoretic question, the incorrect answers of both LLMs did not help me in the least. On my second question, Gemini was wrong, but it did point me to a connection between (1) and (2) (which I should have seen myself), and further investigation led me to negative answers to both (1) and (2). Both Gemini and ChatGPT got (2) wrong. ChatGPT got the answer to (1) right (which it had a 50% chance of, I suppose) but got the argument wrong.

Nonetheless, on my second question Gemini did actually help me, by pointing me to a connection that along with MathOverflow pointed me to the right answer. If you know what you’re doing, you can get something useful out of these tools. But it’s dangerous: you need to be able to extract kernels from truth from a mix of truth and falsity. You can’t trust anything set theoretic the LLM gives, not even if it gives a source.

Hyperreal infinitesimal probabilities and definability

In order to assign non-zero probabilities to such things as a lottery ticket in an infinite fair lottery or hitting a specific point on a target with a uniformly distributed dart throw, some people have proposed using non-zero infinitesimal probabilities in a hyperreal field. Hajek and Easwaran criticized this on the grounds that we cannot mathematically specify a specific hyperreal field for the infinitesimal probability. If that were right, then if there are hyperreal infinitesimal probabilities for such a situation, nonetheless we would not be able to say what they are. But it’s not quite right: there is a hyperreal field that is "definable", or fully specifiable in the language of ZFC set theory.

However, for Hajek-Easwaran argument against hyperreal infinitesimal probabilities to work, we don’t need that the hyperreal field be non-definable. All we need is that the pair (^*R,α) be non-definable, where ^*R is a hyperreal field and α is the non-zero infinitesimal assigned to something specific (say, a single ticket or the center of the target).

But here is a fun fact, much of the proof of which comes from some remarks that Michael Nielsen sent me:

Theorem: Assume ZFC is consistent. Then ZFC is consistent with there not being any definable pair (^*R,α) where ^*R is a hyperreal field and α is a non-zero infinitesimal in that field.

[Proof: Solovay showed there is a model of ZFC where every definable set is measurable. But every free ultrafilter on the powerset of the naturals is nonmeasurable. However, an infinite integer in a hyperreal field defines a free ultrafilter on the naturals—given an infinite integer M, say that a subset A of the naturals is a member of the ultrafilter iff |M| ∈ ^*A. And a non-zero infinitesimal defines an infinite integer—say, as the floor of its reciprocal.]

Given the Theorem, without going beyond ZFC, we cannot count on being able to define a specific hyperreal non-zero infinitesimal probability for situations like a ticket infinite lottery or hitting the center of a target. Thus, if a friend of hyperreal infinitesimal probabilities wants to be able to define one, they must go beyond ZFC (ZFC plus constructibility will do).

Unrestricted quantification and Tarskian truth

It is well-known—a feature and not a bug—that Tarski’s definition of truth needs to be given in a metalanguage rather than the object language. Here I want to note a feature of this that I haven’t seen before.

Let’s start by considering how Tarski’s definition of truth would work for set theory.

We can define satisfaction as a relation between finite gappy sequences of objects (i.e., sets) and formulas where the variables are x₁, .... We do this by induction on formulas.

How does this work? Following the usual way to formally create an inductive definition, we will do something like this:

1. A satisfaction-like relation is a relation between finite sequences of sets and formulas such that:

   1. the relation gets right the base cases, namely, a sequence s satisfies x_n ∈ x_m if and only if the nth entry of s is a member of the mth entry of s, and satisfies x_n = x_m if and only if the nth entry of s is identical to the mth entry

   2. the relation gets right the inductive cases (e.g., s satisfies ∃x_nϕ if and only if for every sequence s′ that includes an nth place and agrees with s on all the places other than the nth place we have s′ satisfying ϕ, etc.)

2. A sequence s satisfies a formula ϕ provided that every satisfaction-like relation holds between s and ϕ.

The problem is that in (2) we quantify over satisfaction-like relations. A satisfaction-like relation is not a set in ZF, since any satisfaction-like relation includes ((a),ϕ₌) for every set a, where (a) is the sequence whose only entry is a at the first location and ϕ₌ is x₁ = x₁. Thus, a satisfaction-like relation needs to be a proper class, and we are quantifying over these, which suggests ontological commitment to these proper classes. But ZF set theory does not have proper classes. It only has virtual classes, where we identify a class with the formula defining it. And if we do that, then (2) comes down to:

3. A sequence s satisfies ϕ if for every satisfaction-like formula F the sentence F(s,ϕ) is true.

And that presupposes the concept of truth. (Besides which, I don’t know if we can define a satisfaction-like formula.) So that’s a non-starter. We need genuine and not merely virtual classes to give a Tarski-style definition of truth for set theory. In other words, it looks like the meta-language in which we give the Tarski-style definition of truth for set theory not only needs a vocabulary that goes beyond the object-language’s vocabulary, but it needs a domain of quantification that goes beyond the object-language’s domain.

Now, suppose that we try to give such a Tarskian definition of truth for a language with unrestricted quantification, namely quantification over literally everything. This is very problematic. For now the satisfaction-like relation includes the pair ((a),ϕ₌) for literally every object a. This relation, then, can neither be a set, nor a class, nor a proper superclass, nor a supersuperclass, etc.

