Talk:Axiom of Subsets Equivalents

No universe in ZF
In ZF set theory the universal set is (deliberately) not defined. So I'm not sure whether these proofs are actually valid. --prime mover 16:42, 9 September 2011 (CDT)


 * In any case, the universal class is virtual and unreal, and it is the collection of all sets $x$ (not classes) such that $x = x$. Any statement involving membership to the universal set $A \in U$ is equivalent to another statement written $\exists x: x = A$ (i.e. that $A$ is a value of $x$).  Membership to the universal class only signifies that it belongs to our universe of discourse, and that we can talk about the class being a member of other classes.  Anyway, ZF allows the capability to talk about classes that are virtual and unreal even if they aren't part of the universe of discourse.  In ZF, we are allowed to talk about classes that don't exist as long as we don't refer to them as values of variables in the same way that we are allowed to refer to the class of Natural Numbers as long as we don't make the mistake of assuming it's the value of some variable before the Axiom of Infinity. -Andrew Salmon 18:03, 9 September 2011 (CDT)