Naturality and definability II

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...

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Autores principales:	Hodges,Wilfrid, Shelah,Saharon
Lenguaje:	English
Publicado:	Universidad de La Frontera. Departamento de Matemática y Estadística. 2019
Materias:	Naturality uniformisability transitive models ZFC set theory
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462019000300009
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spelling	oai:scielo:S0719-064620190003000092020-02-20Naturality and definability IIHodges,WilfridShelah,Saharon 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.info:eu-repo/semantics/openAccessUniversidad de La Frontera. Departamento de Matemática y Estadística.Cubo (Temuco) v.21 n.3 20192019-12-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462019000300009en10.4067/S0719-06462019000300009
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	Naturality uniformisability transitive models ZFC set theory
spellingShingle	Naturality uniformisability transitive models ZFC set theory Hodges,Wilfrid Shelah,Saharon Naturality and definability II
description	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.
author	Hodges,Wilfrid Shelah,Saharon
author_facet	Hodges,Wilfrid Shelah,Saharon
author_sort	Hodges,Wilfrid
title	Naturality and definability II
title_short	Naturality and definability II
title_full	Naturality and definability II
title_fullStr	Naturality and definability II
title_full_unstemmed	Naturality and definability II
title_sort	naturality and definability ii
publisher	Universidad de La Frontera. Departamento de Matemática y Estadística.
publishDate	2019
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462019000300009
work_keys_str_mv	AT hodgeswilfrid naturalityanddefinabilityii AT shelahsaharon naturalityanddefinabilityii
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Naturality and definability II

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