CANTORIAN SET THEORY AND LIMITATION OF SIZE PDF

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This is proved by contradiction. Let F be a function and A be a set. Assume that F A is a proper class. Then there is a function G that maps F A onto V. Therefore, F A is a set. Since the axiom of replacement implies the axiom of separation , the axiom of limitation of size implies the axiom of separation.

The proof starts by proving by contradiction that Ord, the class of all ordinals , is a proper class. Assume that Ord is a set. Therefore, Ord is a proper class.

The function G is a one-to-one correspondence between a subset of Ord and V. This well-ordering defines a global choice function : Let Inf x be the least element of a non-empty set x. First, he proved without using the axiom of union that every set of ordinals has an upper bound. In , William B. Easton used forcing to build a model of NBG with global choice replaced by the axiom of choice. Therefore, the axiom of limitation of size fails in this model. Ord is an example of a proper class that cannot be mapped onto V because as proved above if there is a function mapping Ord onto V, then V can be well-ordered.

The axioms of NBG with the axiom of replacement replaced by the weaker axiom of separation do not imply the axiom of limitation of size.