An odd thought about ZFC

The axioms of ZFC set theory can be divided into (a) the positive axioms, that say that a set with certain properties exist, and (b) two negative axioms that deny the existence of certain sets (Extensionality: given any set, there is no other set with the same members; Regularity: no irregular sets).

The positive axioms divide further into two classes: (i) those that are obvious special cases of naive set theory’s Axiom of Comprehension, and (ii) the Axiom of Choice.

Here is an alternate intellectual history thought experiment. Suppose we never discovered the contradiction in naive set theory or anything like it, maybe because we had a psychological block against thinking about non-self-membered sets, applying Cantor’s Theorem to the universal set, etc. The Axiom of Choice would continue to have an intuitive plausibility, and the “mathematical need” for it, say in the case of the Hahn-Banach Theorem, would likely still arise. And so we would be pulled to adopt it.

This makes me think this. The other positive axioms of ZFC (i.e., the positive axioms of ZF) have an ad hoc feel to them. They are special cases of Comprehension, carefully chosen to both give enough applications of Comprehension and to avoid contradiction (we hope). I feel that much of the plausibility of the other positive axioms of ZFC comes from their being special cases of the highly intuitive—but incoherent—Axiom of Comprehension. And that’s a little suspicious.

Normally one thinks of the Axiom of Choice as the most suspicious of ZFC’s axioms. But here we have a source of suspicion for axioms of ZFC that does not affect Choice.

Well, maybe. Maybe an enemy of Choice could say that both Choice and Comprehension are the fruit of the poisonous tree of principles of plenitude.

An Axiom of Choice strong enough to puzzle

All the main puzzles that follow from the Axiom of Choice (AC)--nonmeasurable sets, Banach-Taski and guessing future coin tosses--need only a weaker version of AC. One weaker version that suffices is this:

(*) There is a choice function for any partition of the interval (0,1) into non-empty countable sets.

Now imagine worlds with point-sized particles that never move, but can perish and come into existence. The world starts at time 0. Each particle has a lifetime between 0 and 1, exclusive. Some locations in the world are never occupied by a particle. Call these "vacant". At all other locations, a particle comes into existence at time 0. Two particles never occupy the same location at the same time. Call such worlds p-worlds.
For each non-vacant location x in a p-world w, there is an associated set L(w,x) of numbers in (0,1), where a number y is in L(w,x) iff some particle at x has lifetime of length y. I now need a crucial metaphysical plenitude assumption:

(**) For any set S such that (a) every member of S is a countable non-empty collection of members of (0,1) and (b) the cardinality of S is at most that of the continuum, there is a p-world w such for each A in S there is a unique location x in w such that L(w,x)=A.

In other words, any set S satisfying (a) and (b) is the set of sets of lifetime lengths for non-vacant locations in some p-world, without duplication.

Given the plenitude assumption, I get the version of AC needed for the paradoxes. For given a partition S of (0,1) into countable sets, there will be a p-world as in (**). Given a member A of S, there will be a unique location x such that L(w,x)=A. Let f(A) be the lifetime of the first particle at x in w. This is our choice function.

So the major paradoxes of AC follow from a plausible plenitude assumption about possible worlds.