I wonder if there is a way of getting around this difficulty by having some kind of a primitive “inductive definition” operator instead of quantifying over satisfaction-like relations.

Another option would be to be a realist about sets but a non-realist about classes, and have some non-realist story about quantification over classes.

I bet people have written on this stuff, as it’s a well-explored area. Anybody here know?

Megethology as mathematics and a regress of structuralisms

In his famous “Mathematics is Megethology”, Lewis gives a brilliant reduction of set theory to mereology and plural quantification. A central ingredient of the reduction is a singleton function which assigns to each individual a singleton of which the individual is the only member. Lewis shows that assuming some assumptions on the size of reality (namely, that it’s very big) there exists a singleton function, and that different singleton functions will yield the same set theoretic truths. The result is that the theory is supposed to be structuralist: it doesn’t matter which singleton function one chooses, just as on structuralist theories of natural numbers it doesn’t matter if one uses von Neumann ordinals or Zermelo ordinals or anything else with the same structure. The structuralism counters the obvious objection to Lewis that if you pick out a singleton function, it is implausible that mathematics is the study of that one singleton function, given that any singleton function yields the same structure.

It occurs to me that there is one hole in the structuralism. In order to say “there exists a singleton function”, Lewis needs to quantify over functions. He does this in a brilliant way using recently developed technical tools where ordered pairs of atoms are first defined in terms of unordered pairs and an ordering is defined by a plurality of fusions, relations on atoms are defined next, and so on, until finally we get functions. However, this part can also be done in a multiplicity of ways, and it is not plausible that mathematics is the study of singleton functions in that one sense of function, given that there are many sense of function that yield the same structure.

Now, of course, one might try to give a formal account of what it is for a construction to have the structure of functions, what it is to quantify not over functions but over function-notions, one might say. But I expect a formal account of quantification over function-notions will presumably suffer from exactly the same issue: no one function-notion-notion will appear privileged, and a structuralist will need to find a way to quantify over function-notion-notions.

I suspect this is a general feature with structuralist accounts. Structuralist accounts study things with a common structure, but there are going to be many accounts of common structure that by exactly the same considerations that motivate structuralism require moving to structuralism about structure, and so on. One needs to stop somewhere. Perhaps with an informal and vague notion of structure? But that is not very satisfying for mathematics, the Queen of Rigor.

A puzzle about consistency

Let T₀ be ZFC. Let T_n be T_n − 1 plus the claim Con(T_n − 1) that T_n − 1 is consistent. Let T_ω be the union of all the T_n for finite n.

Here’s a fun puzzle. It seems that T_ω should be able to prove its own consistency by the following reasoning:

If T_ω is inconsistent, then for some finite n we have T_n inconsistent. But Con(T_n) is true for every finite n.

This sure sounds convincing! It took me a while to think through what’s wrong here. The problem is that although for every finite n, T_ω can prove Con(T_n), it does not follow that T_ω can prove that for every finite n we have Con(T_n).

To make this point perhaps more clear, assume T_n is consistent for all n. Then Con(T_n) cannot be proved from T_n. Thus any finite subset of T_ω is consistent with the claim that for some finite n the theory T_n is inconsistent. Hence by compactness there is a model of T_ω according to which for some finite n the theory T_n is inconsistent. This model will have a non-standard natural number sequence, and “finite” of course will be understood according to that sequence.

Here’s another way to make the point. The theory T_ω proves T_ω consistent if and only if T_ω is consistent according to every model M. But the sentence “T_ω is consistent according to M” is ambiguous between understanding “T_ω” internally and externally to M. If we understand it internally to M, we mean that the set that M thinks consists of the axioms of ZFC together with the ω-iteration of consistency claims is consistent. And this cannot be proved if T_ω is consistent. But if we understand “T_ω” externally to M, we mean that upon stipulating that S is the object in M’s universe whose members^M correspond naturally to the members^V of T_ω (where V is “our true set theory”), according to M, it will be provable that the set S is consistent. But there is a serious problems: there just may be no such object as S in the domain of M and the stipulation may fail. (E.g., in non-standard analysis, the set of finite naturals is never an internal set.)

(One may think a second option is possible: There is such an object as S in M’s universe, but it can’t be referred to in M, in the sense that there is no formula ϕ(x) such that ϕ is satisfied by S and only by S. This option is not actually possible, however, in this case.)

Or so it looks to me. But all this is immensely confusing to me.

Unifying Separation and Choice

Let's round out Axiom of Choice Week. :-)

It’s occurred to me that there is a somewhat pleasant way to integrate the Axioms of Separation and Choice into one axiom schema.

Let’s say that a formula F(x,y) is a partial equivalence (is that the right term?) provided that it’s symmetric and transitive. Now consider this schema (understood to be universally closed over all free variables in F other than x and y):

• If F(x,y) is a partial equivalence, then for any set a there is a subset b such that for every x ∈ b we have F(x,x), and for any x ∈ a such that F(x,x), there is a unique y ∈ b such that F(x,y).

