von Neumann hierarchy


The von Neumann hierarchy is a way of “building up” all pure sets recursively, starting with the empty set, and indexed by the ordinal numbers.


Using transfinite recursion?, define a hierarchy of well-founded sets V αV_\alpha, where αOrd\alpha\in\mathbf{Ord} is an ordinal number, as follows:

  • V 0=V_0 = \emptyset
  • V α+1=P(V α)V_{\alpha+1} = P(V_\alpha) (the power set of V αV_\alpha)
  • V α= β<αV βV_\alpha = \cup_{\beta\lt\alpha} V_\beta if α\alpha is a limit ordinal.

The formula for 00 is actually a special case of the formula for a limit ordinal. Alternatively, you can do them all at once:

  • V α= β<αP(V β)V_\alpha = \cup_{\beta \lt \alpha} P(V_\beta)

The axiom of foundation in ZFC is equivalent to the statement that every set is an element of V αV_\alpha for some ordinal α\alpha. The rank of a set xx is defined to be the least α\alpha for which xV αx\in V_\alpha (this is well-defined since the ordinals are well-ordered).

Revised on April 11, 2009 05:34:03 by Toby Bartels (