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.

Counting with plural quantification

I’ve been playing with the question of what if anything we can say with plural quantification that we can’t say with, say, sets and classes.

Here’s an example. Plural quantification may let us make sense of cardinality comparisons that go further than standard methods. For instance, if our mathematical ontology consists only of sets, we can still define cardinality comparisons for pluralities of sets:

1. Suppose the xx and the yy are pluralities of sets. Then |xx| ≤ |yy| iff there are zz that are an injective function from the xx to the yy.

What is an injective function from the xx to the yy? It is a plurality, the zz, such that each of the zz is an ordered pair of classes, and such that for any a among the xx there is unique b among the yy such that (a,b) is among the zz and for any b among the yy there is at most one a among the xx such that [a,b] is among the zz.

This lets us say stuff like:

2. There are more sets than members of any set.

Or if our mathematical ontology includes sets and classes, we can compare the cardinalities of pluralities of classes using (1), as long as we can define an ordered pair of classes—which we can, e.g., by identifying the ordered pair of a and b with the class of all ordered pairs (i,x) where i = 0 and x ∈ a or where i = 1 and x ∈ b.

This would then let us say (and prove using a variant of Cantor’s diagonal argument, assuming Comprehension for pluralities):

3. There are more classes than sets.

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).

An argument that there are more positive odd numbers than positive even ones

Suppose that we embrace these intuitive theses about sets of natural numbers:

1. If A is a proper subset of B, then B has more members than A.

2. If A has at least as many members as B, then 1 + A has at least as many members as 1 + B, where 1 + C = {1 + c : c ∈ C} is C shifted over by one to the right.

3. Either there are more odd positive numbers than even positive numbers or there are at least as many even positive numbers as odd positive numbers.

Let O be the positive odd numbers and E be the positive even numbers. Write A ≲ B to mean that B has at least as many members as A, and write A ∼ B to mean that they have the same number of members. For a reductio, suppose that O ≲ E. Then 1 + O ≲ 1 + E by (2). But 1 + O = E. Thus, O ≲ E ≲ 1 + E. But 1 + E is all the odd numbers starting with three, which is a proper subset of O, which contradicts (1). So we do not have O ≲ E, and hence by (3):

4. There are more odd positive numbers than even positive numbers!

Of course, the usual Cantorian way of comparing sizes of sets rejects (1).

I think the non-Cantorian’s best bet is either to embrace the conclusion (4) or to deny the special case of totality of comparison in (3). In either case, the non-Cantorian needs to deny the intuitive claim that:

5. There are equally many odd positive numbers and even positive numbers.

Note that (2) does not say or imply that 1 + A has the same number of members as A. Since that would imply that {1, 2, 3, ...} and {2, 3, 4, ...} have the same number of members, that would beg the question against the typical non-Cantorian for whom (1) is a central intuition. One might also wonder whether there is a way of comparing sets of natural numbers that satisfies (1)–(3): the answer is yes (even with (3) generalized to all pairs of sets of naturals).

Causal countability

Say that a set S is causally countable if and only if it is metaphysically possible for someone to causally think through all the items in S. To causally think through the xs is to engage in a step-by-step sequential process of thinking about individual xs such that:

1. Every individual one of the xs is thought about in precisely one step of the process.

2. Each step in the process has at most one successor step.

3. With at most one exception, each step in the process is the successor of exactly one step.

4. The successor of a step causally depends on it.

Causal finitism then ensures that any causally countable set is countable in the mathematical sense. And, conversely, given some assumptions about reality being rational, any countable set is causally countable.

However, causal countability escapes the Skolem paradox, because of causal finitism and how it is anchored in the non-mathematical notion of causation.

Proper classes as merely possible sets

This probably won’t work out, but I’ve been thinking about the Cantor and Russell Paradoxes and proper classes and had this curious idea: Maybe proper classes are non-existent possible sets. Thus, there is actually no collection of all the sets in our world, but there is another possible world which contains a set S whose members are all the sets of our world. When we talk about proper classes, then, we are talking about merely possible sets.