We might call this the Axiom (Schema) of Representatives.

To get the Axiom of Separation, given a formula G(x), let F(x,y) be the formula G(x) ∧ y = x. To get the Axiom of Choice, if c is a set of nonempty disjoint sets, let F(x,y) be ∃d ∈ c(x∈d∧y∈d) and let a = ⋃c (so we need the Union Axiom).

So what?

Nothing earthshaking.

But, first, while there is an advantage to keeping axioms separate for purposes of proving independence results, the more unified our axiomatic system is, the less ad hoc it looks. Unifying Separation and Choice can make us less suspicious about Choice, for instance.

Second, the Axiom Schema of Representatives has nice analogues in some other contexts than set theory. It seems to directly generalize to classes, for instance. Moreover, it extends very nicely to plural quantification to integrate Plural Comprehension with a version of Choice:

• If F(x,y) is a partial equivalence, then there are bs such that (i) for every x among the bs we have F(x,x), and (ii) for any x such that F(x,x), there is a unique y among the bs such that F(x,y).

I don’t know if there is a natural way to extend this to mereology.

One might complain that partial equivalence is less natural than equivalence. I don’t think so. First, it is defined by two instead of three conditions, which makes it seem more natural. Second, examples of partial equivalence relations tend to be more natural than examples of full equivalence relations if our domain is all of reality. For instance, “same color”, “same shape”, “same size”, “same species”, etc., are all partial equivalence relations, since only things with color are the same color as themselves, only things with shape are the same shape as themselves, etc. To form full equivalences, one needs to stipulate awkward relations like “same color or both colorless”.

Fusion and the Axiom of Choice

Assume classical mereology. Then for any formula that has a satisfier, there is a fusion of all of its satisfiers. More precisely, if ϕ is a formula with z not a free variable in ϕ, then the universal closure of the following under all free variables is true:

1. ∃xϕ → ∃zF_ϕ, x(z)

where F_ϕ, x(z) says that z is a fusion of the satisfiers of ϕ with respect to the variable x (there is more than one account of what exactly the “fusion” is). This is the fusion axiom schema.

Stipulate that a region of physical space is a fusion of points.

Question: Is there a nonmeasurable region of (physical) space?

Assuming the language for formulas in our classical mereology is sufficiently rich, the answer is positive. For simplicity, suppose that physical space is Euclidean (the non-Euclidean case is handled by working in a small neighborhood which is diffeomorphic to a neighborhood of a Euclidean space). Let ψ be the isomorphism between the points of physical space and the mathematical space R³. Let ϕ be the formula ψ(x) ∈ y. Applying (1), we conclude that for any subset a of R³, there is a set of points of physical space that correspond to a under ψ. If we let a be one of the standard nonmeasurable subsets of R³, we get an affirmative answer to our question.

But now we have an interesting question:

2. What grounding or explanatory relation is there between the existence of a nonmeasurable region of physical space and the existence of a nonmeasurable subset of mathematical space?

The two simplest options are that one is explanatorily prior to the other. Let’s explore these.

Suppose the existence of a nonmeasurable physical region depends on the existence of the nonmeasurable set. Well, it is a bit strange to think of a concrete object—a region of physical space—as partly grounded in the existence of a set. This doesn’t sound quite right to me.

What about the other way around? This challenges the fairly popular doctrine that complex things entities are a free lunch given simples. For if the existence of the nonmeasurable region is prior to the existence of an abstract set, it seems that we actually have quite a significant metaphysical “effect” of this complex object.

Moreover, if the existence of the nonmeasurable region is not grounded in the existence of nonmeasurable set, whether or not there is grounding running the other way, we have a difficult question of why there is in fact a nonmeasurable region. Without relying on nonmeasurable sets, it doesn’t seem we can get the nonmeasurable region out of the axioms of classical mereology. It seems we need some sort of a mereological Axiom of Choice. How exactly to formulate that is difficult to say, but one version that is enough for our purposes would be that given any formula ρ(x,y) that expresses a non-empty equivalence relation on the simples satisfying ϕ, there is an object z such that if ϕ(x) then there is exactly one simple x′ such that ρ(x,x′) and x′ is a part of z, and every object that meets z meets some simple satsifying ϕ.

But my intuition is that a mereological Axiom of Choice would badly violate the doctrine that complex objects are a free lunch. If all we had in the way of complex-object-forming axioms were reflexivity, transitivity and fusion, then it would not be crazy to say that complex objects are a fancy way of talking about simples. But the “indeterministic” nature of the Axiom of Choice does not, I think, allow one to say that.

Modality and the Axiom of Choice

Suppose that the set theory of our world is a Solovay model, where we don’t have the Axiom of Choice (AC), and where every subset of the reals is Lebesgue measurable. Now imagine that God picks out a line in space, and defines the Vitali equivalence relation for points on that line (where two points are equivalent if and only if the distance between them is a rational number). It is then surely within God’s power to create a particle of some unexemplified type T at exactly one point in every equivalence class. There is nothing incoherent about that! But if God did that, then there would surely be a set of the points containing a particle of type T. And that set would be a nonmeasurable Vitali set.

