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Set theory with and without urelements and categories of interpretations. (English) Zbl 1108.03051
The author proves the synonymity of two set theories, ZF and ZFU, in the sense that there are interpretations \(I:\text{ZF}\to \text{ZFU}\) and \(J: \text{ZFU}\to \text{ZF}\) with the property that \(I\circ J\) and \(J\circ I\) are “identities”. (The theory ZFU carries an extra axiom dictating that the set of ur-elements is countable. The success of the present work crucially depends on this.) One can think of common-sense interpretations \(I':\text{ZF}\to \text{ZFU}\) and \(J': \text{ZFU}\to \text{ZF}\). (For instance, \(J'\) picks, as members of the domain, those elements whose transitive closures contain no ur-elements.) But neither has the inverse to produce identity. So the author resorts to ingeneous constructions to obtain the above \(I\) and \(J\). (A. Visser embarked on classifying theories by manners of mutual interpretability in [Categories of theories and interpretations. Logic Group Preprint Series, Vol. 228, Univ. Utrecht (2004) (per bibl.); see also Lect. Notes Log. 26, 284–341 (2006; Zbl 1107.03066)]. The strictest condition is synonymity. \(I'\) and \(J'\) satisfy a weaker condition of homoty of ZF and ZFU. But synonymity was an open question.)

MSC:
03E30 Axiomatics of classical set theory and its fragments
03F25 Relative consistency and interpretations
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[1] Jech, T., Set Theory , Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. · Zbl 1007.03002
[2] Visser, A., Categories of Theories and Interpretations , vol. 228 of Logic Group Preprint Series , Faculteit Wijsbegeerte, Universiteit Utrecht, 2004.
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