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”.