So what?

Well, prima facie, there are three possibilities about the existence of nonmeasurable sets:

1. Necessarily, there are no nonmeasurable sets.

2. Necessarily, there are nonmeasurable sets.

3. It is contingent whether there are nonmeasurable sets.

My argument strongly suggests that if there are no nonmeasurable sets, it is nonetheless possible that there are nonmeasurable sets. Hence, (1) is ruled out.

So we have an argument for the disjunction of (2) and (3).

Now, I think a lot of people have the intuition that mathematical facts are necessary. If so, then (3) is ruled out. They will see this as an argument for (2).

I don’t see it that way myself: I am quite open to contingent mathematical truths.

More generally, the argument shows that:

4. For any set of disjoint nonempty subsets of the reals, it is possible that there is a choice function.

Again, if the existence of pure sets is not a contingent matter, we conclude AC is true for all subsets of the reals.

An odd thought about ZFC

The axioms of ZFC set theory can be divided into (a) the positive axioms, that say that a set with certain properties exist, and (b) two negative axioms that deny the existence of certain sets (Extensionality: given any set, there is no other set with the same members; Regularity: no irregular sets).

The positive axioms divide further into two classes: (i) those that are obvious special cases of naive set theory’s Axiom of Comprehension, and (ii) the Axiom of Choice.

Here is an alternate intellectual history thought experiment. Suppose we never discovered the contradiction in naive set theory or anything like it, maybe because we had a psychological block against thinking about non-self-membered sets, applying Cantor’s Theorem to the universal set, etc. The Axiom of Choice would continue to have an intuitive plausibility, and the “mathematical need” for it, say in the case of the Hahn-Banach Theorem, would likely still arise. And so we would be pulled to adopt it.

This makes me think this. The other positive axioms of ZFC (i.e., the positive axioms of ZF) have an ad hoc feel to them. They are special cases of Comprehension, carefully chosen to both give enough applications of Comprehension and to avoid contradiction (we hope). I feel that much of the plausibility of the other positive axioms of ZFC comes from their being special cases of the highly intuitive—but incoherent—Axiom of Comprehension. And that’s a little suspicious.

Normally one thinks of the Axiom of Choice as the most suspicious of ZFC’s axioms. But here we have a source of suspicion for axioms of ZFC that does not affect Choice.

Well, maybe. Maybe an enemy of Choice could say that both Choice and Comprehension are the fruit of the poisonous tree of principles of plenitude.

Truth directed scoring rules on an infinite space

A credence assignment c on a space Ω of situations is a function from the powerset of Ω to [0, 1], with c(E) representing one’s degree of belief in E ⊆ Ω.

An accuracy scoring rule s assigns to a credence assignment c on a space Ω and situation ω the epistemic utility s(c)(ω) of having credence assignment c when in truth we are in ω. Epistemic utilities are extended real numbers.

The scoring rule is strictly truth directed provided that if credence assignment c₂ is strictly truer than c₁ at ω, then s(c₂)(ω)>s(c₁)(ω). We say that c₂ is strictly truer than c₁ if and only if for every event E that happens at ω, c₂(E)≥c₁(E) and for every event E that does not happen at ω, c₂(E)≤c₁(E), and in at least one case there is strict inequality.

A credence assignment c is extreme provided that c(E) is 0 or 1 for every E.

Proposition. If the probability space Ω is infinite, then there is no strictly truth directed scoring rule defined for all credences, or even for all extreme credences.

In fact, there is not even a scoring rule that strictly truth directed when restricted to extreme credences, where an extreme credence is one that assigns 0 or 1 to every event.

This proposition uses the following result that my colleague Daniel Herden essentially gave me a proof of:

Lemma. If PX is the power set of X, then there is no function f : PX → X such that f(A)≠f(B) whenever A ⊂ B.

Now, we prove the Proposition. Fix ω ∈ Ω. Let s be a strictly truth directed scoring rule defined for all extreme credences. For any subset A of PΩ, define c_A to be the extreme credence function that is correct at ω at all and only the events in A, i.e., c_A(E)=1 if and only if ω ∈ E and E ∈ A or ω ∉ E and E ∉ A, and otherwise c_A(E)=0. Note that c_B is strictly truer than c_A if and only if A ⊂ B. For any subset A of PΩ, let f(A)=s(c_A)(ω).

Then f(A)<f(B) whenever A ⊂ B. Hence f is a strictly monotonic function from PPΩ to the reals. Now, if Ω is infinite, then the reals can be embedded in PΩ (by the axiom of countable choice, Ω contains a countably infinite subset, and hence PΩ has cardinality at least that of the continuum). Hence we have a function like the one the Lemma denies the existence of, a contradiction.

Note: This suggests that if we want strict truth directedness of a scoring rule, the scoring rule had better take values in a set whose cardinality is greater than that of the continuum, e.g., the hyperreals.

Proof of Lemma (essentially due to Daniel Herden): Suppose we have f as in the statement of the Lemma. Let ON be the class of ordinals. Define a function F : ON → A by transfinite induction:

• F(0)=f(⌀)

• F(α)=f({F(β):β < α}) whenever α is a successor or limit ordinal.

