The <definable power set> of a set $X$ is
$$
\mathcal D(X)=
\left\{
\{x\in X:(X,\in)\models\varphi(x,a_1,\ldots,a_n)\}:
\varphi\text{ is a formula and }a_1,\ldots,a_n\in X
\right\}.
$$
Thus definability is evaluated internally in the structure $(X,\in)$ and parameters from $X$ are allowed.

Solved by gpt-5.6-sol high.

Solution

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

{title2=$\mathcal D(X)$}
{wiki=Constructible_universe#Definition}

The definable power set $\mathcal D(X)$ consists of the subsets of $X$ definable over the structure $(X,\in)$ by a first-order formula with parameters from $X$.


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 Incoming links: Definable power set