Here is the story about the Russell Paradox. There can be a set R whose members are all the actual world’s non-self-membered sets. (In fact, since by the Axiom of Foundation, none of the actual world’s sets are self-membered, R is a set whose members are all the actual world’s sets.) But R is not itself one of the actual world’s sets, but a set in another possible world.

The story about Cantor’s Paradox that this yields is that there can be a cardinality greater than all the cardinalities in our world, but there actually isn’t. And in world w₂ where such a cardinality exists, it isn’t the largest cardinality, because its powerset is even larger. But there is a third world which has a cardinality larger than any in w₂.

It’s part of the story that there cannot be any collections with non-existent elements. Thus, one cannot form paradoxical cross-world collections, like the collection of all possible sets. The only collections there are on this story are sets. But we can talk of collections that would exist counterfactually.

The challenge to working out the details of this view is to explain why it is that some sets actually exist and others are merely possible. One thought is something like this: The sets that actually exist at w are those that form a minimal model of set theory that contains all the sets that can be specified using the concrete resources in the world. E.g., if the world contains an infinite sequence of coin tosses, it contains the set of the natural numbers corresponding to tosses with heads.

Counting infinitely many headaches

If the worries in this post work, then the argument in this one needs improvement.

Suppose there are two groups of people, the As and the Bs, all of whom have headaches. You can relieve the headaches of the As or of the Bs, but not both. You don’t know how many As or Bs there are, or even whether the numbers are finite or finite. But you do know there are more As than Bs.

Obviously:

1. You should relieve the As’ headaches rather than the Bs’, because there are more As than Bs.

But what does it mean to say that there are more As than Bs? Our best analysis (simplifying and assuming the Axiom of Choice) is something like this:

2. There is no one-to-one function from the As to the Bs.

So, it seems:

3. You should relieve the As’ headache rather than the Bs’, because there is no one-to-one function from the As to the Bs.

For you should be able to replace an explanation by its analysis.

But that’s strange. Why should the non-existence of a one-to-one function from one set or plurality to another set or plurality explain the existence of a moral duty to make a particular preferential judgment between them?

If the number of As and Bs is finite, I think we can do better. We can then express the claim that there are more As than Bs by an infinite disjunction of claims of the form:

4. There exist n As and there do not exist n Bs,

which claims can be written as simple existentially quantified claims, without any mention of functions, sets or pluralities.

Any such claim as (4) does seem to have some intuitive moral force, and so maybe their disjunction does.

But in the infinite case, we can’t find a disjunction of existentially quantified claims that analysis the claim that there are more As than Bs.

Maybe what we should say is that “there are more As than Bs” is primitive, and the claim about there not being a one-to-one function is just a useful mathematical equivalence to it, rather than an analysis?

The thoughts here are also related to this post.

A really weird place in conceptual space regarding infinity

Here’s a super-weird philosophy of infinity idea. Maybe:

1. The countable Axiom of Choice is false,

2. There are sets that are infinite but not Dedekind infinite, and

3. You cannot have an actual Dedekind infinity of things, but

4. You can have an actual non-Dedekind infinity of things.

If this were true, you could have actual infinites, but you couldn’t have Hilbert’s Hotel.

Background: A set is Dedekind-infinite if and only if it is the same cardinality as a proper subset of itself. Given the countable Axiom of Choice, one can prove that every infinite set is Dedekind infinite. But we need some version of the Axiom of Choice for the proof (assuming ZF set theory is consistent). So without the Axiom of Choice, there might be infinite but not Dedekind-infinite sets (call them “non-Dedekind infinite”). Hilbert’s Hotel depends on the fact that its rooms form a Dedekind infinity. But a non-Dedekind infinity would necessarily escape the paradox.

Granted, this is crazy. But for the sake of technical precision, it’s worth noting that the move from the impossibility of Hilbert’s Hotel to the impossibility of an actual infinite depends on further assumptions, such as the countable Axiom of Choice or some assumption about how if actual Dedekind infinities are impossible, non-Dedekind ones are as well. These further assumption are pretty plausible, so this is just a very minor point.

I think the same technical issue affects the arguments in my Infinity, Causation and Paradox book (coming out in August, I just heard). In the book I pretty freely use the countable Axiom of Choice anyway.

