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.

Holes and substantivalism

Suppose substantivalism about space is correct. Imagine now that the following happens to a slice of swiss cheese: the space where the holes were suddenly disappears. I don’t mean that the holes close up. I mean that the space disappears: all the points and regions that used to be in the hole are no longer there (and any air that used to be there is annihilated). The surfaces of the cheese that faced the hole now are at an edge of space itself.

The puzzle now is that in this story we have an inconsistent triad:

1. There is no intrinsic change in the cheese.
2. The slice of cheese no longer has holes.
3. Changing with respect to whether you have holes is intrinsic.

Here are my arguments for the three claims. There is no intrinsic change in the slice of cheese as something outside the cheese has changed—space has been annihilated. The slice of cheese no longer has holes, as it makes sense to talk of the size or shape or volume of a hole, but there is no size or shape or volume where there is no space. And changing with respect to whether you have holes is change of shape, and changes of shape are intrinsic.
It seems that the above story forces you to reject one of the following:

4. Substantivalism about space
5. Intrinsicness of shape.

But there is another way out. Deny (3). Whether you have holes is not intrinsic. What is intrinsic is your topological genus with respect to your internal space and similar topological properties.

Note, also, a lesson relevant to the famous Lewis and Lewis paper on holes: the counting of holes should not involve the counting of regions, but the computation of a numerical invariant, namely the genus.

Connected and scattered objects

Intuitively, some physical objects, like a typical organism, are connected, while other physical objects, like a typical chess set spilled on a table, are disconnected or scattered.

What does it mean for an object O that occupies some region R of space to be connected? There is a standard topological definition of a region R being connected (there are no open sets U and V whose intersections with R are non-empty such that R ⊆ U ∪ V), and so we could say that O is connected if and only if the region R occupied by it is connected.

But this definition doesn’t work well if space is discrete. The most natural topology on a discrete space would make every region containing two or more points be disconnected. But it seems that even if space were discrete, it would make sense to talk of a typical organism as connected.

If the space is a regular rectangular grid, then we can try to give a non-topological definition of connectedness: a region is connected provided that any two points in it can be joined by a sequence of points such that any two successive points are neighbors. But then we need to make a decision as to what points count as neighbors. For instance, while it seems obvious that (0,0,0) and (0,0,1) are neighbors (assuming the points have integer Cartesian coordinates), it is less clear whether diagonal pairs like (0,0,0) and (1,1,1) are neighbors. But we’re doing metaphysics, not mathematics. We shouldn’t just stipulate the neighbor relation. So there has to be some objective fact about the space that decides which pairs are neighbors. And things just get more complicated if the space is not a regular rectangular grid.

Perhaps we should suppose that a physical discrete space would have to come along with a physical “neighbor” structure, which would specify which (unordered, let’s suppose for now) pairs of points are neighbors. Mathematically speaking, this would turn the space into a graph: a mathematical object with vertices (points) and edges (the neighbor-pairs). So perhaps there could be at least two kinds of regular rectangular grid spaces, one in which an object that occupies precisely (0,0,0) and (1,1,1) is connected and another in which such an object is scattered.

But we can’t use this graph-theoretic solution in continuous spaces. For here is something very intuitive about Euclidean space: if there is a third point c on the line segment between the two points a and b, then a and b are not neighbors, because c is a better candidate for being a’s neighbor than b. But in Euclidean space, there is always such a third point, so no two points are neighbors. Fortunately, in Euclidean space we can use the topological notion.

But now we have a bit of a puzzle. We have a topological notion of a physical object being connected for objects in a continuous space and a graph theoretic notion for objects in a discrete space. Neither notion reduces to the other. In fact, we can apply the topological one to objects in a discrete space, and conclude that all objects that occupy more than one point are scattered, and the graph theoretic one to objects in Euclidean space, and also conclude that all objects that occupy more than one points are scattered.

Maybe we should have a disjunctive notion: an object is connected if and only if it is graph-theoretically connected in a space with a neighbor-relation or topologically connected in a space with a topological structure.

