What numbers could be

Benacerraf famously argued that no set theoretic reduction can capture the natural numbers. While one might conclude from this that the natural numbers are some kind of sui generis entities, Benacerraf instead opts for a structuralist view on which different things can play the role of different numbers.

The argument that no set theoretic reduction captures the negative numbers is based on thinking about two common reductions. On both, 0 is the empty set ⌀. But then the two accounts differ in how the successor sn of a number n is formed:

1. sn = n ∪ {n}

2. sn = {n}.

On the first account, the number 5 is equal to the set {0, 1, 2, 3, 4}. On the second account, the number 5 is equal to the singleton {{{{{⌀}}}}}. Benacerraf thinks that we couldn’t imagine a good argument for preferring one account over another, and hence (I don’t know how this is supposed to follow) there can’t be a fact of the matter about why one account—or any other set-theoretic reductive account—is correct.

But I think there is a way to adjudicate different set-theoretic reductions of numbers. Plausibly, there is reference magnetism to simpler referrents of our terminology. Consider an as consisting of a set of natural numbers, a relation <, and two operations + and ⋅, satisfying some axioms. We might then say that our ordinary language arithmetic is attracted to the abstract entities that are most simply defined in terms of the fundamental relations. If the only relevant fundamental relation is set membership ∈, then we can ask which of the two accounts (a) and (b) more simply defines <, + and ⋅.

If simplicity is brevity of expression in first order logic, then this can be made a well-defined mathematical question. For instance, on (a), we can define a < b as a ∈ b. One provably cannot get briefer than that. (Any definition of a < b will need to contain a, b and ∈.) On the other hand, on (b), there is no way to define a < b as simply. Now it could turn out that + or ⋅ can be defined more simply on (b), in a way that offsets (a)’s victory with <, but it seems unlikely to me. So I conjecture that on the above account, (a) beats (b), and so there is a way to decide between the two reductions of numbers—(b) is the wrong one, while (a) at least has a chance of being right, unless there is a third that gives a simpler reduction.

In any case, on this picture, there is a way forward in the debate, which undercuts Benacerraf’s claim that there is no way forward.

I am not endorsing this. I worry about the multiplicity of first-order languages (e.g., infix-notation FOL vs. Polish-notation FOL).

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.

Presentism and counting future sufferings

I find it hard to see why on presentism or growing block theory it’s a bad thing that I will suffer, given that the suffering is unreal. Perhaps, though, the presentist or growing blocker can say that is a primitive fact that it is bad for me that a bad thing will happen to me.

But there is now a second problem for the presentist. Suppose I am comparing two states of affairs:

1. Alice will suffer for an hour in 10 hours.
2. Bob will suffer for an hour in 5 hours and again for an hour in 15 hours.

Other things being equal, Alice is better off than Bob. But why?

The eternalist can say:

1. There are more one-hour bouts of suffering for Bob than for Alice.

Maybe the growing blocker can say:

2. It will be the case in 16 hours that there are more bouts of suffering for Bob than for Alice.

(I feel that this doesn’t quite explain why it’s B is twice as bad, given that the difference between B and A shouldn’t be grounded in what happens in 16 hours, but nevermind that for this post.)

But what about the presentist? Let’s suppose preentism is true. We might now try to explain our comparative judgment by future-tensing (1):

3. There will be more bouts of suffering for Bob than for Alice.

But what does that mean? Our best account of “There are more Xs than Ys” is that the set of Xs is bigger than the set of Ys. But given presentism, the set of Bob’s future bouts of suffering is no bigger than the set of Alice’s future bouts of suffering, because if presentism is true, then both sets are empty as there are no future bouts of suffering. So (3) cannot just mean that there are more future bouts of suffering for Bob than for Alice. Perhaps it means that:

4. It will be the case that the set of Bob’s bouts of suffering is larger than the set of Alice’s.

This is true. In 5.5 hours, there will presently be one bout of suffering for Bob and none for Alice, so it will then be the case that the set of Bob’s bouts of suffering is larger than the set of Alice’s. But while it is true, it is similarly true that:

5. It will be the case that the set of Alice’s bouts of suffering is larger than the set of Bob’s.

For in 10.5 hours, there will presently be one bout for Alice and none for Bob. If we read (3) as (4), then, we have to likewise say that there will be more bouts of suffering for Alice than for Bob, and so we don’t have an explanation of why Alice is better off.

