Sh:301

• Hodges, W., & Shelah, S. (2019). Naturality and definability II. Cubo, 21(3), 9–27. arXiv: math/0102060 DOI: 10.4067/s0719-06462019000300009 MR: 4077584
• Abstract:
  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.
• Version 2020-07-14 (20p) published version (19p)

@article{Sh:301,
 author = {Hodges, Wilfrid and Shelah, Saharon},
 title = {{Naturality and definability II}},
 journal = {Cubo},
 fjournal = {Cubo. A Mathematical Journal},
 volume = {21},
 number = {3},
 year = {2019},
 pages = {9--27},
 issn = {0716-7776},
 mrnumber = {4077584},
 mrclass = {08A35 (03E35 18A15)},
 doi = {10.4067/s0719-06462019000300009},
 note = {\href{https://arxiv.org/abs/math/0102060}{arXiv: math/0102060}},
 arxiv_number = {math/0102060}
}