Answer by Justina Colmena for Taking a proper class as a model for Set Theory
The class $V$ of all sets is not a model of $ZFC$, because it is a proper class, not a set.A model of $ZFC$ is a set (or small class) $U\in V$ which satisfies all the axioms of $ZFC$ when these axioms...
View ArticleAnswer by Asaf Karagila for Taking a proper class as a model for Set Theory
Yes, that is true. But note that in its nature statements like $\operatorname{Con}(T)$ are meta-theoretic statements. So when we say that $V$ is a model of $\sf ZF$, we mean that in the meta-theory it...
View ArticleAnswer by Joel David Hamkins for Taking a proper class as a model for Set Theory
What is shown in the cases you mention is not that the model is a model of ZFC, made as a single statement, but rather the scheme of statements that the model satisfies every individual axiom of ZFC,...
View ArticleTaking a proper class as a model for Set Theory
When I am reading through higher Set Theory books I am frequently met with statements such as '$V$ is a model of ZFC' or '$L$ is a model of ZFC' where $V$ is the Von Neumann Universe, and $L$ the...
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