### Naturality and Definability II

by Hodges and Shelah. [HoSh:301]

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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