Wednesday, January 31, 2024

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.

1 comment:

Heavenly Philosophy said...

Interestingly, Joshua Rasmussen also thinks that the axioms of ZFC are arbitrary and ad hoc, so he just rejects them. I think he created his own axioms or just prefers positive set theory over ZFC.