I claim that this function is one-to-one.

Let H_α = {F(δ):δ < α}.

Suppose F is one-to-one on β for all β < α. If α is a limit ordinal, then it follows that F is one-to-one on α. Suppose instead that α is a successor of β. I claim that F is one-to-one on α, too. The only possible failure of injectivity on α could be if F(β)=F(γ) for some γ < β. Now, F(β)=f(H_β) and F(γ)=f(H_γ). Note that H_γ ⊂ H_β since F is one-to-one on β. Hence f(H_β)≠f(H_γ) by the assumption of the Lemma. So, F is one-to-one on ON by transfinite induction.

But of course we can’t embed ON in a set (Burali-Forti).

A superpower

Imagine Alice claimed she could just see, with reliability, which unprovable large cardinal axioms are true. We would be initially sceptical of her claims, but we could imagine ways in which we could come to be convinced of her having such an ability. For instance, we might later be able to prove a lot of logical connections between these axioms (say that axiom A₁₂ implies axiom A₁₄) and then find that Alice’s oracular pronouncements matched these logical connections (she wouldn’t, for instance, affirm A₁₂ while denying A₁₄) to a degree that would be very hard to explain as just luck.

Suppose, then, that we have come to be convinced that Alice has the intuitive ability to just see which large cardinal axioms are true. This would be some sort of uncanny superpower. The existence of such a superpower would sit poorly with naturalism. An intuition like Ramanujan’s about the sums of series could be explained by naturalism—we could simply suppose that his brain unconsciously sketched proofs of various claims. But an intuition about large cardinal axioms wouldn’t be like that, since these axioms are not provable.

Now as far as we know, there is no one exactly like Alice who just has reliable intuitions about large cardinal axioms. But our confidence in the less abstruse axioms of Zermelo-Fraenkel set theory—intuitive axioms like the axiom of replacement—commits us to thinking that either we in general, or those most expert in the matter, are rather like Alice with respect to these less abstruse axioms. The less abstruse axioms are just as unprovable as the more abstruse ones that Alice could see. Therefore, it seems, if Alice’s reliable intuition provided an argument against naturalism, our own (or our experts’) intuition about the more ordinary axioms, an intuition which we take to be reliable, gives us an argument against naturalism. Seeing the axiom of replacement to be true is just as much a superpower as would be Alice’s seeing that, say, measurable cardinals exist (or that they do not exist).

Is our universe of sets minimal?

Our physics is based on the real numbers. Physicists use the real numbers all over the place: quantum mechanics takes place in a complex Hilbert space, and the complex numbers are isomorphic to pairs of real numbers, while relativity theory takes place in a manifold that is locally isomorphic to a Lorentzian four-dimensional real space.

The real numbers are one of an infinite family of mathematical objects known as real closed fields. Other real closed fields than the real numbers could be used in physics instead—for instance, the hyperreals—and I think we would have the same empirical predictions. But the real numbers are simpler and more elegant: for instance, they are the only Dedekind-complete and the minimal Cauchy-complete real closed field.

At the same time, the mathematics behind our physics lives within a set theoretic universe. That set theoretic universe is generally not assumed to be particularly special. For instance, I know of no one who assumes that our set theoretic universe is isomorphic to Shepherdson’s/Cohen’s minimal model of set theory. On the contrary, it is widely assumed that our set theoretic universe has a standard transitive set model, which implies that it is not minimal, and few people seem to believe the Axiom of Constructibility which would hold in a minimal model.

This seems to me be rationally inconsistent. If we are justified in thinking that the mathematics underlying the physical world is based on a particularly elegant real closed field even though other fields fit our empirical data, we would also be justified in thinking it’s based on a particularly elegant universe of sets even though other universes fit our empirical data.

(According to Shipman, the resulting set theory would be one equivalent to ZF + V=L + “There is no standard model”.)

Set theory and physics

Assume the correct physics has precise particle positions (similar questions can be asked in other contexts, but the particle position context is the one I will choose). And suppose we can specify a time t precisely, e.g., in terms of the duration elapsed from the beginning of physical reality, in some precisely defined unit system. Consider two particles, a and b, that exist at t. Let d be the distance between a and b at t in some precisely definable unit system.

Here’s a question that is rarely asked: Is d a real number?

This seems a silly question. How could it not be? What else could it be? A complex number?

Well, there are at least two other things that d could be without any significant change to the equations of physics.

First, d could be a hyperreal number. It could be that particle positions are more fine-grained than the reals.

Second, d could be what I am now calling a “missing number”. A missing number is something that can intuitively be defined by an English (or other meta-language) specification of an approximating “sequence”, but does not correspond to a real number in set theory. For instance, we could suppose for simplicity that d lies between 0 and 1 and imagine a physical measurement procedure that can determine the nth binary digit of d. Then we would have an English predicate M_d(n) which is true just in case that procedure determined the n binary digit to be 1. But it could turn out that in set theory there is no set whose members are the natural numbers n such that M_d(n). For the axioms of set theory only guarantee the existence of a set defined using the predicates of set theory, while M_d is not a predicate of set theory. The idea of such “missing numbers” is coherent, at least if our set theory is coherent.

