Naturality and Definability Ii

In two papers [3] and [4] we noted that in common practice many algebraic constructions are defined only ‘up to isomorphism’ rather than explicitly. We mentioned some questions raised by this fact, and we gave some partial answers. The present paper provides much fuller answers, though some questions remain open. Our main result, Theorem 4, says that there is a transitive model of Zermelo-Fraenkel set theory with choice (ZFC) in which every explicitly definable construction is ‘weakly natural’ (a weakening of the notion of a natural transformation). A corollary is that there are models of ZFC in which some well-known constructions, such as algebraic closure of fields, are not explicitly definable. We also show (Theorem 2) that there is no transitive model of ZFC in which the explicitly definable constructions are precisely the natural ones. Most of this work was done when the second author visited the first at Queen Mary, London University under SERC Visiting Fellowship grant GR/E9/639 in summer 1989, and later when the two authors took part in the Mathematical Logic year at the Mittag-Leffler Institute in Djursholm in September 2000. The second author proposed the approach of section 3 on the first occasion and the idea behind the proof of Theorem 4 on the second. Between 1975 and 2000 the authors (separately or together) had given some six or seven false proofs of versions of Theorem 4 or its negation. The authors thank Ian Hodkinson for his invaluable help (while research assistant to Hodges under SERC grant GR/D/33298) in unpicking some of the earlier false proofs.