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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.

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