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Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets that are models of ZFC (assuming ZFC is consistent). However, these set models are non-standard. In particular, they do not use the normal membership relation and they are not
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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).
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other than itself. In particular it is not possible to use the method of inner models to prove that any given statement true in the minimal model (such as the
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gave another construction of the minimal model as the strongly constructible sets, using a modified form of Gödel's constructible universe.
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of the minimal model can be named; in other words there is a first-order sentence
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Shepherdson, J. C. (1953), "Inner models for set theory. III",
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Shepherdson, J. C. (1952), "Inner models for set theory. II",
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implies that the minimal model (if it exists as a set) is a
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32:. The minimal model was introduced by Shepherdson (
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is the unique element of the minimal model for which
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The existence of a minimal model cannot be proved in
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304:(2), Association for Symbolic Logic: 145–167,
268:(4), Association for Symbolic Logic: 225–237,
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179:(1963), "A minimal model for set theory",
70:V that is a standard model of ZF, and the
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158:The minimal model of set theory has no
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77:is the set of ordinals that occur in
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117:set. More precisely, every element
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213:"Inner models for set theory. I"
196:10.1090/S0002-9904-1963-10989-1
105:of ZFC, and also satisfies the
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230:Association for Symbolic Logic
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298:The Journal of Symbolic Logic
262:The Journal of Symbolic Logic
221:The Journal of Symbolic Logic
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55:, even assuming that ZFC is
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211:Shepherdson, J. C. (1951),
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166:) is not provable in ZFC.
107:axiom of constructibility
101:. This set is called the
111:Löwenheim–Skolem theorem
342:Constructible universe
182:Bull. Amer. Math. Soc.
44:) and rediscovered by
164:continuum hypothesis
68:von Neumann universe
109:V=L. The downward
89:constructible sets
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87:is the class of
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24:is the minimal
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177:Cohen, Paul J.
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232:: 161–190,
189:: 537–540,
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170:References
57:consistent
18:set theory
115:countable
336:Category
81:, then L
326:0057828
318:2268947
290:0053885
282:2266609
254:0045073
246:2266389
205:0150036
72:ordinal
66:in the
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314:JSTOR
278:JSTOR
242:JSTOR
228:(3),
216:(PDF)
42:1953
38:1952
34:1951
306:doi
270:doi
234:doi
191:doi
91:of
61:set
53:ZFC
30:ZFC
28:of
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