Perhaps, though, instead of counting bouts of suffering, the presentist can count intervals of time during which there is suffering. For instance:

6. The set of hour-long periods of time during which Bob is suffering is bigger than the set of hour-long periods of time during which Alice is suffering.

Notice that the times here need to be something like abstract ersatz times. For the presentist does not think there are any future real concrete times, and so if the periods were real and concrete, the two sets in (6) would be both empty.

And now we have a puzzle. How can fact (6), which is just a fact about sets of abstract ersatz times, explain the fact about how Bob is (or is going to be) worse off than Alice? I can see how a comparative fact about sets of sufferings might make Bob worse off than Alice. But a comparative fact about sets of abstract times should not. It is true that (6) entails that Bob is worse off than Alice. But (6) isn’t the explanation of why.

Our best explanation of why Bob is worse off than Alice is, thus, (1). But the presentist can’t accept (1). So, presentism is probably false.

Two questions about sets

Here are two curious philosophical questions about set theory and its applicability outside mathematics.

Question 1: Suppose that every person has a perfectly well-defined mass. Is there a set of everybody mass, say the set of all real numbers x such that x is someone’s mass in kilograms?

The standard ZFC axioms are silent on this. They do say that for any predicate F in the language of set theory there is a set of all real numbers x satisfying F. But "mass" and "kilogram" are not parts of the language of set theory.

Question 2: What does it mean to say that there are finitely many horses?

An obvious answer is that if H is the set of all horses, then H is in one-to-one correspondence with some natural number. But the standard ZFC axioms only give us sets of sets, not sets of physical things like horses. If the correct set theory has ur-elements, elements that aren’t sets, maybe there is a set of all horses—but maybe not even then.

I suppose we could go metalinguistic. Begin by describing the set S of first-order logic sentences (sentences can be thought of as sets, even if sets are pure, i.e., have only sets as members) that say "There are no horses", "There is at most one horse", "There are at most two horses",.... And then say, using language beyond set theory, that at least one sentence in S is true.

But the metalinguistic approach won’t solve the seemingly related problem of what it means to say that there are countably many horses.

How likely are you to be in a random finite subset of an infinite set?

Suppose that out of a set of infinitely many people, including you, a finite subset is chosen at random. How likely are you to be in that subset? Intuitively, not very likely. And the larger the infinity, the less likely.

But how do you pick out a finite subset at random? Here’s a natural way. First, pick out a subset at random, by flipping a fair coin for each person in the original set, and including a person in the subset if the comes up heads. Almost surely, this will generate an infinite subset (a consequence of the law of large numbers). But suppose this experiment is repeated—perhaps uncountably infinitely often—until the set picked out is finite. (This construction requires that the set of potential repetitions be well-ordered.) Or maybe you just get lucky, and to everybody’s surprise the set picked out is finite.

So now we have a method for picking out a finite subset at random (though it may take some luck). How likely are you to be in that finite subset?

Well, think about it step-by-step. Before you learned that the set picked out by the heads was finite, your probability that you were in the set was the probability that your coin landed heads, i.e., 1/2. Then you learn that the set of people for whom heads was rolled is finite. But this fact tells you nothing about your coin toss. For the claim that the set of people with heads is finite is logically equivalent to the claim that the set of people other than you with heads is finite. And the latter claim tells you nothing about your coin toss.

So, your probability needs to stay at 1/2.

Thus, the probability that a random finite subset of the infinitely many people includes you is finite. This is a little counterintuitive when the infinity is countable. And it becomes far more counterintuitive the larger this infinity gets. It is a stupendously implausible claim when that infinity is large, say ℶ_ω.

Causal finitism blocks the story by making it impossible for you to find out that the set of people who got heads is finite.