It seems reasonable to say that d is indeed a real number, and to say similar things about any other quantities that can be similarly physically specified. But what guarantees such a match between set theory and physics? I see four options:

1. Luck: it’s just a coincidence.

2. Our set theory governs physics.

3. Physics governs our set theory.

4. There is a common governor to our set theory and physics.

Option 1 is an unhappy one. Option 4 might be a Cartesian God who freely chooses both mathematics and physics.

Option 2 is interesting. On this story, there is a Platonically true set theory, and then the laws of physics make reference to it. So it’s then a law of physics that distances (say) always correspond to real numbers in the Platonically true set theory.

Option 3 comes in at least two versions. First, one could have an Aristotelian story on which mathematics, including some version of set theory, is an abstraction from the physical world, and any predicates that we can define physically are going to be usable for defining sets. So, physics makes sets. Second, one could have a Platonic multiverse of universes of sets: there are infinitely many universes of sets, and we simply choose to work within those that match our physics. On this view, physics doesn’t make sets, but it chooses between the universes of sets.

The Math Tea argument

The Math Tea argument is an argument that there are real numbers that can’t be defined. The idea is this: there are only countably many definitions of real numbers (e.g., π^e or "The middle root of the polynomial x³ − 5x² + 2x + 4"), and uncountably many real numbers, so there are real numbers that have no definitions.

Elegant as this argument is, it has crucial set-theoretic flaws. For instance, there is no guarantee that there is a set of all the definable real numbers. The axioms of set theory tell us that for any predicate F in the language of set theory there is a set of all the numbers that satisfy F. But the predicate "is definable" is in English, not in set theory.

We can, however, argue for the following weaker claim. Assume set theory is true. Then either:

1. There is a real number that cannot be defined in the language of set theory, or

2. "A real number is missing": there is an English language formula F(n) whose only semantic predicate is set-theoretic satisfaction such that there is no real number x whose nth digit after the decimal point is 1 if F(n) and is 0 if not F(n).

Here is the argument. A formula of set-theory defines a real number if it has exactly one free variable and is satisfied by precisely one real number. Say that F(n) if and only if the nth formula of set theory (in lexicographic ordering) defining a real number defines a real number that does not have a 1 in the nth place after the decimal point. The only semantic predicate in F(n) is set-theoretic satisfaction. Suppose (2) is false. Then there is a real number x whose nth digit after the decimal point is 1 if F(n) and is 0 if not F(n). If x can be defined in the language of set theory by a formula ϕ, then suppose ϕ is the nth real-number-defining formula. Then F(n) if and only if x does not have a 1 in the nth place. But x has a 1 in the nth place if and only if F(n). Contradiction! So, x cannot be defined, and hence (1) is true.

Logically speaking, if ZF is consistent, ZFC is consistent both with (1) (this follows by letting the digits of x be defined by the set of all set-theoretic truths and noting that if ZF is consistent, we can consistently suppose there is a set of all set-theoretic truths, but that set of course cannot be defined) and with the denial of (1).

But philosophically speaking, we might reasonably say that (2) would imply that "there aren’t enough real numbers", which sounds wrong, so it seems more reasonable to accept (1) instead.

Is there a set of all set-theoretic truths?

Is there a set of all set-theoretic truths? This would be the set of sentences (in some encoding scheme, such as Goedel numbers) in the language of set theory that are true.

There is a serious epistemic possibility of a negative answer. If ZF is consistent, then there is a model M of ZFC such that every object in M is definable, i.e., for every object a of M, there is a defining formula ϕ(x) that is satisfied by a and by a alone in M (and if there is a transitive model of ZF, then M can be taken to be transitive). In such a model, it follows from Tarski’s Indefinability of Truth that there is no set of all set-theoretic truths. For if there were such a set, then that set would be definable, and we could use the definition of that set to define truth. So, if ZF is consistent, there is a model M of ZFC that does not contain a set of all the truths in M.

Interestingly, however, there is also a serious epistemic possibility of a positive answer. If ZF is consistent, then there is a model M of ZFC that does contain a set of all the truths in M. Here is a proof. If ZF is consistent, so is ZFC. Let ZFC^T be a theory whose language is the language of set theory with an extra constant T, and whose axioms are the axioms of ZFC with the schemas of Separation and Replacement restricted to formulas of ZFC (i.e., formulas not using T), plus the axiom:

1. ∀x(x ∈ T → S(x))

where S(x) is a sentence saying that x is the code for a sentence (this is a syntactic matter, so it can be specified explicitly), and the axiom schema that has for every sentence ϕ with code n:

2. ϕ ↔ n ∈ T.

Any finite collection of the axioms of ZFC^T is consistent. For let M be a model of ZFC (if ZF is consistent, so is ZFC, so it has a model). Then all the axioms of ZFC will be satisfied in M. Furthermore, for any finite subset of the additional axioms of ZFC^T, there is an interpretation of the constant T under which those axioms are true. To see this, suppose that our finite subset contains (1) (no harm throwing that in if it’s not there) and the instances ϕ_i ↔ n_i ∈ T of (2) for i = 1, ..., m. It is provable from ZF and hence true in M that there is a set t such that x ∈ t if and only if x = n₁ and ϕ₁, or x = n₂ and ϕ₂, …, or x = n_m and ϕ_m.