Set size and paradox

Some people want to be able to compare the sizes of sets in a way that respects the principle:

1. If A is a proper subset of B, then A ≤ B but not B ≤ A.

They do this in order to escape what they think are paradoxical consequences of the Cantorian way of comparing sizes. But from one paradox they fall into another. For the following can be proved without the Axiom of Choice:

2. If there is a transitive and reflexive relation ≤ between sets of reals (or just countable sets of reals) that satisfies (1), then the Banach-Tarski Paradox holds.

And the Banach-Tarski Paradox is arguably more paradoxical than the paradoxes of infinity that (1) is supposed to avoid.

Degrees of freedom

The number of degrees of freedom in a system is the number of numerical parameters that need to be set to fully determine the system. Scientists have an epistemic preference for theories that posit systems with fewer degrees of freedom.

But any system with n real-valued degrees of freedom can be redescribed as a system with only one real-valued degree of freedom, where n is finite or countable. For instance, consider a three-dimensional system which is fully described at any given time by a position (x, y, z) in three-dimensional space. We can redescribe x, y and z by real-valued variable X, Y and Z in the interval from 0 and 1, for instance by letting X = 1/2 + π⁻¹arctan x and so on. Now write out these new variables in decimal:

• X = 0.X₁X₂X₃...
• Y = 0.Y₁Y₂Y₃...
• Z = 0.Z₁Z₂Z₃...

Finally, let:

• W = 0.X₁Y₁Z₁X₂Y₂Z₂X₃Y₃Z₃....

Then W encodes all the information about X, Y and Z, which in turn encode all the information about (x, y, z) and hence about our system at a given time. (This obviously generalizes to any finite number of degrees of freedom. For a countably infinite one, things are slightly more complicated, but can still be done.)

There is a lesson here, even if not a particularly deep one. The epistemic preference for theories that have fewer degrees of freedom cannot be separated from the the epistemic preference for simpler theories. For of course rewriting a theory that made use of (x, y, z) in terms of W is in practice going to make for a significantly messier theory. So we cannot replace a simplicity preference by a preference for a low number of degrees of freedom.

Objection: Instead of a simplicity preference, we may a priori specify that laws of nature be given by differential equations in terms of the variables involved. But when, say, x, y and z vary smoothly over time, it is very unlikely that W will do so as well.

Response: But one can find a replacement for W that is smoothly related to x, y and z up to any desired degree of precision, and hence we can give a differential-equation based theory that fits the experimental data pretty much equally well but has only one degree of freedom.

Universal countable numerosity: A hypothesis worth taking seriously?

Here’s a curious tale about sets and possible worlds: What sets there are varies between metaphysically possible worlds and for any possible world w₁, the sets at w₁ satisfy the full ZFC axioms and there is also a possible world w₂ at which there exists a set S such that:

1. At w₂, there is a bijection of S onto the natural numbers (i.e., a function that is one-to-one and whose range is all of the natural numbers).

2. The members of S are precisely the sets that exist at w₁.

Suppose that this tale is true. Then assume S5 and this further principle:

3. If two sets A and B are such that possibly there is a bijection between them, then they have the same numerosity.

(Here I distinguish between “numerosity” and “cardinality”: to have the same cardinality, they need to actually have a bijection.) Then:

4. Necessarily, all infinite sets have the same numerosity, and in particular necessarily all infinite sets have the same numerosity as the set of natural numbers.

For if A and B are infinite sets in w₁, then at w₂ they are subsets of the countable-at-w₂ set S, and hence at w₂ they have a bijection with the naturals, and so by (3) they have the same numerosity.

Given the tale, there is then an intuitive sense in which all infinite sets are the same size. But it gets more fun than that. Add this principle:

5. If two pluralities are such that possibly there is a bijection between them, then the two pluralities have the same numerosity.

(Here, a bijection between the xs and the ys is a binary relation R such that each of the xs stands in R to a unique one of the ys, and vice versa.) Then:

6. Necessarily, the plurality of sets has the same numerosity as the plurality of natural numbers.

For if the xs are the plurality of sets of w₁, then there will be a world w₂ and a countable-at-w₂ set S such that the xs are all and only the members of S. Hence, there will be a bijection between the xs and the natural numbers at w₂, and hence at w₁ they will have the same numerosity by (5).

