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- Minimal_model_(set_theory) abstract "In set theory, a minimal model is a minimal standard model of ZFC.Minimal models were introduced by (Shepherdson 1951, 1952, 1953).The existence of a minimal model cannot be proved in ZFC, even assuming that ZFC is consistent, but follows from the existence of a standard model as follows. If there is a set W in the von Neumann universe V which is a standard model of ZF, and the ordinal κ is the set of ordinals which occur in W, then Lκ is the class of constructible sets of W. If there is a set which is a standard model of ZF, then the smallest such set is such a Lκ. This set is called the minimal model of ZFC, and also satisfies the axiom of constructibility V=L. The downward Löwenheim–Skolem theorem implies that the minimal model (if it exists as a set) is a countable set. More precisely, every element s of the minimal model can be named; in other words there is a first order sentence φ(x) such that s is the unique element of the minimal model for which φ(s) is true.Cohen (1963) gave another construction of the minimal model as the strongly constructible sets, using a modified form of Godel's constructible universe. Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets which are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, they do not use the normal element relation and they are not well founded.If there is no standard model then the minimal model cannot exist as a set. However in this case the class of all constructible sets plays the same role as the minimal model and has similar properties (though it is now a proper class rather than a countable set).".
- Minimal_model_(set_theory) wikiPageID "18374778".
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- Minimal_model_(set_theory) wikiPageOutDegree "9".
- Minimal_model_(set_theory) wikiPageRevisionID "661311108".
- Minimal_model_(set_theory) wikiPageWikiLink Axiom_of_constructibility.
- Minimal_model_(set_theory) wikiPageWikiLink Category:Constructible_universe.
- Minimal_model_(set_theory) wikiPageWikiLink Constructible_universe.
- Minimal_model_(set_theory) wikiPageWikiLink Inner_model.
- Minimal_model_(set_theory) wikiPageWikiLink Löwenheim–Skolem_theorem.
- Minimal_model_(set_theory) wikiPageWikiLink Standard_model_(set_theory).
- Minimal_model_(set_theory) wikiPageWikiLink Von_Neumann_universe.
- Minimal_model_(set_theory) wikiPageWikiLink ZFC.
- Minimal_model_(set_theory) wikiPageWikiLink Zermelo–Fraenkel_set_theory.
- Minimal_model_(set_theory) wikiPageWikiLinkText "Minimal model (set theory)".
- Minimal_model_(set_theory) wikiPageWikiLinkText "minimal model".
- Minimal_model_(set_theory) hasPhotoCollection Minimal_model_(set_theory).
- Minimal_model_(set_theory) last "Shepherdson".
- Minimal_model_(set_theory) wikiPageUsesTemplate Template:Citation.
- Minimal_model_(set_theory) wikiPageUsesTemplate Template:Harvs.
- Minimal_model_(set_theory) wikiPageUsesTemplate Template:Harvtxt.
- Minimal_model_(set_theory) year "1951".
- Minimal_model_(set_theory) year "1952".
- Minimal_model_(set_theory) year "1953".
- Minimal_model_(set_theory) subject Category:Constructible_universe.
- Minimal_model_(set_theory) hypernym Model.
- Minimal_model_(set_theory) type Person.
- Minimal_model_(set_theory) comment "In set theory, a minimal model is a minimal standard model of ZFC.Minimal models were introduced by (Shepherdson 1951, 1952, 1953).The existence of a minimal model cannot be proved in ZFC, even assuming that ZFC is consistent, but follows from the existence of a standard model as follows. If there is a set W in the von Neumann universe V which is a standard model of ZF, and the ordinal κ is the set of ordinals which occur in W, then Lκ is the class of constructible sets of W.".
- Minimal_model_(set_theory) label "Minimal model (set theory)".
- Minimal_model_(set_theory) sameAs m.04f385d.
- Minimal_model_(set_theory) sameAs Q6865324.
- Minimal_model_(set_theory) sameAs Q6865324.
- Minimal_model_(set_theory) wasDerivedFrom Minimal_model_(set_theory)?oldid=661311108.
- Minimal_model_(set_theory) isPrimaryTopicOf Minimal_model_(set_theory).