Moreover, any such set can be proved in ZF to satisfy:

3. ∀x(x ∈ t → S(t)).

Interpreting T to be that set t in M will make the finite subset of the additional axioms true.

So, by compactness, ZFC^T has an interpretation I in some model M. In M there will be an object t such that t = I(T). That object t will be a set of all the truths in M that do not contain the constant T. Now consider the interpretation I⁻ of ZFC in M, which is I without any assignment of a value to the constant T (since T is not a constant of ZFC). Then ZFC will be true in M under I⁻. Moreover, the object t in M will be a set of all the truths in M.

So, if ZF is consistent, then there is a model of ZFC with a set of all set-theoretic truths and a model of ZFC without a set of all set-theoretic truths.

The latter claim may seem to violate the Tarski Indefinability of Truth. But it doesn’t. For that set of all truths will not itself be definable. It will exist, but there won’t be a formula of set theory that picks it out. There is nothing mathematically new in what I said above, but it is an interesting illustration of how one can come close to violating Indefinability of Truth without actually violating it.

Now, what if we take a Platonic view of the truths of set theory? Should we then say that there really is a set of all set-theoretic truths? Intuitively, I think so. Otherwise, our class of all sets is intuitively “missing” a subset of the set of all sentences. I am inclined to think that the Axioms of Separation and Replacement should be extended to include formulas of English (and other human languages), not just the formulas expressible in set-theoretic language. And the existence of the set of all set-theoretic truths follows from an application of Separation to the sentence “n is the code for a sentence of set theory that is true”.

Arbitrariness and contingency

I’ve come to be impressed by the idea that where there is apparent arbitrariness, there is probably contingency in the vicinity.

The earth and the moon on average are 384400 km apart. This looks arbitrary. And here the fact itself is contingent.

Humans have two arms and two legs. This looks arbitrary. But it is actually a necessary truth. However there is contingency in the vicinity: it is a contingent fact that humans, rather than eight-armed intelligent animals, exist on earth.

Ethical obligations have apparent arbitrariness, too. For instance, we should prefer mercy to retribution. Here, there are two possibilities. First, perhaps it is contingent that we should prefer mercy to just retribution. The best story I know which makes that work out is Divine Command Theory: God commands us to prefer mercy to just retribution but could have commanded the opposite. Second, perhaps it is necessary that we should prefer mercy to retribution, because our nature requires it, but it is contingent that we rather than beings whose nature carries the opposite obligation exist.

Now here is where I start to get uncomfortable: mathematics. When I think about the vast number of possible combinations of axioms of set theory, far beyond where any intuitions apply, axioms that cannot be proved from the standard ZFC axioms (unless these are inconsistent), it’s all starting to look very arbitrary. This pushes me to one of three uncomfortable positions:

• anti-realism about set theory

• Hamkins’ set-theoretic multiverse

• contingent mathematical truth.

Independence of FOL-validity

A sentence ϕ of a dialect of First Order Logic is FOL-valid if and only if ϕ is true in every non-empty model under every interpretation. By the Goedel Completeness Theorem, ϕ is valid if and only if ϕ is a theorem of FOL (i.e., has a proof from no axioms beyond any axioms of FOL). (Note: This does not use the Axiom of Choice since we are dealing with a single sentence.)

Here is a meta-logic fact that I think is not as widely known as it should be.

Proposition: Let T be any consistent recursive theory extending Zermelo-Fraenkel set theory. Then there is a sentence ϕ of a dialect of First Order Logic such that according to some models of T, ϕ is FOL-valid (and hence a theorem of FOL) and according to other models of T, ϕ is not FOL-valid (and hence not a theorem of FOL).

Note: The claim that ϕ is FOL-valid according to a model M is shorthand for the claim that a certain complex arithmetical claim involving the Goedel encoding of ϕ is true according to M.

The Proposition is yet another nail in the coffins of formalism and positivism. It tells us that the mere notion of FOL-theoremhood has Platonic commitments, in that it is only relative to a fixed family of universes of sets (or at least a fixed model of the natural numbers or a fixed non-recursive axiomatization) does it make unambiguous sense to predicate FOL-theoremhood and its lack. Likewise, the very notion of valid consequence, even of a finite axiom set, carries such Platonic commitments.

Proof of Proposition: Let G be a Rosser-tweaked Goedel sentence for T with G being Σ₁ (cf. remarks in Section 51.3 here). Then G is independent of T. In ZF, and hence in T, we can prove that there is a Turing machine Q that halts if and only if G holds. (Just make Q iterate over all natural numbers, halting if the number witnesses the existential quantifier at the front of the Σ₁ sentence G.) But one can construct an FOL-sentence ϕ such that one can prove in ZF that ϕ is FOL-valid if and only if Q halts (one can do this for any Turing machine Q, not just the one above). Hence, one can prove in T that ϕ is FOL-valid if and only if I holds.