So if my curious tale is true, not only does each infinite set have the same numerosity, but the plurality of sets has the same numerosity as each of these infinite sets.

We can now say that a set or plurality has countable numerosity provided that it is either finite or has the same numerosity as the naturals. Then the conclusion of the tale is that each set (finite and infinite), as well as the plurality of sets, has countable numerosity. I.e., universal countable numerosity.

But hasn’t Cantor proved this is all false? Not at all. Cantor proved that this is false if we put “cardinality” in place of “numerosity”, where cardinality is defined in terms of actual bijections while numerosity is defined in terms of possible bijections. And I think that possible bijections are a better way to get at the intuitive concept of the count of members.

Still, is my curious tale mathematically consistent? I think nobody knows. Will Brian, a colleague in the Mathematics Department, sent me a nice proof which, assuming my interpretation of its claims is correct, shows that if ZFC + “there is an inaccessible cardinal” is consistent, then so is my tale. And we have no reason to doubt that ZFC + “there is an inaccessible cardinal” is consistent. So we have no reason to doubt the consistency of the tale.

As for its truth, that's a different matter. One philosophically deep question is whether there could in fact be so much variation as to what the sets are in different metaphysically possible worlds.

More on comparing infinities

Some people don't want to say that there are just as many even natural numbers as natural numbers. But suppose that you and I will spend eternity singing numbers in harmony. You will sing every natural number in sequence: 1, 2, 3, ..., with a long pause for applause in between. And while you sing n, I will sing 2n. We will vary the speed of our singing to ensure that we take equal amounts of time. Clearly:

1. The number of natural numbers = the number of your performances.
2. The number of your performances = the number of my performances.
3. The number of my performances = the number of even natural numbers.
4. So, the number of natural numbers = the number of even natural numbers.

Premises (1)-(3) are obviously true, and I don't understand what "the same number" relation could mean if it's not transitive.

Might all infinities be the same size?

A lot of people find Cantor's discovery that there are different infinities paradoxical. To be honest, there are many counterintuitive things involving infinities, but this one doesn't strike me as particularly counterintuitive. Nonetheless, I want to explore the possibility that while Cantor's Theorem is of course true, it doesn't actually show that infinity comes in different sizes. Cantor's Theorem says that if A is a set, then there is no pairing (i.e., bijection) between the members of A and those of the power set PA. It follows that there are different (cardinal, but in an intuitive rather than mathematical sense) sizes of infinity given this Pairing Principle:

1. PP: Two sets A and B have the same size if and only if there is a pairing between them.

Given PP and Cantor's Theorem, if A is an infinite set, then A is a different size from PA. But of course PA is infinite if A is, so there are infinite sets of different size.

A number of people have disputed the sufficiency part of PP, since it gives rise to the counterintuitive consequence that the set of primes and the set of integers have the same size as you can pair them up. But you really shouldn't both complain that there are different infinities and that PP makes the primes and the integers have the same infinite size. I am going to leave the sufficiency of PP untouched, but suggest that the pairing condition might not be necessary for sameness of size, and I will offer an alternative. That alternative seems to leave open the possibility that all infinities are the same.

To think about this, start with this thought experiment. Imagine that there is a possible world w that has some but not all of the actual world's sets, but that it still has enough sets to satisfy the ZFC axioms just as (I shall suppose) the sets of the actual world do. The set membership relation in w will be the same as in the actual world in the sense that if A is a set that exists both w and the actual world, then A has exactly the same members in both worlds (and in particular, all the actual world members of A exist in w). Then here is something that might well happen. We have two sets A and B that exist both in the actual world and in w. In the actual world, there is a pairing f between A and B. A pairing is just a set of ordered pairs satisfying some additional constraints (the first element is always from A and the second is always from B, and each element of A occurs as the first element of exactly one pair, and each element of B occurs as the second element of exactly one pair). It might, then, be the case that although A and B exist in w, f does not--it exists in the actual world but not in the impoverished world w. It might even be the case that no pairing between A and B exists in the impoverished world. In that case, we have something very interesting: A and B satisfy the pairing condition in PP in the actual world but fail to satisfy it in w. If we are to satisfy the ZFC in w, this can only happen if both A and B are infinite.

