Naturality and Definability II

Details Naturality and Definability II Naturality and Definability II

2001-02-07

arxiv.org/abs/math/0102060v1

Mathematics

Logic

Scientific paper

In two papers 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 says that there is a transitive model of Zermelo-Fraenkel set theory with choice (ZFC) in which every fully 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 that there is no model of ZFC in which the explicitly definable constructions are precisely the natural ones.

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