Thus, in T it is provable that ϕ is FOL-valid if and only if G holds. But T is a consistent theory (otherwise one could formalize in T the proof of its inconsistency). Since G is independent of T, it follows that the FOL-validity of ϕ is as well.

Something that has no reasonable numerical epistemic probability

I think I can give an example of something that has no reasonable (numerical) epistemic probability.

Consider Goedel’s Axiom of Constructibility. Goedel proved that if the Zermelo-Fraenkel (ZF) axioms are consistent, they are also consistent with Constructibility (C). We don’t have any strong arguments against C.

Now, either we have a reasonable epistemic probability for C or we don’t.

If we don’t, here is my example of something that has no reasonable epistemic probability: C.

If we do, then note that Goedel showed that ZF + C implies the Axiom of Choice, and hence implies the existence of non-measurable sets. Moreover, C implies that there is a well-ordering W on the universe of all sets that is explicitly definable in the language of set theory.

Now consider some physical quantity Q where we know that Q lies in some interval [x − δ, x + δ], but we have no more precise knowledge. If C is true, let U be the W-smallest non-measurable subset of [x − δ, x + δ].

Assuming that we do have a reasonable epistemic probability for C, here is my example of something that has no reasonable epistemic probability: C is false or Q is a member of U.

Optimalism about necessity

There are many set-theoretic claims that are undecidable from the basic axioms of set theory. Plausibly, the truths of set theory hold of necessity. But it seems to be arbitrary which undecidable set-theoretic claims are true. And if we say that the claims are contingent, then it will be arbitrary which claims are contingent. We don’t want there to be any of the “arbitrary” in the realm of necessity. Or so I say. But can we find a working theory of necessity that eliminates the arbitrary?

Here are two that have a hope. The first is a variant on Leslie-Rescher optimalism. While Leslie and Rescher think that the best (narrowly logically) scenario must obtain, and hence endorse an optimalism about truth, we could instead affirm an optimalism about necessity:

1. Among the collections of propositions, that collection of propositions that would make for the best collection of all the necessary truths is in fact the collection of all the necessary truths.

And just as it arguably follows from Leslie-Rescher optimalism that there is a God, since it is best that there be one, it arguably follows from this optimalism about necessity that there necessarily is a God, since it is best that there necessarily be a God. (By the way, when I once talked with Rescher about free will, he speculatively offered me something that might be close to optimalism about necessity.)

Would that solve the problem? Maybe: maybe the best possible—both practically and aesthetically—set theory is the one that holds of necessary truth.

I am not proposing this theory as a theory of what necessity is, but only of what is in fact necessary. Though, I suppose, one could take the theory to be a theory of what necessity is, too.

Alternately, we could have an optimalist theory about necessity that is theistic from the beginning:

2. A maximally great being is the ground of all necessity.

And among the great-making properties of a maximally great being there are properties like “grounding a beautiful set theory”.

I suspect that (1) and (2) are equivalent.

More remarks on Aristotelian set theory

If we have an Aristotelian picture of abstracta, we should expect that what mathematical objects exist differs between possible worlds.

For the Aristotelian, abstract objects are abstractions from concrete things. So we shouldn’t expect the same full panoply of sets regardless of what concrete things there are. For instance, suppose that the universe contains exactly three point particles, A, B and C. Then we can immediately abstract from these particle positions distance ratios like AB : BC, AC : AB and AC : BC. These ratios are then represented by real numbers. So we are going to have these real numbers. More sophisticated abstractive processes may well generate other real numbers: for instance, we will have a real number representing the ratio of the height of the triangle drawn from A to the base BC. And given a real number, we might be able to use purely abstract processes to generate further real numbers: given a and b, we may generate a + b and ab, say. But there is no reason to think that these abstract processes will generate the same collection of real numbers regardless of what the three particle positions we start with are.

So, what real numbers exist should vary between possible worlds. But every real number defines a subset of the natural numbers (just write the real number in binary, and let the nth bit decide if n is in the subset or not). If the real numbers vary between possible worlds, so do the subsets of the natural numbers. In particular, we should expect that in different possible worlds, a different set counts as ``the power set’’ of the natural numbers.

Furthermore, what bijections there are between sets will vary between possible worlds. Thus, if we see the question of whether two sets have the same count of members as having the same answer in every world where the two sets exist, we cannot take the standard Cantorian account of the size of a set. Instead, we may want to generate the concept of sameness of size from bijections in different worlds. Thus, we may try to say that two sets A and B are the same size at level 0 provided that there is a bijection between A and B. Then we say that A and B are the same size at level n provided that possibly there is a set C that is the same size as A at level p and the same size as B at level q and n ≥ 1 + p + q. Finally, we say that A and B are the same size simpliciter provided that they are the same size at some finite level. This is complicated, and I haven’t checked under what assumptions it generates a transitive relation (it’s plausibly reflexive and symmetric).

Anyway, the point is this: It is an interesting and not easy philosophical project to work out the set-theoretic consequences of Aristotelianism. This could make a good dissertation.