Things might go even further. We might suppose that w only contains sets that are countable in the actual world. The mathematical (much less metaphysical!) possibility of such a scenario cannot be proved from ZFC if ZFC is consistent, but it follows from the Standard Model Hypothesis which a lot of set theorists find plausible. If w only contains sets that are actually countable, then any infinite sets in w will have a pairing in the actual world. There is, thus, an important sense in which from the broader point of view of the actual world, all infinite sets in w have the same size. But w is impoverished. There are pairings that exist in the actual world but don't exist in w, and so applying PP inside w will yield the conclusion that the infinite sets in w come in different sizes. However, intuitively, it still seems true to say that these sets in w are all the same size, but w just doesn't have enough pairings to see this.

Here's one way to argue for this interpretation of the hypothesis. Plausibly:

2. Pairing-Sufficiency: If there is a pairing between sets A and B, they are the same size.
3. Absolute-Size: If two sets are the same size in one possible world, they are the same size in any world in which they both exist.

Pairing-Sufficiency is one half of PP. Given Pairing-Sufficiency and Absolute-Size, if two sets have a pairing in any possible world, including the actual one, they have the same size in every world, including w. Thus, in w all the infinite sets in fact have the size, but you wouldn't know that if your tools were restricted to the pairings in w.

Thinking about the above scenario suggests a modification of PP to a Possible Pairing Principle:

4. PPP: Two sets A and B have the same size if and only if possibly there is a pairing between them.

Given that the members of a set cannot vary between possible worlds, I think PPP is at least as plausible as PP. Moreover, I think that if there are any cases (like my hypothesis that a world like w is possible) where PPP and PP come apart, we should side with PPP. Here's why. I think we go for PP as an abstraction from our general method of comparing the sizes of pluralities by pairing. (One imagines a pre-numerate people trading goats and spears in 1:1 ratio by lining up each goat with a spear.) But the natural abstraction from our general method is that if one could pair up the two sets, then and only then they are the same size. So PPP is the natural hypothesis. The only reason to go for PP is, I think, acceptance of PPP plus an additional hypothesis such as that what pairings there are doesn't vary between possible worlds.

If PPP (or just (2) and (3)) is true and my w hypothesis is a genuine metaphysical possibility, then it is metaphysically possible that all infinite sets are the same size--i.e., it could be that the actual world is relevantly like w. Furthermore, we clearly don't have relevant empirical evidence to the contrary. So, if all this works, it is epistemically possible that all infinite sets are of the same size. (Of course, the most controversial part of all this is the idea that what sets there are might differ--even in the case of pure sets--between worlds.)

But perhaps this won't satisfy the people who find size differences between infinities paradoxical. For they might find it paradoxical enough that there could be infinities of different sizes, something that was definitely a part of my story (remember that I started with two worlds, one in which there were differently sized infinities and an impoverished w with all infinities of the same size according to PPP).

I think I might be able to do something to satisfy them, while at the same avoiding the biggest problem with the above story, namely the assumption that what pure sets there are differs between worlds. Here's my trick. In the above, I assumed that pairings were all sets. But in line with the Platonism suffusing all of the above arguments, let's try something. Let's allow that there are pairings that aren't sets. Those pairings would be binary relations satisfying the right formal axioms. But here I mean "relations" in the Platonic philosopher's sense, not in the mathematician's sense where a relation is a set of ordered pairs. Let's suppose, further, that corresponding to any set of ordered pairs, there is a relation which relates all and only those pairs which are found in the set. In my earlier story, I made sense of the idea that two sets in w might not have a pairing in w and yet might be the same size by adverting to pairings that exist in another world (the actual one--and then at the end I flipped things around so that w was actual). But now we do the same thing by distinguishing between mathematical pairings--namely, sets of ordered pairs satisfying the right axioms--and metaphysical pairings--namely, Platonic binary relations satisfying analogous axioms. If my earlier story is coherent (I mean that to be a weaker condition than "metaphysically possible"), then so is this one: In w, there are infinite sets that do not have a mathematical pairing, but every pair of infinite sets possibly has a metaphysical pairing. But now this story doesn't rely on varying what pure sets exist between worlds. The story appears compatible with the idea that pure sets are the same in every world. But there are, nonetheless, metaphysical pairings that do not correspond to mathematical pairings, and PPP should be interpreted with respect to the metaphysical pairings, not just the mathematical ones. Note, too, that what metaphysical pairings hold between sets might differ between possible worlds, without any variation in sets. For some of the pairings may correspond to extrinsic relations. Here is an extrinsic relation that could turn out to be a metaphysical pairing, depending on what I actually was thinking yesterday: x is related to y if and only if x and y came up in one of my thoughts yesterday in this order.