Counterparts and singletons

Here is an interesting problem for Lewis. Lewis says that sets are necessary beings, and hence count as existing in all worlds. Very plausibly then:

1. If A is a set and w₁ and w₂ are worlds, then A in w₂ is a counterpart of A in w₁.

After all, if identity isn’t good enough for being a counterpart, nothing is. Note that (1) does not say that A at w₂ is the only counterpart of A in w₁. To handle some identical twin scenarios, Lewis may need to allow a world to have more than one counterpart of an object.

Let α be Aristotle. Let A = {α} be the singleton of α. Lewis is now committed to the truth of:

2. Possibly α is not a member of A.

For Lewis’s criterion for whether F(a, b) is possible is whether there is a world w with counterparts a′ and b′ of a and b respectively such that F(a′,b′) holds at w. Let w be a non-actual world where there is a counterpart β of Aristotle. Since individuals are world-bound, β ≠ α. Moreover set membership is necessary, so:

3. β is not a member of A at w.

Since β is a counterpart of α and A is a counterpart of A by (1), it follows that (2) is true. But (2) seems clearly wrong: it is impossible for Aristotle not to be a member of A.

Here’s what seems to me to be the best way out for Lewis: Require pairwise counterparts rather than individual counterparts (in fact, I vaguely remember that Lewis may do that somewhere) for possibility claims involving two objects. Thus that β is not a member of A and β and A are individually counterparts of α and A isn’t enough to make it be that possibly α is not a member of A. One would need β and A to be pairwise counterparts of α and A. But perhaps they’re not. Perhaps, rather, it is β and B = {β} that are pairwise counterparts of α and A. However, this greatly complicates the counterpart relation as well as Lewis’s identification of properties with sets.

Contingent pure sets?

It is widely held that some sets exist contingently. The standard examples are sets that have contingent entities among their members (or the members of their members or ...), such as the singleton set of me or the set of all actual cows. I wonder if such examples exhaust the contingently existing sets. Could there be contingently existing pure sets, sets whose members all the way down are sets?

Well, they're not going to be sets whose existence can be proved from the axioms of set theory, if these axioms are necessary truths. But one interesting class of potential candidates could be sets defined in non-set-theoretic terms. For instance, suppose that in the actual world a coin is actually tossed an infinite number of times, with the occasions numbered 1,2,3,.... Then, if probability theory is to be applicable to the real world, we need to suppose something like the hypothesis that there is a set of all natural numbers corresponding to occasions when the coin landed heads. But would that set exist if the coin had landed in a radically different sequence or not been tossed at all? I used to assume that of course the answer to a question like that would be affirmative. I still think it's likely to be affirmative. But the matter is far from clear to me now.

The Axiom of Separation

The Axiom of Separation in Zermelo-Fraenkel (ZF) set theory implies that, roughly, for any set A and any unary predicate F(x), there is a subset B of all the x in A such that F(x). But only roughly. Technically the axiom only implies this for predicates definable in the language of set theory. We philosophers tend to forget that technical fact when we use set theory, much as we tend to blithely extend set theory to allow for ur-elements (elements that are not themselves sets). But if we are going to be realists about sets (which I am not saying we should be), we should have a real worry about what predicates can be legitimately used in the Axiom of Separation. (That's one of the lessons of this post.)

Consider the predicate L(x) which holds if and only if someone likes x. This is definitely not formulated in the language of set theory, so ZF set theory gives us no guarantee that there is, say, the set of all real numbers that satisfy L(x). If it turns out that there are only finitely many numbers that are liked, then we have no worries: for any real numbers x₁,...,x_n, there is a set that contains them and only them (this follows from the Axiom of Pairs plus the Axiom of Union). There will be other special cases where things work out, say when all but finitely many numbers are liked. But in general there is no guarantee from the axioms of ZF that there is a set of all liked numbers.

One might use this to try to get out of some paradoxes of infinity, by limiting the applicability of set theory. That's a strategy worth exploring further, but risky. For the above observations also severely limit the physical applicability of set theory. Suppose, for instance, that at each of infinitely many points in spacetime there is a well-defined temperature. It is usual then to suppose that there is a function T from the spacetime manifold to the real numbers such that T(z)=u if and only if the temperature at z (or, more precisely, at the point of spacetime corresponding to the point z in the mathematical manifold that models spacetime) is u. And we need there to be such a function T to be able to make physical predictions.

