
Van Neumann Hierarchy
Posted:
Apr 25, 2014 11:13 PM


The Van Neumann hierarchy is defined with transfinite induction as V_eta = \/{ P(V_xi)  xi < eta }.
Within ZF, using regularity and replacement, how does one show that for every set A, there's some ordinal eta, with A in V_eta.
It would suffice to show for every transitive set A, there's some ordinal eta, with A in V_eta.
With that, since every set is a subset of it's transitive closure, every set would be inside the Van Neumann hierarchy.

