 Solution

ID: past-exam-of-the-analytic-topology-and-axiomatic-set-theory-course-of-the-university-of-oxford/2010/b/6/d/i/solution

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

This recursion on ordinals is available without assuming the Axiom of foundation for all sets. The stages are increasing and transitive, by transfinite induction using their definitions.

Work internally in the specified model of ZF without the Axiom of foundation, and assume the membership induction theorem. Apply it to

φ (x) \equiv \exists α \in On (x \in V_{α}) .

Suppose every member

y

x

lies in some stage. Assign to each such

y

the least ordinal

α_{y}

for which

y \in V_{α_{y}}

β

be their supremum, taking

β = 0

for the empty collection. Monotonicity gives

x \subseteq V_{β}

, and therefore

x \in V_{β + 1}

This proves progressivity of

φ

, so membership induction gives it for every set of the model. Every stage already consists of sets in that model. Consequently

V^{*} = α \in On ⋃ V_{α} .

The equality is internal to the given model; its ordinals and power sets are computed there.

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