Naturality and Definability, I

Eilenberg and Mac Lane [1] explained the notion of a 'natural' embedding by giving a categorical definition. Starting from their examples, we argue that one could equally well explain natural as meaning 'uniformly definable in set theory'. But do the categorically natural embeddings coincide with the uniformly definable ones? This is partly a technical question about whether certain well-known algebraic constructions are definable in set theory. It can also be seen as a test of the adequacy of ZFC as a foundation for algebra. After laying out the groundwork in Section 1, we analyse some examples (such as divisible hulls of abelian groups) in Section 2, and in Section 3 we formalise the main problem. The first partial answer is in Section 4: naturality does imply uniform definability with parameters, provided only a set of isomorphism types are involved. The proof also partially answers a model-theoretic question of Gaifman about definable operations. In Section 5 we prove a partial converse: uniform definability can imply naturality if urelements are allowed. Section 6 puts an upper bound on possible strengthenings of this result by showing that if uniform definability implies definable naturality in a model M of set theory, then M has a global choice function. In part II of this paper we hope to give an example of an unnatural but uniformly definable construction. The results of Section 6 are due to the second author. The first author takes responsibility for the rest; he thanks the Royal Society for a grant which took him to Jerusalem for three weeks in 1981, where the results in the paper were assembled in discussion between the authors.