We can now suppose that this story works in every possible world. Thus, assuming the coherence of the Standard Model Hypothesis, we have a mathematically coherent story--whether the metaphysics works is another question (the story is too Platonic for my taste, and I don't share the motivation anyway)--on which (a) all infinite sets are really of the same size (and hence of the same size as the natural numbers), (b) what pure sets there are does not differ between worlds, and (c) Cantor's Theorem and all the axioms of ZF or ZFC are true. If we were to go for such a view, we would want to distinguish between sets being of the size metaphysically speaking and their having the same mathematical cardinality. The latter relationship would be defined by a version of the PP with "pairings" restricted to the mathematical ones. And then mathematics could go on as usual.

Cardinality paradoxes

Some people think it is absurd to say, as Cantorian mathematics does, that there are no more real numbers from 0 to 100 than from 0 to 1.

But there is a neat argument for this:

1. If the number of points on a line segment that is 100 cm long equals the number of points on a line segment that is 1 cm long, then the number of real numbers from 0 to 100 equals the number of real numbers from 0 to 1.
2. The number of points on a line that is 100 cm long equals the number of distances in centimeters between 0 and 100 cm.
3. The number of points on a line that is 1 meter long equals the number of distances in meters between 0 and 1 meter.
4. The number of distances in centimeters between 0 and 100 cm equals the number of real numbers between 0 and 100.
5. The number of distances in meters between 0 and 1 meters equals the number of real numbers between 0 and 1.
6. A line is 100 cm if and only if it is 1 meter long.
7. Equality in number is transitive.
8. So, the number of points on a line that is 100 cm is equals the number of points on a line that is 1 meter long.
9. So, the number of distances in centimeters between 0 and 100 cm equals the number of distances in meters between 0 and 1 meters.
10. So, the number of real numbers between 0 and 100 equals the number of real numbers between 0 and 1.

Cardinality and worlds

For every initial ordinal k, there is a possible world with exactly k photons. But there is no set of all initial ordinals (proof: suppose there is such a set; the union of the members of any set of ordinals is an ordinal; so the union of the initial ordinals is an ordinal; it must have the same cardinality as some initial ordinal in the set; but for any ordinal in the set, there is a larger one in the set). So there is no set of all possible worlds.

This argument doesn't use the Axiom of Choice and hence it improves on the argument I gave here.

Non-conglomerability

This result is probably known, and probably not optimal. A conditional probability function P is conglomerable provided that for any partition {H_i} (perhaps infinite and maybe even uncountable) of the state space if P(A|H_i)≥r for all i, then P(A)≥r.

Theorem. Assume the Axiom of Choice. Suppose P is a full conditional probability function (i.e., Popper function) P on an uncountable space such that:

1. all singletons are measurable
2. the function satisfies this regularity condition for all elements x and y: P({x}|{x,y})>0
3. there is a partition of the probability space into two disjoint subsets A and B with the same cardinality such that P(A)>0 and P(B)>0

Then P is not conglomerable.

Conditions (2) and (3) are going to be intuitively satisfied for plausible continuous probabilities, like uniform and Gaussian ones. So in those cases there is no hope for a conglomerable conditional probability.

Sketch of proof: Let Q be a hyperreal-valued unconditional probability corresponding to P, so that P(X|Y)=Q(X∩Y)/Q(Y). The regularity condition (2) implies that that there is a hyperreal α such that Q(F)/α is finite, non-zero and non-infinitesimal for each finite set F. (Just let α=Q({x_0}) for any fixed x₀.) Let R(F) be the standard part of Q(F)/α for any finite set α. Then P(F|G)=R(F∩G)/R(G) for any finite sets F and G. Moreover, R is finitely additive and non-zero on every singleton.

