Plural quantification and the continuum hypothesis

Some people, including myself, are concerned that plural quantification may be quantification over sets in logical clothing rather than a purely logical tool or a free lunch. Here is a somewhat involved argument in this direction. The argument has analogues for mereological universalism and second-order quantification (and is indeed a variant of known arguments in the last context).

The Continuum Hypothesis (CH) in set theory says that there is no set whose cardinality is greater than that of the integers and less than that of the real numbers. In fact, due to the work of Goedel and Cohen, we know that CH is independent of the axioms of Zermelo-Fraenkel-Choice (ZFC) set theory assuming ZFC is consistent, and indeed ZFC is consistent with a broad variety of answers to the question of how many cardinalities there are between the integers and the reals (any finite number is a possible answer, but there can even be infinitely many). While many of the other axioms of set theory sound like they might be just a matter of the logic of collections, neither CH nor its denial seems like that. Indeed, these observations may push one to think that there many different universes of sets, some with CH and others with an alternative to CH, rather than a single privileged concept of “true sets”.

Today I want to show that plural quantification, together with some modal assumptions, allow one to state a version of CH. I think this pushes one to think analogous things about plural quantification as about sets: plural quantification is not just a matter of logic (vague as this statement might be) and there may even be a plurality of plural quantifications.

This is well-known given a pairing function. But I won’t assume a pairing function, and instead I will do a bunch of hard work.

The same approach will give us a version of CH in Monadic Second Order logic and in a mereology with arbitrary fusions.

Let’s go!

Say that a possible world w is admissible provided that:

1. w is a multiverse of universes

2. for any two universes u and v and item a in u, there is a unique item b in v with the same mass as a

3. the items in each universe are well-ordered by mass

4. for each item in each universe there is an item in the same universe with bigger mass

5. for each item c in a universe u if a is not of the least mass in u, a has an immediate predecessor with respect to mass in u.

The point of (3)–(5) is to ensure that each universe has a least-mass item and that there are only countably many items. If we assumed that masses are real numbers, we would just need (3) and (4).

Say that pluralities of items xx in a universe u and yy in a universe v of an admissible world correspond provided that for all natural numbers i, if a is in u and b is in v and a and b have equal mass, then a is among xx if and only if b is among yy. Two individual items correspond provided that they have equal mass.

Say that an admissible possible world w is big provided that:

6. at w: there is a plurality xx of items such that (a) for any universe u and any plurality yy of items in u, there is a universe v such that the subplurality of items from xx that are in v corresponds to yy and (b) there are no distinct universes u and v each with an item in common with xx such that the subpluralities of xx consisting of items in u and v, respectively, correspond to each other.

The bigness condition ensures that we have at least continuum-many universes.

Say that the head of a universe u is the item u in the universe that has least mass. Say that two items are neighbors provided that they are in the same universe. We can identify universes with their heads.

Say that a plurality hh of heads of universes is countable provided that:

• There is a plurality xx of items such that each of the hh has exactly one neighbor among the xx and no two items of xx correspond.

The plurality xx defines a mapping of each head in hh to one of its neighbors, and the above condition ensures each distinct pair of heads is mapped to non-corresponding neighbors, and that ensures there are countably many hh.

Say that a plurality hh of heads of universes is continuum-sized provided that:

• There is a plurality xx of items such that each head z among the hh has a neighbor among the xx, and for any universe u and any plurality yy of the items of u, there is a unique head z among the hh such that the plurality of its neighbors corresponds to z.

The plurality xx basically defines a bijection between hh and the subpluralities of any fixed universe.

Given pluralities gg and hh of heads of universes, say pluralities xx and yy of items define a mapping from gg to hh provided that:

• There are pluralities xx and yy of items such that for each item a from gg, if uu is the plurality of a’s neighbors among the xx, then there is a unique item b among the hh such that the plurality vv of b’s neighbors among the yy corresponds to uu.

If a and b are as above, we say that b is the value of a under the mapping defined by xx and yy. Here’s how this works: xx defines a map of heads in gg to pluralities of their respective neighbors and yy defines a map of some of the heads in hh to pluralities of their respective neighbors, and then the correspondence relation can be used to match up heads in gg with heads in hh.

