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.

A natural extension of Lebesgue measure, and Brown's argument against the Continuum Hypothesis

The Lebesgue measure on the reals has the following property: it assigns zero probability to every singleton. This is intuitively what we would expect of a uniform measure of the reals by the following line of thought: If I throw a uniformly distributed dart at the interval [0,1], the probability that I will land on any singleton is infinitely smaller than one, i.e., zero. Here is a generalization of this line of thought: If A is a subset of [0,1] of lower cardinality than [0,1] (i.e., lower cardinality than c, the cardinality of the continuum), then the probability that the dart will land in A is infinitely smaller than one, i.e., zero, since A is intuitively infinitely smaller than [0,1] (the union of infinitely many disjoint copies of A will still be smaller than [0,1], assuming the Axiom of Choice).

One might at this point ask: Does Lebesgue measure respect this intuition? If the Continuum Hypothesis is true, then of course it does. For then all subsets of lower cardinality than [0,1] are countable, and all countable subsets have null measure, since the singletons have null measure. Without the Continuum Hypothesis this may or may not be true. See the references here. However, even without the Continuum Hypothesis, but given Choice, it can be easily shown (see the previous link) that any Lebesgue measurable subset of lower cardinality than the continuum has null measure. But ZFC is consistent with the claim that all subsets of the reals of lower cardinality than the continuum have null measure as well as with the claim that some are non-measurable and hence do not have null measure.

Nonetheless, given Choice, the Lebesgue measure on the reals extends in a very natural way to a translation-invariant measure m* on a σ-algebra F* that contains all subsets of the reals that have cardinality less than that of the continuum. We can call this the lower-cardinality-nulling extension of the Lebesgue measure.

Because of the intuitions in the first paragraph, it seems to me that the lower-cardinality-nulling extension of Lebesgue measure (or some extension thereof) is the measure we should work with for confirmation theoretic purposes where normally Lebesgue measure is used. Also, while right now I don't know if there could be a translation-invariant extension of Lebesgue measure that assigns non-zero measure to some subset of cardinality less than c, it is easy to see that if there is such an extension, then it assigns measure 1 to some subset of [0,1] of lower cardinality, and that is surely intuitively unacceptable for the uniform results of dart throws and the like. Hence, every acceptable translation-invariant extension of Lebesgue measure assigns zero to every set of cardinality less than c, and since there is a translation-invariant such extension, so we have a good intuitive argument in favor of working with such a measure.

If this is right then, then the Brown two-dart argument against the Continuum Hypothesis (see references and helpful critique here) is misguided. For we should take the measure governing dart throws to be a lower-cardinality-nulling extension of Lebesgue measure. And once we do that, then the Brown two-dart argument works just as well without assuming the Continuum Hypothesis. Hence whatever problem it identifies is not specific to the Continuum Hypothesis.

Appendix: Construction of the lower-cardinality-nulling extension of the Lebesgue measure. Let F* be the σ algebra consisting of all the subsets of R that differ from a member F by a set whose cardinality is less than c. Suppose A is a member of F*. Then A can be represented as (U∪N₁)−N₂ where U is in F* and N₁ and N₂ have cardinality less than c. Let m*(A)=m(U). To see that m* is well-defined, suppose that (U∪N₁)−N₂=(V∪M₁)−M₂, where V is in F and M₁ and M₂ have cardinality less than c. Then U and V differ by sets of cardinality less than c, and hence their symmetric difference has null Lebesgue measure, and so m(U)=m(V), and m*(A) is well-defined. Now a countable union of sets A_n of cardinality less than c has cardinality less than c. This follows from the fact that (using the Axiom of Choice) c has uncountable cofinality, so that there is a cardinality K such that |A_n|≤K<c for all n, and hence, again by Choice, the union of the A_n must have cardinality at most K, if K is infinite, and at most countable if K is finite. Since F* is clearly closed under complements, it's a σ-algebra. To check that m* is a measure, we need only check it's countably additive. This easily follows once again from the fact that a countable union of sets of cardinality less than c has cardinality less than c. And translation invariance is obvious.

The above argument only uses the claim that measurable sets of cardinality less than c are null. This is true for n-dimensional Lebesgue measure. (Proof: Suppose that A in Rⁿ is a bounded measurable set of cardinality less than c. Let A' be a projection of A onto any one axis. Then A' is a measurable one-dimensional set of cardinality less than c, and hence null. For large enough L, A will be a subset of the cartesian product of A' and an (n−1)-dimensional ball of radius L, and so A will also be null. But if all bounded measurable sets of cardinality less than c are null, then so are the unbounded ones.) And so the cardinality-nulling extension works in n-dimensions as well.