One solution is to extend the Axiom of Separation to include some or even all predicates not in the language of set theory. This is the solution that is typically implicitly used by philosophers. The Axiom of Separation has a lot of intuitive force thus extended, but we need to be careful since we know that the incoherent Axiom of Comprehension also had a lot of intuitive force.

A second option would be to have physics make set-theoretic claims. Thus, a theory positing that at each point of spacetime there is a temperature would also posit that there exists a corresponding function from the mathematical manifold that models spacetime to the real numbers. I think this would be quite an interesting option: it would mean that physics actually places constraints on what the universe of set theory is like.

Perhaps if we are not Platonists about sets, things are easier. But I am not sure. Things might just be murkier rather than easier.

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.

From properties to sets

If we have abundant properties in our ontology, do we need to posit a second kind of entities, the sets?

Properties are kind of like sets. If P is a property, write x∈P if and only if x has P. A whole bunch of the Zermelo-Fraenkel axioms then are quite plausible. But not all. The most glaring failure is extensionality. The property of being human and the property of being a member of a globally dominant primate species have the same instances, but are not the same property.

We can get extensionality by a little trick and an axiom. Assume the following Axiom of Choice for Properties:

1. If R is any symmetric and transitive relation, then there is a property P such that (a) if x has P, then x stands in R to itself, and (b) for all x if x stands in R to itself, there exists a unique y such that x stands in R to y and y has P.

Like the ordinary Axiom of Choice, this is a kind of principle of plenitude. Apply (1) to the relation C of coextensionality that holds between two properties if and only if they have the same instances. This generates a property S₁ that is had only by properties and is such that for any property P there exists exactly one property Q such that P and Q are coextensive and Q has S₁. In other words, S₁ selects a unique property coextensive with a given property.

To a first approximation, then, we can think of those entities that have S₁ as sets. Then every set is a property, but not every property is a set. We certainly have extensionality, with the usual restriction to allow for urelements (i.e., extensionality only applies to sets). All the other axioms of Zermelo-Fraenkel with urelements minus Separation, Foundation and Choice are pretty plausibly true (they follow from plausible analogues for properties on an abundant view of properties). We get Choice for sets for free from (1).

Unfortunately, we cannot have Separation, however. For the property S₁ is coextensive to some set U by our assumptions. And the members of U will just be the instances of S₁, i.e., all the sets. And so we have a universal set, and of course a universal set plus Separation implies Comprehension, and hence the Russell Paradox.

So matters aren't so easy. The Axiom of Foundation is also not so clear. Might there not be a self-instancing property?

Thus the above simple approach gives us too many sets. But there is a solution to this problem, and this is simply to postulate the following second axiom about properties:

2. There is a property S₂ of properties such that (a) concreteness has S₂, and (b) all the axioms of Zermelo-Fraenkel Set Theory with Urelements minus Extensionality are satisfied when we stipulate that (i) a set is anything that has S₂ and (ii) A∈B if and only if A is an instance of B.

This axiom is fairly plausible, I think.

Now suppose that S₁ is as before, and let S₂ be any property satisfying (2). Then let S be the conjunction of S₁ and S₂. It is easy to see that if we take our sets to be those properties that have S, we will have all of Zermelo-Fraenkel with Choice and Urelements (ZFCU). Or at least so it seems to me—I haven't written out formal proofs, and maybe I need some further plausible assumptions about what abundant properties are like.

Of course, we cannot expect S₁ and S₂ to be unique. So there will be multiple candidates for sets. That's fine with me.

The big question is whether (1) and (2) are true. But if the theoretical utility of positing sets is a reason to think sets exist, then theoretical utility plus parsimony plus the reasons to believe in properties are a reason to think (1) and (2) are true.

