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I learned functions in terms of sets. Domain and codomain are sets. Function is a set of ordered pairs between them.

How could we go the other way? A set can be "defined" by the predicate that tests membership, but then how do we model the predicates? Some formalism like the lambda calculus?

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Not sure of details to make it a mathematical foundation but:

A category can be defined in terms of its morphisms without mentioning objects and a topos has predicates as morphisms into the subobject classifier.

i think predicates are functions returning booleans

lambda calculus would provide a computational way to determine the truth value of the predicate, any computable predicate that is.

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