Topics Mathematics

Question I have been studying axiomatic set theory as a foundation of mathematics and am stuck on the definition of a relation as a subset of a Cartesian product. I have two problems. The first is that a large number of relations seem to be presupposed prior to this definition: the truth-functional relations of logic, for example, or the relations of set-membership and subset. Doesn't this make the definition circular? Second, in specifying which subset of the Cartesian product is intended, a polyadic predicate is usually invoked; but isn't a polyadic predicate a relation, thus giving a second circularity? Furthermore, these are vicious circles, not harmless ones.

Accepted:
February 19, 2006

Comments

All theorems about relations in axiomatic set theory are proven just from the axioms, using the rules of first-order logic. Thus, no facts about relations are presupposed in these proofs--at least, not if by "presupposed" you mean "used to justify a step in a proof."

But perhaps this is not the sense of "presupposed" that you have in mind. Perhaps what you are thinking is that in order to really understand what's going on in the development of the theory of relations in axiomatic set theory, you have to have an intuitive understanding of what a relation is. Or perhaps you mean that no one would believe that the rules of first-order logic represent correct reasoning, or that the axioms of set theory are true statements about sets, if they weren't familiar with certain relations, such as the truth-functional relations of logic or the set-membership relation. You may be right about this.

Is this a problem for the idea that axiomatic set theory is a foundation for mathematics? Not necessarily. When people say that set theory is the foundation of mathematics, I don't think they mean that someone with no understanding of logic or sets could come to understand mathematics by simply reading a completely formalized development of mathematics in axiomatic set theory. They just mean that all the theorems of mathematics can be justified by proofs that assume only the axioms of set theory. This is a claim about the justification of mathematics, not a claim about how people acquire understanding of mathematics.