A problem for easy ontology arguments

Consider this "easy ontology" argument:

1. There are no unicorns.
2. So, there are zero unicorns.
3. So, there is a zero.

This seems fine. Now consider the parallel:

4. Every leprechaun is a fairy.
5. So, the set of leprechauns is a subset of the set of fairies.
6. So, there is a set of leprechauns.
7. If there is a set of leprechauns, it's empty. (There aren't any leprechauns!)
8. So, there is an empty set.

That seems fine as well. So far so good. But now:

9. Every non-self-membered set (set a that isn't one of its own members) is a set.
10. So, the set of non-self-membered sets is a subset of the set of all sets.
11. So, there is a set of non-self-membered sets (the Russell set).

But of course (11) yields a contradiction (just ask if the Russell set is a member of itself).

What to do? One move is to make the easy ontology arguments defeasible. This isn't in the spirit of the game. The other is to add to the premises of the easy ontology argument a coherence premise: that there is a coherent theory of zero, of the empty set and of the Russell and universal sets. The coherence premise will be false in the Russell case but will be true in the other cases. But the point is one that should make us take easy ontology less easily. (I wouldn't be surprised if this was in the easy ontology literature, with which I have little familiarity.)

Ordering subsets of the line and the Banach-Tarski paradox

Suppose we want to rank the subsets of the interval [0,1] by size. Consider this:

1. There is a total, transitive and reflexive ordering (i.e., a total preorder) ≤ of subsets of the interval [0,1] such that (a) if A⊆B, then A≤B and (b) if A and B are disjoint non-empty sets with A≤B, then A<A∪B,

where X<Y means that X≤Y but not conversely.

Condition (a) is intuitively correct for any notion of size comparison. Condition (b), then, is like a regularity condition in probability theory. Any regular probability is going to yield a comparison like in (1).

One can get such an ordering by applying Szpilrajn's Theorem to get a total ordering of the subsets of [0,1] that extends subset inclusion. Note, however, that Szpilrajn's Theorem uses a version of the Axiom of Choice. It's an interesting question whether one can have (1) without any Axiom of Choice.

Now, the Banach-Tarski Paradox—that a solid ball can be disassembled into a finite number of subsets that can be reassembled into two solid balls each of the same size as the original—also uses a version of the Axiom of Choice. It would be nice to have (1) while avoiding the Banach-Tarski Paradox. One might even think Bayesian epistemology requires that one be able to hold on to (1) while avoiding the Banach-Tarski Paradox, since the latter spells doom for rigid-motion invariant probabilities in a three-dimensional region. But without using any version of the Axiom of Choice one can prove:

Theorem. If (1) is true, then the Banach-Tarski Paradox holds.

This essentially follows from Note 1 in Pawlikowski's paper, together with the fact that the solid ball has the same cardinality as [0,1] (so an order on the subsets of [0,1] as in (1) transfers to an order on the subsets of the ball that satisfies (1)), and the following pleasant observation:

2. If ≤ is as in (1) and A₁,A₂,A₃,A₄ are non-empty sets, then if B=A₁∪A₂∪A₃∪A₄, then at most one of the A_i satisfies the condition B−A_i<A_i and at least one of the A_i satisfies the condition A_i<B−A_i.

Say that X~Y iff X≤Y and Y≤X. To prove (2), suppose first that A_i>B−A_i and j≠i. Then A_j⊆B−A_i. Moreover, A_i⊆B−A_j. Thus, B−A_j≥A_i>B−A_i≥A_j, and so we do not have A_j>B−A_j. Next, suppose A_i does not satisfy A_i≤B−A_i. Thus, by totality of ≤, we have A_i>B−A_i. But this can happen for at most one i. So there will be three distinct i such that A_i≤B−A_i. To obtain a contradiction, suppose all three inequalities are non-strict and suppose i and j are two of the three indices. Then A_i~B−A_i and A_j~B−A_j. Observe that A_j is a subset of B−A_i, and hence A_j≤A_i. By the same argument, A_i≤A_j, and so A_j~A_i. The same will go for the third index, so there are three indices i, j and k such that A_i~B−A_i~A_j~B−A_j~A_k~B−A_k. But B−A_k contains the union of A_i and A_j and so by (1) we have A_i<B−A_k, which is a contradiction.

