Internal approach to external sets and universes

Kanovei, Vladimir; Reeken, Michael

doi:10.1007/bf01057803

Cited by 16 publications

(22 citation statements)

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“…For this purpose, the formulation of SMA"" above could be viewed as rather artificial in the choice of Ordg as the index domain and the related requirement that internal infinite sets are equinumerous to Ordg. However, as in the KST case above, we are going to prove the consistency of the statement (6) there is n E *w \ w such that [0, n) is non-equinumerous to *w 21) with NCT, which should contradict any reasonable formulation of SMA. This result is obtained as a rather simple corollary, of the consistency of (6) with KST which we proved above.…”

Section: Independencementioning

confidence: 90%

“…In order to fulfill the NST definition of internal sets, we first define I I ' = { z E II : (3 y E 9) (z E y)}. One easily demonstrates (see e. g. KANOVEI and REEKEN [6,Part 11) that 1 ' is an elementary extension of S in the sense of NST-Transfer. Define st t iff z E S. Let W = UnEw P"(lI').We have to restrict W the same way as in the previous subsection and towards the same goal.…”

Section: Independencementioning

confidence: 99%

“…Let W' be the set of all those elements z E W which satisfy (3 y E S) (11x11 y). The structure (HI'; E, st ) interprets NST + (6). The class part can be attached as in the end of Subsection 6.3.…”

Section: Independencementioning

confidence: 99%

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Special Model Axiom in Nonstandard Set Theory

Kanovei

Reeken

1999

Mathematical Logic Qtrly

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We demonstrate that the special model axiom SMA of Ross admits a natural formalization in Kawdi's nonstandard set theory KST but is independent of KST. As an application of our methods to classical model theory, we present a short proof of the consistency (with ZFC) of the existence of a IC+ like /+saturated model of PA for a given cardinal 6.Mathematics Subject Classification: 03H05, 03C50.Keywords: Nonstandard set theory, Special model axiom. IntroductionLet K be an infinite cardinal. The Special Model Axiom SMA, is the following statement (see Ross [12] and also JIN [5]): If 2l is an infinite internally presented structure of a first-order language L, cardL < K , and X = card%, then U has the form of the union U = U,<.21a of an elementary chain {U,},

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Section: Independencementioning

confidence: 90%

Section: Independencementioning

confidence: 99%

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Special Model Axiom in Nonstandard Set Theory

Kanovei

Reeken

1999

Mathematical Logic Qtrly

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“…Аксиома Ограниченности вытекает из ограниченности этого вложения. [21], [22].) (ii) Допустим, что ϕ st выводима в BST.…”

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Проблемы Теоретико-Множественного Нестандартного Анализа

Kanovei¹,

Любецкий²,

Lyubetskii³

2007

Uspekhi Mat. Nauk

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В статье излагаются главные результаты последних лет в области теоретико-множественного нестандартного анализа (нестандартные теории множеств и классов). Показано, каким образом универсумы более простых теорий (начиная с обычной теории множеств Цермело-Френкеля ZFC) могут быть расширены до универсумов более сложных нестандартных теорий множеств и классов. Заключительный параграф излагает основные положения булевозначного анализа как части теоретико-множественного нестандартного анализа. Библиография: 66 названий.

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“…In the mid-1990s we formulated Hrbaček set theory HST [14], based on earlier theories in [8,9]. This theory accumulated achievements of different nonstandard set theories and inhibited their faults.…”

Section: Introductionmentioning

confidence: 99%

Effective Cardinals in the Nonstandard Universe

Kanovei¹,

Reeken²

2006

Mathematical Logic in Asia

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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.

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Internal approach to external sets and universes

Cited by 16 publications

References 15 publications

Special Model Axiom in Nonstandard Set Theory

Special Model Axiom in Nonstandard Set Theory

Проблемы Теоретико-Множественного Нестандартного Анализа

Effective Cardinals in the Nonstandard Universe

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