Naturality and definability II

  • Wilfrid Hodges Herons Brook, Sticklepath, Devon EX20 2PY, England.
  • Saharon Shelah Institute of Mathematics, Hebrew University, Jerusalem, Israel.
Keywords: Naturality, uniformisability, transitive models, ZFC set theory

Abstract

We regard an algebraic construction as a set-theoretically defined map taking structures A to structures B which have A as a distinguished part, in such a way that any isomorphism from A to A' lifts to an isomorphism from B to B'. In general the construction defines B up to isomorphism over A. A construction is uniformisable if the set-theoretic definition can be given in a form such that for each A the corresponding B is determined uniquely. A construction is natural if restriction from B to its part A always determines a map from the automorphism group of B to that of A which is a split surjective group homomorphism. We prove that there is no transitive model of ZFC (Zermelo-Fraenkel set theory with Choice) in which the uniformisable constructions are exactly the natural ones. We construct a transitive model of ZFC in which every uniformisable construction (with a restriction on the parameters in the formulas defining the construction) is ‘weakly’ natural. Corollaries are that the construction of algebraic closures of fields and the construction of divisible hulls of abelian groups have no uniformisations definable in ZFC without parameters.

References

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[7] T. Jech, Set theory (Academic Press, New York, 1978).
[8] G. Melles, ‘Classification theory and generalized recursive functions’, D.Phil. dissertation, University of California at Irvine, 1989.
Published
2020-01-20
How to Cite
Hodges, W., & Shelah, S. (2020). Naturality and definability II. CUBO, A Mathematical Journal, 21(3), 09–27. https://doi.org/10.4067/S0719-06462019000300009