Corollary 1. The claim that every partial order extends to a total order implies the Banach-Tarski Paradox.

And since it's well known that the Banach-Tarski Paradox cannot be proved in Zermelo-Fraenkel (ZF) set theory unless ZF is inconsistent:

Corollary 2. Claim (1) cannot be proved in ZF, unless ZF is inconsistent.

I have some thoughts on how one might perhaps be able to improve Corollary 1 to show the result under the weaker assumption that every set has a total order. (The existence of Lebesgue nonmeasurable sets can be proved from that.)

Philosophical implications: I think standard Bayesianism requires both (1) and the falsity of Banach-Tarski. So standard Bayesianism fails. Moreover, we get an argument for the possibility of incommensurability in decision theory. For if (1) is false, then there can be incommensurability (denying (1) requires affirming that there are some partially preordered sets that have no total preorder whose strict order relation extends the strict order relation of the original preorder), and if Banach-Tarski is true, there can be incommensurability (incommensurability in probabilities implies incommensurability in choices).

Reducing sets

I find sets to be very mysterious candidates for abstract entities. I think it's their extensionality that seems strange to me. And anyway, if one can reduce entities to entities that we anyway want to have in our ontology, ceteris paribus we should. I want to describe a three-step procedure—with some choices at each step—for generating sets. I will use plural quantification quite a lot in this. I am assuming that one can make sense of plural quantification apart from sets.

Step 1: The non-empty candidates. The non-empty candidates, some of which will end up counting as sets in the next step, will be entities that stand in "packaging" relation to a plurality of objects, such that for any plurality, or at least for enough pluralities, there is a candidate that packages that plurality. There are many options for the non-empty candidates and the packaging relation.

Option A: Plural existential propositions, of the form <The Xs exist>, where a plural existential proposition p packages a plurality, the Xs, provided that it attributes existence to the Xs and only to the Xs.

Option B: Plural existential states of affairs (either Armstrong or Plantinga style), i.e., states of affairs of the Xs existing, where a plural existential state of affairs e packages a plurality, the Xs, provided that it is a state of affairs of the Xs existing. I got this option from Rob Koons.

Option C: This family of options generates the candidates in two sub-steps. The first is to have candidates that stand in a packaging relation to individuals, such that each candidates packages precisely one individual. Call these "singleton candidates". For brevity if x is a singleton candidate that packages y, I will say x is a singleton of y. The second step is to take our non-empty candidates to be mereological sums of singleton candidates, and to say that a mereological sum m packages the Xs if and only if m is a mereological sum of Ys such that each of the Ys is a singleton of one of the Xs and each of the Xs is packaged by exactly one of the Ys. We need the singleton packaging relation to satisfy the condition (*) that a mereological sum of singletons of the Xs has no singletons as parts other than the singletons of the Xs. (In particular, no singleton of y can be a part of any singleton of x if x and y are distinct.)

We get different instances of Option C by considering different singleton candidates. For instance, we could have the singleton candidates be individual essences, and a singleton candidate then packages precisely the entity that it is an individual essence of. I got this from Josh Rasmussen. Or we might use variants of Options A and B here: maybe a proposition attributing existence to x or a state of affairs of x existing will be our candidate singleton. (Whether the state of affairs option here differs from Option B depends on whether the state of affairs of a plurality existing is something different from the mereological sum of the states of affairs of the individuals in the plurality existing.)

There are many other ways of packaging pluralities.

Step 2: The empty candidate. We also need an empty candidate, which will be some entity that differs from the non-empty candidates of Step 1. Ideally, this will be an entity of the same sort as the non-empty candidates. For instance, if our non-empty candidates are propositions, we will want our empty candidate to be a proposition, say some contradictory proposition.

Step 3: Pruning the candidates. The basic idea will be that x is a member of a candidate y if and only if y is one of the non-empty candidates and y packages a plurality that has x in it. But the above is apt to give us too many candidates for them all to be sets. There are at least two reasons for this. First, on some of the options, there won't be a unique candidate packaging any given plurality. For instance, there might be more than one proposition attributing existence to the same plurality. Thus, the propositions <The Stagirite and Tully exist> and <Aristotle and Cicero exist> will be different propositions if Millianism is false, but both attribute existence to the same plurality. Second, some of the candidates will be better suited as candidates for proper classes than for sets and some candidates may be unsuitable either as sets or as proper classes. For instance, there might be a proposition that says that the plural existential propositions exist. Such a proposition packages all the candidates, including itself, and will not be a good set or proper class on many axiomatizations.

