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Topic: Van Neumann Hierarchy
Replies: 13   Last Post: Apr 28, 2014 10:45 AM

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William Elliot

Posts: 1,593
Registered: 1/8/12
Van Neumann Hierarchy
Posted: Apr 25, 2014 11:13 PM
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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.




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