On Rasmussen and Bailey's "How to build a thought"

[Revised 11/21/2025 to fix a few issues.]

Rasmussen and Bailey prove that under certain assumptions it follows that there are possible thoughts that are not grounded in anything physical.

I want to offer a version of the argument that is slightly improved in a few ways.

Start with the idea that an abstract object x is a “base” for types of thoughts. The bases might be physical properties, types of physical facts, etc. I assume that in all possible worlds exactly the same bases abstractly exist, but of course what bases obtain in a possible world can vary between worlds. I also assume that for objects, like bases, that are invariant between worlds, their pluralities are also invariant between worlds.

Consider these claims:

1. Independence: For any plurality xx of bases, there is a possible world where it is thought that exactly one of the xx obtains and there is no distinct plurality yy of bases such that it is thought that exactly one of the yy obtains.

2. Comprehension: For any formula ϕ(x) with one free variable x that is satisfied by at least one base, there is a plurality yy of all the bases that satisfy ϕ(x).

3. Plurality: There are at least two bases.

4. Basing: Necessarily, if there is a plurality xx of bases and it is thought that exactly one of the xx obtains, then there obtains a base z such that necessarily if z obtains, it is thought that exactly one of the xx obtains.

By the awkward locution “it is thought that p”, I mean that something or some plurality of things thinks that p, or there is a thinkerless thought that p. The reason for all these options is that I want to be friendly to early-Unger style materialists who think that there no thinkers. :-)

Theorem: If Independence, Comprehension, Plurality and S5 are true, Basing is false.

Here is how this slightly improves on Rasmussen and Bailey:

• RB’s proofs use the Axiom of Choice twice. I avoid this. (They could avoid it, too, I expect.)

• I don’t need a separate category of thoughts to run the argument, just a “it is thought that exactly one of the xx exists” predicate. In particular, I don’t need types of thoughts, just abstract bases.

• RB use the concept of a thought that at least one of the xx exists. This makes their Independence axiom a little bit less plausible, because one might think that, say, someone who thinks that at least one of the male dogs exists automatically also thinks that at least one of the dogs exists. One might also reasonably deny this, but it is nice to skirt the issue.

• I replace grounding with mere entailment in Basing.

• I think RB either forgot to assume Plurality or are working with a notion of plurality where empty collections are possible.

Some notes:

• RB don’t explicitly assume Comprehension, but I don’t see how to prove their Cantorian Lemma 2 without it.

• Independence doesn’t fit with the necessary existence of an omniscient being. But we can make the argument fit with theism by replacing “it is thought” with “it is non-divinely thought”.

• I think the materialist could just hold that there are pluralities xx of bases such that no one could think about them.

Proofs

Write G(z,xx) to mean that z is a base, the xx are a plurality of bases, and necessarily if z obtains it is thought that exactly one of the xx obtains.

The Theorem follows from the following lemmas.

Lemma 1: Given Independence, Basing and S5, for every plurality of bases xx there is a z such that G(z,xx) and for every other plurality of bases yy it is not the case that G(z,yy).

Proof: Let w be a possible world like in Independence. By Basing, at w there obtains a base z such that G(z,xx). By S5 and the bases and pluralities thereof being the same at all worlds, we have G(z,xx) at the actual world, too. Suppose now that we actually have G(z,yy) with yy other than xx. Then at w, it is thought that exactly one of yy exists. But that contradicts the choice of w. Thus, actually, we have G(z,xx) but not G(z,yy).

Lemma 2: Assume Comprehension and Plurality. Then there is no formula ϕ(z,xx) open only in z and xx such that for every plurality of bases xx there is a z such that ϕ(z,xx) while for every other plurality of bases yy it is not the case that ϕ(z,yy).

Proof: Suppose we have such a ϕ(z,xx). Say that z is an admissible base provided that there is a unique plurality of bases xx such that ϕ(z,xx). I claim that there is an admissible base z such that z is not among any xx such that ϕ(z,xx). For suppose not. Then for all admissible bases z, z is among all xx such that ϕ(z,xx). Let a and b be distinct bases. Let ff, gg and hh be the pluralities consisting of a, of b, and of both a and b respectively. Then the above assumptions show that we must have ϕ(a,ff), ϕ(b,gg) and either ϕ(a,hh) or ϕ(b,hh), and either of these options violates our assumptions on ϕ. By Comprehension, then, let yy be the plurality of all admissible bases z such that z is not among any xx such that ϕ(z,xx). Let z be an admissible base such that ϕ(z,yy). Is z among the yy? If it is, then it’s not. If it is not, then it is. Contradiction!

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).

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.

Preference structures had by no possible agent

Say that a preference structure is a total, transitive and reflexive relation (i.e., a total preorder) on centered worlds--i.e., world-agent pairs <w,x>. Then there is a preference structure had by no possible agent. This is in fact just an easy adaptation of the proof of Cantor's Theorem.

Let c be my own centered world <@,Pruss>. We now define a preference structure Q as follows. If agent x at world w, where <w,x> is not the same as <@,Pruss>, prefers her own centered world <w,x> to c, then we say that c is Q-preferable to <w,x>; otherwise, we say that <w,x> is Q-preferable to c. Then we say that all the centered worlds that according to the preceding are Q-preferable to c are Q-equivalent and all the centered worlds we said to be less Q-preferable than c are also Q-equivalent. Thus, Q ranks centered worlds into three classes: those less good than c, those better than c and finally c itself.

But now note that no possible agent has Q as her preference structure. First of all, I at the the actual world do not have Q as my preference structure--that's empirically obvious, in that the worlds do not fall into three equipreferability classes for me. And if <w,x> is different from <@,Pruss>, then x's preference-order at w (if any) between c and <w,x> differs from what Q says about the order.

So what? Well, I think this provides a slight bit of evidence for the idea that agents choose under the guise of the good.