That’s not too bad, but it makes the notion of the connectedness of a physical object be a rather unnatural and gerrymandered notion. Maybe that’s how it has to be.

Or maybe only one of the two kinds of spaces is actually a possible physical space. Perhaps physical space must have a topological structure. Or maybe it must have a graph-theoretic structure.

Here’s a different suggestion. Given a region of space R, we can define a binary relation c_R where c_R(a, b) if and only if the laws of nature allow for a causal influence to propagate from a to b without leaving R. Then say that a region of space R is connected provided that any two distinct points can be joined by a sequence of points such that successive points are c_R-related in one order or the other (i.e., if d_i and d_i+1 are successive points then c_R(d_i, d_i+1) or c_R(d_i+1, d_i)).

On this story, if we have a universe with pervasive immediate action at a distance, like in the case of Newtonian gravity, all physical objects end up connected. If we have a discrete universe with a neighbor structure and causal influences can propagate between neighbors and only between them, we recover the graph-theoretic notion.

Disconnected bodies and lives

We can imagine what it is like for a living to have a spatially disconnected body. First, if we are made of point particles, we all are spatially disconnected. Second, when a gecko is attacked, it can shed a tail. That tail then continues wiggling for a while in order to distract the pursuer. A good case can be made that the gecko’s shed tail remains a part of the gecko’s body while it is wiggling. After all, it continues to be biologically active in support of the gecko’s survival. Third, there is the metaphysical theory on which sperm remains a part of the male even after it is emitted.

But even if all these theories are wrong, we should have very little difficulty in understanding what it would mean for a living thing to have a spatially disconnected body.

What about a living thing having a temporally disconnected life? Again, I think it is not so difficult. It could be the case that when an insect is frozen, it ceases to live (or exist), but then comes back to life when defrosted. And even if that’s not the case, we understand what it would mean for this to be the case.

But so far this regarded external space and external time. What about internally spatially disconnected bodies and internally temporally disconnected lives? The gecko’s tail and sperm examples work just as well for internal as well as external space. So there is no conceptual difficulty about a living thing having a disconnected body in its inner space.

But it is much more difficult to imagine how an organism could have an internal-time disconnect in its life. Suppose the organism ceases to exist and then comes back into existence. It seems that its internal time is uninterrupted by the external-time interval of non-existence. An external-time interval of non-existence seems to be simply a case of forward time-travel, and time-travel does not induce disconnectes in internal time. Granted, the organism may have some different properties when it comes back into existence—for instance, its neural system might be damaged. But that’s just a matter of an instantaneous change in the neural system rather than of a disconnect in internal time. (Note that internal time is different from subjective time. When we go under general anesthesia, internal time keeps on flowing, but subjective time pauses. Plants have internal time but don’t have subjective time.)

This suggests an interesting apparent difference between internal time and internal space: spatial discontinuities are possible but temporal ones are not.

This way of formulating the difference is misleading, however, if some version of four-dimensionalism is correct. The gecko’s tail in my story is four-dimensional. This four-dimensional thing is connected to the four-dimensional thing that is the rest of the gecko’s body. There is no disconnection in the gecko from a four-dimensional perspective. (The point particle case is more complicated. Topologically, the internal space will be disconnected, but I think that’s not the relevant notion of disconnection.)

This suggests an interesting pair of hypotheses:

• If three-dimensionalism is true, there is a disanalogy between internal time and internal space with respect to living things at least, in that internal spatial disconnection of a living thing is possible but internal temporal disconnection of a living thing is not possible.

• If four-dimensionalism is true, then living things are always internally spatiotemporally connected.

But maybe these are just contingent truths Terry Pratchett has a character who is a witch with two spatially disconnected bodies. As far as the book says, she’s always been that way. And that seems possible to me. So maybe the four-dimensional hypothesis is only contingently true.

And maybe God could make a being that lives two lives, each in a different century, with no internal temporal connection between them? If so, then the three-dimensional hypothesis is also only contingently true.