We now say that the mapping defined by xx and yy is injective provided that distinct items in gg never have the same value under the mapping. (This is only going to be possible if gg is continuum-sized.)

If there are xx and yy that define an injective mapping from gg to hh, then we say that |gg| ≤ |hh|. If we have |gg| ≤ |hh| but not |hh| ≤ |gg|, we say that |gg| < |hh|.

The rest is easy. The Continuum Hypothesis for the heads in big admissible w says that there aren’t pluralities of heads gg and hh such that gg is not countable, hh is continuum-sized, and |gg| < |hh|.

We can also get analogues of the finite alternatives to the Continuum Hypothesis. For instance, an analogue to 2^ℵ₀ = ℵ₃ says that there are pluralities bb, cc and dd of heads such that bb is not countable, dd is continuum sized and |bb| < |cc| < |dd|, but there are not pluralities aa, bb, cc and dd with aa not countable, dd continuum-sized and |aa| < |bb| < |cc| < |dd.

Megethology as mathematics and a regress of structuralisms

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

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

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

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

Mereology, plural quantification and free lunches

It is sometimes claimed that arbitrary mereological fusions and plural quantification are a metaphysical free lunch, just a new way of talking without any deep philosophical (or at least metaphysical) commitments.

I think this is false.

Consider this Axiom of Choice schema for mereology:

1. If for every x and y such that ϕ(x) and ϕ(y), either x = y or x and y don’t overlap, and if every x such that ϕ(x) has a part y such that ψ(y), then there is a z such that for every x such that ϕ(x), there is common part y of x and z such that ψ(y).

Or this Axiom of Choice schema for pluralities:

2. If for all xx and yy such that ϕ(xx) and ϕ(yy) either xx and yy are the same or have nothing in common, then there are zz that have exactly one thing in common with every xx such that ϕ(xx).

If arbitrary mereological fusions and plural quantification are a metaphysical free lunch, just a handy way of talking, then whether (1) or (2) is correct is just a verbal question.

But (1) and (2) respectively imply mereological and plural Banach-Tarski paradoxes:

3. If z is a solid ball made of points, then it has five pairwise non-overlapping parts, of which the first two can be rigidly moved to be pairwise non-overlapping and compose another ball of the same size as z, and the last three can likewise be so moved.

4. If the xx are the points of a solid ball, then there are aa, bb, cc, dd and ee which have nothing pairwise in common and such that together they make up xx, and there are rigid motions that allow one to move aa and bb into pluralities that have nothing in common but make up a solid ball of the same size as xx and to move cc, dd and ee into pluralities that have nothing in common and make up another solid ball of the same size.

Conversely, assuming ZF set theory is consistent, there is no way to prove (3) and (4) if we do not have some extension to the standard axioms of mereology or plurals like the Axiom of Choice. The reason is that we can model pluralities and mereological objects with sets of points in three-dimensional space, and either (3) or (4) in that setting will imply the Banach-Tarski paradox for sets, while the Banach-Tarski paradox for sets is known not to be provable from ZF set theory without Choice.

But whether (3) or (4) is true is not a purely verbal question.

One reason it’s not a purely verbal question is intuitive. Banach-Tarski is too paradoxical for it or its negation to be a purely verbal thing.

Another is a reason that I gave in a previous post with a similar argument. Whether the Banach-Tarski paradox holds for sets is not a purely verbal question. But assuming that the Axiom of Separation can take formulas involving mereological terminology or plural quantification, each of (3) and (4) implies the Banach-Tarski paradox for sets.

Consciousness and plurality

One classic critique of Descartes’ “cogito ergo sum” is that perhaps there can be thought without a subject. Perhaps the right thing to say about thought is feature-placing language like “It’s thinking”, understood as parallel to “It’s raining” or “It’s sunny”, where there really is no entity that is raining or sunny, but English grammar requires a subject so we through in an “it”.

There is a more moderate option, though, that I think deserves a bit more consideration. Perhaps thought has an irreducibly plural subject, and in a language that expresses the underlying metaphysics better, we should say “The neurons are (collectively) thinking” or maybe even “The particles are (collectively) thinking.” On this view, thought is a relation that holds between a plurality of objects, without these objects making up a whole that thinks. This, for instance, is a very natural view for physicalist who is a compositional nihilist (i.e., thinks that only simples exists).

