Effective Cardinals in the Nonstandard Universe

Abstract

We study the structure of effective cardinals in the nonstandard set universe of Hrbaček set theory HST. Some results resemble those known in descriptive set theory in the domain of Borel reducibility of equivalence relations. * This project was partially supported by DFG grant 436 RUS 17/68/05. † IITP, Moscow, kanovei@mccme.ru and vkanovei@math.uni-wuppertal.de. Support of RFBR 03-01-00757 acknowledged. -Contact author. ‡ Dept. Math., University of Wuppertal, reeken@math.uni-wuppertal.de. 1 See [5, 29] on the precursorial history of infinitesimal analysis.1 Structure of the nonstandard universeThe language of Hrbaček set theory HST contains two basic predicates, the membership ∈ and the standardness st, hence it is called the st-∈-language.The axioms of HST describe a set universe À where the following classes are defined,so that Ë ⊆ Á, Á is an elementary extension of Ë in the ∈-language, Ë (and Á as well) satisfies ZFC in the ∈-language, the class Á is transitive, and the universe À is a von Neumann superstructure over Á. The universe À satisfies all ZFC axioms except for Regularity (weakened to Regularity over Á), Choice (weakened to Standard Size Choice) and Power Set axioms. The axioms of Separation and Replacement are accepted in the st-∈-language.Metamathematically, HST is equiconsistent with ZFC, and HST is a conservative extension of ZFC in the sense that any ∈-formula Φ is a theorem of ZFC iff Φ st (the relativization of Φ to Ë) is a theorem of HST. See [17] on 1 ∃ st and ∀ st are shorthands for "there is a standard", "for all standard". 2 A set x is well-founded iff its transitive closure has no infinite ∈-decreasing chains.3 axioms, metamathematics, basic set theoretic structures, and the structure of hyperreals in the HST universe.Convention 1.1 We argue in HST below unless otherwise stated.