I am not going anywhere with this. Just thinking about the options. And not sure what to think.

The deep question for the philosophy of spacetime

There is more than one way of putting this point, so the assumptions I will make are not at all essential, and I don't even endorse the assumptions. Assume absolutism about spacetime. On one reading of absolutism, there is then a location relation between objects and points or regions of spacetime (on another reading there is an object- or point-valued location determinable). Depending on the version of absolutism, the location relation may correspond to the predicate is wholly located at, is at least partly located at or is exactly located at (I may be leaving out some options).

Now the deep question is this: What is it that makes a relation between objects and points or regions of a topological space be a location relation? (The question can also be put on relationism. Then the question is what is it that makes a family of relations between objects be a family of spatial, or spatiotemporal, relations.)

There are two extreme answers.

Location monism: There is just one location relation. In a Newtonian and in an Einsteinian world and in a 12-dimensional discrete universe, one and the same relation relates objects to points or regions of a topological space, obviously a very different topological space in each case.

Location functionalism: Any natural (sufficiently natural? perfectly natural?) relation between objects and points in a topological space, where the topological space is either concrete and cosntituted as a topological space by natural relations, or abstract as in this post, is a location relation. What the axioms are will depend on which location relation one takes as fundamental as well as on difficult metaphysical issues. Supposing that the relation is being exactly located at, and the spatial relata are regions, then the axioms might be very lax. In fact they might be nothing but:

• If xLR and x is a part of y, then there is a unique region R' that contains R such that yLR'.
• If yLR and x is a part of y, then there is a unique region R' that is contained in R such that xLR'.

(If Thomistic part nihilism is true, do this with virtual parts.) On this view, a relation of being exactly located at any phase space with a topology will count as a location relation as long as the relation is in fact natural. One might additionally add some more axioms, such as that no object is exactly located at two distinct regions (though I myself am inclined to deny that as it's incompatible with my best account of transsubstantiation), but the result about phase spaces will remain true. If one wants to rule them, one can either insist that in fact there is no phase space location in which is natural or disallow abstract topological spaces as relata, despite the benefits of allowing them. One might also add an axiom that makes this be a location in spacetime, by using causation. For instance, we might require the topological space to have a partial ordering on its points (we might add something about how the ordering should play nice with the topology), which we will call "at least as late as", and then extend this to a relation between regions: R' is at least as late as R provided that every point y of R' is at least as late as some point x of R and every point x of R has some point y of R' such that y is at least as late as x. Then add:

• Normally, if event E causes event E', and E and E' are exactly located at R and R' respectively, then R' is at least as late as R.

Monism and functionalism are extreme theories because functionalism classifies as locational as many relations as anybody could possibly reasonably want to do that to and monism classifies as locational as few as anybody who thinks location is real reasonably could.

I incline to functionalism here.

Relativity Theory and abortion

In an earlier post, I offered the principle that just as the laws of physics should be invariant under change of reference frame, so should the laws of morality. One consequence of that is that various inside-outside distinctions are not going to be significant of themselves. Here is an interesting little consequence of that: Any argument that abortion is permissible based on the fact that abortion takes place inside the body of the woman cannot be right, since the claim that the fetus is inside the woman's body is not invariant under coordinate transformations.

This doesn't mean that the argument is thoroughly refuted. But it does mean that the mere geometrical fact that the fetus is within the woman's body is insignificant. There may, however, be more significant related invariant facts about dependence, burden, etc. The invariance move does not, thus, settle the discussion, but moves it forward, by forcing the pro-choicer arguer to give a fuller story about the distinction in invariant terms, which terms non-coincidentally are going to be descriptively richer, thereby deepening the debate.

Of course, one doesn't need relativity theory to show the problem with the principle that one can do what one likes as long as it is within the confines of one's body. One can also proceed by counterexample. If one accidentally swallowed Whoville, one would not be permitted to follow that up with a drink of something intended to kill all the Whos.