It seems to me that it is hard to reject this view if one’s only data is the fact of consciousness, as it is for Descartes. What kills the three-dimensionalist version of this view, in my opinion, is that it cannot do justice to the identity of the thinker over time, since there would be different pluralities of neurons or particles engaged in the thinking over time. And a four-dimensionalist version cannot do justice to the identity of the thinker in counterfactual scenarios. However, this data isn’t quite as self-evident as what Descartes wants.

In any case, I think this is a view that non-naturalists like me need to take pretty seriously.

Unifying Separation and Choice

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

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

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

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

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

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

So what?

Nothing earthshaking.

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

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

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

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

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

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.

What plurals are there?

Plural quantification is meant to be a logical way of avoiding some technical and/or conceptual difficulties with sets and second-order quantification. Instead of quantifying over one thing, one quantifies over pluralities. Thus, a theist might say: For all xs, God thinks of the xs in their interrelationship.

What plurals are there? Intuitively, for any finite list of objects, there is a plurality of precisely those objects. After all, we can easily have a sentence about any finite plurality of things we have names for: Alice, Bob and Carl like each other. But what furthe pluralities are there?

An expansive proposal is plural comprehension: the axiom schema that says that for any formula F with free variables that include y, for any values of the free variables other than y, there are xs such that y is one of the xs iff F. Unlike the comprehension schema in naive set theory, there does not seem to be any direct Russell-type paradox for plural comprehension, because the xs are not in general an object, but multiple objects.

But plural comprehension on its own does not seem to quite settle what plurals there are. Suppose we have a plurality of nonempty disjoint sets. We can for instance ask: Is there a plurality of objects that includes exactly one object from each of these sets? If (a) there is a set of these disjoint sets, and (b) the Axiom of Choice holds for sets, then the answer is affirmative by plural comprehension. But of course whether the Axiom of Choice holds for sets is itself not philosophically settled, and further not every plurality of sets is such that there is a set of the sets in the plurality.

Observations of this sort show that plural quantification is not as metaphysically innocent as it may seem. You might have hoped that there is no further metaphysical commitment in allowing for plural quantification than in singular quantification. But we can now have substantive questions about what pluralities there are even after we have fixed what singular objects there are, even if we assume plural comprehension. For instance, suppose we think that the objects are the physical objects of the world plus the elements of a model of ZF set theory with ur-elements and with the negation of the Axiom of Choice. We can know what all the objects are, and it still not be decided what pluralities there are. For in the case of a set of disjoint nonempty sets that lacks a choice set, as far as I can tell, there still might be a "choice plurality" (a plurality that has exactly one object from each of the disjoint sets) or there might not be one. (And if you say, well, the Axiom of Choice is obviously true, I may try to come back with a similar issue regarding Choice for proper classes.)

Or I might make a similar point about the Continuum Hypothesis (CH). The following story seems quite coherent. Every uncountable subset of the real numbers is in a bijection with the set of reals (i.e., CH is true), but there is an uncountable plurality of real numbers not in bijection with the plurality of reals. (It's easy to define bijections of pluralities in terms of pluralities of pairs.) But it's also coherent that CH is true, but there is no such uncountable plurality of reals--i.e., that CH is true for sets but its analogue for pluralities is false.

We might try to get out of this by insisting that, necessarily, the right set theory has to have a stronger version of the Schema of Separation that allows for formulas free plural variables and for the plural-membership relation. But that's conceding that the theory of pluralities is metaphysically non-innocent, because now what pluralities there are will constrain what objects there are!

So the question of what restrictions we put on plurals is a really substantive question.

Next note that following point. There seem to be two particularly simple and non-arbitrary answers to the Special Composition Question which asks which pluralities compose a whole: nihilism (there are no non-trivial cases of composition) and universalism (every plurality composes a whole). But once we have realized that it is a substantive question what pluralities there are, it seems that what objects there are and affirming universalism, even with mereological essential thrown in, doesn't settle the question of what wholes there are. There is substantial metaphysics to be done to figure out what pluralities there are!

I say the above with a caution: there are various technicalities I am glossing over, and I wouldn't be surprised if some of them turned out to be really important.