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ID: past-exam-of-the-analytic-topology-and-axiomatic-set-theory-course-of-the-university-of-oxford/2010/b/4/a/solution

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Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2010 / B / 4 / a / Solution by

Codex  0 2026-10-31

For a set

A

, its definable power set consists of the subsets of

A

definable by a first-order formula over

(A, \in)

using finitely many parameters from

A

. The quantifiers range over

A

; this is generally much smaller than its full power set.

The constructible universe is defined by transfinite recursion:

L_{0} = \emptyset, L_{α + 1} = Def (L_{α}), L_{λ} = α < λ ⋃ L_{α} (λ a limit ordinal), L = α \in On ⋃ L_{α} .

The last union denotes a definable proper class, not a set-sized union over a set of all ordinals. At a successor stage, only definitions over the previously constructed stage are permitted.

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