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In constructing the sets in the cumulative hierarchy from the empty set, why is the power set operation even needed?

Let $V_{0} = \emptyset$, and inductively define $V_{\alpha+1} = V_{\alpha} \cup \{V_{\alpha}\}$. By using the union operator we can keep on constructing larger and larger sets. Why do we even need the power set operator here?

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This way you only get the ordinals. Note that the construction only yields sets which are transitive and all of their elements are transitive.

But not all sets are ordinals, not all sets are even sets of ordinals. Even if every set is equipotent to an ordinal (assuming choice, anyway).

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  • $\begingroup$ So in some sense, the power set operator is more powerful? $\endgroup$ – Maxis Jaisi Feb 16 '17 at 7:47
  • $\begingroup$ Well... Obviously. Just look at Cantor's theorem. $\endgroup$ – Asaf Karagila Feb 16 '17 at 7:48

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