Since A² has the same cardinality as A, there is a function f from B to the subsets of A with the property that f(b) and f(c) are disjoint if b and c are distinct and every f(b) is uncountable. Choose a finite number c such that P(A)<c/(1+c). For each b in B, choose a finite subset F_b of f(b) such that R(F_b)>cR({b}). Such a finite subset exists as R is finitely additive and the sum of an uncountable number of non-zero positive numbers is always infinity. Let H be the union of the F_b as b ranges over B. Then A−H has at most the cardinality of B. Let h be a one-to-one function from A−H to B. For each b in B, let G_b=F_b if there is no a in A−H such that h(a)=b; otherwise, let G_b=F_b∪{a} for such an a. Let H_b={b}∪G_b. Then R(G_b)>cR({b}) and so R(G_b)/R(H_b)>c/(1+c). Then P(A|H_b)=P(G_b|H_b)=R(G_b)/R(H_b)>c/(1+c). But the H_b are a partition of our probability space, and P(A)<c/(1+c), so we have a violation of conglomerability.

A quick argument for the bijection principle

The bijection principle says that if we have two sets A and B and we can pair up all the objects of the two sets, then the the sets have the same number of members.

Some people don't like the bijection principle because it leads to the counterintuitive conclusion that there are as many primes as natural numbers.

Here's an argument for the bijection principle. Let's run the argument directly for the above controversial case—that should be enough of an intuition pump to get the general principle. Take infinitely many pieces of paper that are red on one side and blue on the other. Number the pieces of paper 1,2,3,..., putting the numerals down on the red sides. Then on the piece of paper numbered n on the red side, write down the nth prime on the blue side. Then:

1. There are just as many natural numbers as red sides.
2. There are just as many red sides as blue sides.
3. There are just as many blue sides as prime numbers.
4. So, there are just as many natural numbers as prime numbers.

It's very hard to deny that 4 follows from 1-3, and it's very hard to deny any of 1-3.

A cardinality objection to unrestricted modal profiles

The modal profile of an object tells us which worlds the object exists in and what it consists of in those worlds.

The unrestricted modal profiles (UMP) thesis says that for any map f that assigns to some worlds w a concrete object f(w) in w and that assigns nothing to other worlds, there is a possible concrete object O_f such that O_f exists in all and only the worlds w to which f assigns an object and has the property that in w, O_f is wholly composed of f(w) (or of parts of f(w) that compose f(w)).

One can think of UMP as the next step after unrestricted composition (UC) which holds that for any concrete objects there is an object composed of them. UC is not enough to guarantee the existence of ordinary objects like tables and chairs, since there is no guarantee that the modal profile of a UC-guaranteed object composed of the particles in a table will match the modal profile of a table. But UC+UMP, plus a thesis about the physical world being made of temporal parts of particles, will give us what we need here.

However, UMP is false.

1. There is a set of all actual concrete objects.
2. If UMP is true, then for any cardinality K, there are at least K actual concrete objects.
3. So, if UMP is true, there is no set of all actual concrete objects. (By 2)
4. So, UMP is not true. (By 1 and 3)

Now, claim (1) is very plausible. Claim (3) follows from (2) since for any set there is a greater cardinality by Cantor's Theorem and so it's impossible to have a set whose cardinality is at least as large as every cardinality.

That leaves (2). Assume UMP. Suppose K is any infinite cardinality (we don't need to worry about the finite case). For any K, there is a possible world w with at least K concrete objects (say, K photons). Let x be any concrete object in the actual world @. Then there will be at least K maps f with the property that f(@)=x and f(w) is a concrete object in w and f assigns nothing to any other world. To each such map f there corresponds at least one distinct object O_f in the actual world. (Distinct as difference in modal profiles implies non-identity of objects by Leibniz's Law.) So there are at least K concrete objects in the actual world.

Leibniz and the Gaifman-Hales Theorem

The basic idea behind Leibniz's characteristique is that all concepts are generated out of simple concepts, and there are no non-trivial logical relations between the simple concepts.