Why do sets have their members essentially?

It is often said that sets have their members essentially. Why? Normally the finger gets pointed at the axiom of extensionality which says that two sets that have exactly the same members are equal. The axiom of extensionality does not seem to imply by itself that sets have their members essentially. Consider a set theory without ur-elements (i.e., every member of a set is itself a set). Suppose that in the actual world there are two sets, Bob and Jane, where Bob is has no members and Jane's only member is Bob. Then imagine that in every other worlds all the sets are exactly like in the actual world, except that Jane has no members and Bob has Jane as its only member. I do not see—though I could be missing something—how this violates any of the axioms of set theory. If the axiom of extensionality were extended to a transworld context (if A in w₁ has the same members as B does in w₂, then A=B), we would get a contradiction, but the axiom of extensionality is an intraworld axiom.

But nonetheless I have reason to think that sets have their members essentially if the axiom of extensionality is correct, namely that what seem to me to be the two most plausible intuitive kinds of reasons to think sets need not have their members essentially conflict with the axiom of extensionality. We can argue for this by considering paradigmatic cases of the two reasons.

First kind of reasons: Losing members. Let S be the set { Quine, Wittgenstein, Socrates }. Then, the intuition goes, in a world w* where Quine never comes into existence, S only has two members, namely Wittgenstein and Socrates.

But in the actual world there is also the set T={ Wittgenstein, Socrates }. And we have every reason to think that T will exist in w* and have the same two members there that it actually has. Thus, in w*, the sets S and T have the same members, namely Wittgenstein and Socrates, and so in w* we have S=T. Since identity is necessary, it follows that actually S=T. But that's absurd since s has three members and T has two. So we need to reject

Second kind of reasons: Sets defined by descriptions. Let M be the set of all mathematicians. Intuitively, in a world where Hilbert becomes a biologist instead of a mathematician, M still exists, but Hilbert isn't a member of M. The intuition here is that in every world, or at least every world with mathematicians, M has as its members all and only the mathematicians.

But by the same token, if B is the set of all biologists, then in every world where there are biologist B will have as its members all and only the biologists. But there is a world where all and only biologists are mathematicians. In that world, then M=B. But since identity is necessary, it follows that actually M=B, and hence that Darwin was a mathematician and Hilbert a biologist, which is absurd.

Maybe you have some other reasons to doubt the essentiality of set membership?

Reducing sets to propositions

Lewis tried to reduce propositions to sets.  I think that doesn't work.  But maybe the other way around does.  Plausibly, for any xs there is a proposition that those xs exist. One can then identify collections with those propositions that affirm the existence of one or more things. Then x is a member of a collection c if and only if c represents, perhaps inter alia, x’s existing. We can then pick out our favorite axioms of set theory, and stipulate that any collection of collections that satisfies these axioms counts as a universe of sets. We will have to make a decision when we defined a collection as a proposition that affirms the existence of one or more things whether nonexistent things are permitted. If they are, and if propositions are all necessary beings, then we will have sets of nonexistent things. If they are not, then we won’t.

How many universes in a multiverse?

Suppose that all universes, or all universes which are "good enough", exist. For any object x, let U_x be a universe which contains exactly one conscious non-divine being, where this being is an angel who spends most of his time thinking virtuously and pleasurable precisely about x. This kind of world is surely "good enough". Thus, it seems there are at least as many universes in the multiverse as there are sets, since for any set S there is a universe U_S. That's not a problem yet—after all, maybe the collection of universes is a proper class. But now for any proper class K, we can imagine the universe U_K. Now the number of universes is really creeping up on us—it seems that we have at least one universe per set or proper class. Maybe we can introduce some higher order way of counting, beyond sets and proper classes. Sure. But the same problem will occur again.