 Solution

ID: past-exam-of-the-analytic-topology-and-axiomatic-set-theory-course-of-the-university-of-oxford/2009/b/4/i/solution

V_{0} = \emptyset, V_{α + 1} = P (V_{α}), V_{λ} = β < λ ⋃ V_{β} (λ a nonzero limit) .

The recursion theorem of ZF supplies each set stage. The whole universe is

V = ⋃_{α \in Ord} V_{α}

x \in V_{α}

exactly when the rank of

x

is less than

α

, where

rank (x) = sup {rank (z) + 1 : z \in x}

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