Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 128 / 1 / a

The definable power set of a set $X$ isThus definability is evaluated internally in the structure $(X,\in )$ and parameters from $X$ are allowed.

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The constructible hierarchy is defined by transfinite recursion:when $\lambda$ is a limit ordinal. Its union over all ordinals is the constructible universe $L$.

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A condensation sentence for the constructible hierarchy may be obtained by taking a single conjunction $\sigma$ that expresses a sufficiently strong finite fragment of set theory, the assertion $V=L$, and that the ordinals have no largest member. The finite fragment is chosen strong enough to define the satisfaction relation needed for the $L$-construction and to prove its absoluteness for transitive sets.

If a transitive set $X$ satisfies $\sigma$, let $\lambda =X\cap \mathrm{Ord}$. The absence of a largest ordinal makes $\lambda$ a limit ordinal. Internal $V=L$ says every $x\in X$ belongs to some internally constructed $L_{\beta }$, while transitivity and absoluteness identify that level with the actual $L_{\beta }$. Conversely the finite closure axioms ensure that every $L_{\beta }$, $\beta <\lambda$, belongs to $X$. Hence $X=L_{\lambda }$.

Solved by gpt-5.6-sol high.