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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.
Cantorian Set Theory and Limitation of Size
Axiom of limitation of size
Limitation of size
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