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Tytuł: Non-Trivial Universes and Sequences of Universes
Autorzy: Coghetto, Roland
Słowa kluczowe: Tarski-Grothendieck set theory
Grothendieck universe
universe hierarchy
Data wydania: 2022
Data dodania: 29-gru-2022
Wydawca: DeGruyter Open
Źródło: Formalized Mathematics, Volume 30, Issue 1, Pages 53-66
Abstrakt: Universe is a concept which is present from the beginning of the creation of the Mizar Mathematical Library (MML) in several forms (Universe, Universe_closure, UNIVERSE) [25], then later as the_universe_of, [33], and recently with the definition GrothendieckUniverse [26], [11], [11]. These definitions are useful in many articles [28, 33, 8, 35], [19, 32, 31, 15, 6], but also [34, 12, 20, 22, 21], [27, 2, 3, 23, 16, 7, 4, 5]. In this paper, using the Mizar system [9] [10], we trivially show that Grothendieck’s definition of Universe as defined in [26], coincides with the original definition of Universe defined by Artin, Grothendieck, and Verdier (Chapitre 0 Univers et Appendice “Univers” (par N. Bourbaki) de l’Exposé I. “PREFAISCEAUX”) [1], and how the different definitions of MML concerning universes are related. We also show that the definition of Universe introduced by Mac Lane ([18]) is compatible with the MML’s definition. Although a universe may be empty, we consider the properties of non-empty universes, completing the properties proved in [25]. We introduce the notion of “trivial” and “non-trivial” Universes, depending on whether or not they contain the set ω (NAT), following the notion of Robert M. Solovay. The following result links the universes U₀ (FinSETS) and U₁ (SETS): GrothendieckUniverse ω = GrothendieckUniverse U₀ = U₁ . Before turning to the last section, we establish some trivial propositions allowing the construction of sets outside the considered universe. The last section is devoted to the construction, in Tarski-Grothendieck, of a tower of universes indexed by the ordinal numbers (See 8. Examples, Grothendieck universe, [24]). Grothendieck’s universe is referenced in current works: “Assuming the existence of a sufficient supply of (Grothendieck) univers”, Jacob Lurie in “Higher Topos Theory” [17], “Annexe B – Some results on Grothendieck universes”, Olivia Caramello and Riccardo Zanfa in “Relative topos theory via stacks” [13], “Remark 1.1.5 (quoting Michael Shulman [30])”, Emily Riehl in “Category the ory in Context” [29], and more specifically “Strict Universes for Grothendieck Topoi” [14].
Afiliacja: Roland Coghetto - cafr-MSA2P asbl, Rue de la Brasserie 5, 7100 La Louvi`ere, Belgium
DOI: 10.2478/forma-2022-0005
ISSN: 1426-2630
e-ISSN: 1898-9934
metadata.dc.identifier.orcid: 0000-0002-4901-0766
Typ Dokumentu: Article
Właściciel praw: © 2022 The Author(s)
CC BY-SA 3.0 license
Występuje w kolekcji(ach):Formalized Mathematics, 2022, Volume 30, Issue 1

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