Today I was thinking about how to model this mathematically. Concepts presumably form a Boolean algebra. But infinities are very important to Leibniz. For instance, to each individual there corresponds a complete individual concept, which is an infinite concept specifying everything the individual does. So an ordinary Boolean algebra with binary conjunction and disjunction won't be good enough for Leibniz. We need concepts to form a complete Boolean algebra, one where an arbitrary set of elements has a conjunction and a disjunction.

So we want the space of concepts to be a complete Boolean algebra. We also want it to be generated by—built up out of—the set of simple concepts. Finally, we don't want there to be any nontrivial logical relations between the simple concepts. We want the theory to be entirely formal. This is one of Leibniz's basic intuitions. It seems to me that the way to formalize this condition is to say that the complete Boolean algebra is freely generated by the simple concepts.

Pity, though. The Gaifman-Hales Theorem implies that if there are infinitely many simple concepts, there is no complete Boolean algebra generated by them (this assumes a quite weak version of the Axiom of Choice, namely that every infinite set contains a countably infinite subset).

It looks, thus, like the Leibniz project is provably a failure.

Perhaps not, though. Apparently if one relaxes the requirement that the complete Boolean algebra be a set and allows it to be a proper class, but keeps the idea that the simple concepts form a set, one can get a complete Boolean algebra freely generated by the simple concepts.

Still, it's interesting that from an infinite set of simple concepts, one generates a proper class of concepts.

Countable additivity

One of the Kolmogorov axioms of probability says that if A₁,A₂,... is a countable sequence of disjoint sets, then P(A₁∪A₂∪...)=P(A₁)+P(A₂)+.... I once (when writing my Philosophia Christi paper on fine and coarse tuning) thought that while we had intuitive reason to accept unrestricted additivity (where we do not restrict to countably many sets) and we had intuitive reason to accept finite additivity, there was no in-between reason for accepting countable additivity. Since unrestricted additivity is unsupportable (if you pick a random number between 0 and 1, the probability of picking any particular number is zero, and the sum of the uncountably many probabilities of particular numbers will be zero, but the probability of the union of these singleton sets is one), I thought we should go for finite additivity.

When I thought this, I was wrong, because at the time I didn't know about the phenomenon of non-conglomerability which countable additivity seems to be needed for ruling out. Non-conglomerability is where you have a measure P of probabilities (maybe not technically a probability measure), a set E of events where each event in E has non-zero probability and it is certain that exactly one of the events in E will happen, and an event A such that P(A|B)>x for all B in E but P(A)<x. In such a case, your probability of A is less than x even though you know that no matter which event in B will happen, you will have P(A)>x. This is pathological.

It is well-known that countable additivity entails conglomerability. I like proving this with a two-step argument. The first step is an easy argument that if the set E of events is countably additive, then because P(A)=P((A∩B₁)∪(A∩B₂)∪...)=P(A∩B₁)+P(A∩B₂)+..., if we have P(A|B)>x for all B in E, then P(A)>x as well.

The second step in the proof is that if the members of a set E of disjoint events each have non-zero probabilities, then E has only countably many events in it. This step allows us to rule out non-conglomerability using only countable additivity. This step follows from the following fact about real numbers

1. If E is a set and f is a function that assigns to each member of E a non-negative number such that for any finite sequence x₁,...,x_n of distinct members of E we have f(x₁)+...+f(x_n)≤1, then all but countably many members of E are zero,

by letting f=P and using the finite additivity of P.

If we need countable additivity precisely to rule out non-conglomerability, then we have an explanation of why it only has to be countable additivity. The reason has to do with the property (1) of real numbers, which property in turn follows from the fact that the real numbers are Archimedean—for every pair of positive real numbers x and y, there is a finite natural number n such that nx>y.

In other words, we have countable additivity in the probability calculus precisely because the probability values have a countable-like, i.e., Archimedean, structure. (Another way of seeing the countable structure of the reals: they are the completion of the rationals.)

And if generalize the values of the probability function to a larger, non-Archimedean field, we will need to require something stronger than countable additivity in order to avoid non-conglomerability.