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.

Some issues concerning eliminative structuralism for second-order arithmetic

Eliminative structuralist philosophers of mathematics insist that what mathematicians study is structures rather than specific realizations of these structures, like a privileged natural number system would be. One example of such an approach would be to take the axioms of second-order Peano Arithmetic PA2, and say an arithmetical sentence ϕ is true if and only if it is true in every standard model of PA2. Since all such models are well-known to be isomorphic, it follows that for every arithmetical sentence ϕ, either ϕ or  ∼ ϕ is true, which is delightful.

The hitch here is the insistence on standard (rather than Henkin) models, since the concept of a standard depends on something very much like a background set theory—a standard model is a second-order model where every subset of Dⁿ is available as a possible value for the second-order n-ary variables, where D is the first-order domain. Thus, such an eliminative structuralism in order to guarantee that every arithmetical sentence has a truth value seems to have to suppose a privileged selection of subsets, and that’s just not structural.

One way out of this hitch is to make use of a lovely internal categoricity result which implies that if we have any second-order model, standard or not, that contains two structures satisfying PA2, then we can prove that any arithmetical sentence true in one of the two structures is true in the other.

But that still doesn’t get us entirely off the hook. One issue is modal. The point of eliminative structuralism is to escape from dependence on “mathematical objects”. The systems realizing the mathematical structures on eliminative structuralism don’t need to be systems of abstract objects: they can just as well be systems of concrete things like pebbles or points in space or times. But then what systems there are is a contingent matter, while arithmetic is (very plausibly) necessary. If we knew that all possible systems satisfying PA2 would yield the same truth values for arithmetical sentences, life would be great for the PA2-based eliminative structuralist. But the internal categoricity results don’t establish that, unless we have some way of uniting PA2-satisfying systems in different possible worlds in a single model. But such uniting would require there to be relations between objects in different worlds, and that seems quite problematic.

Another issue is the well-known issue that assuming full second-order logic is “too close” to just assuming a background set-theory (and one that spans worlds, if we are to take into account the modal issue). If we could make-do with just monadic second-order logic (i.e., the second-order quantifiers range only over unary entities) in our theory, things would be more satisfying, because monadic second-order logic has the same expressiveness as plural quantification, and we might even be able to make-do with just first-order quantification over fusions of simples. But then we don’t get the internal categoricity result (I am pretty sure it is provable that we don’t get it), and we are stuck with assuming a privileged selection of subsets.