Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 4 / 8E (Numbers and Sets)

Monomials are indexed by ${\mathbb{N}}^n$, a countable set. A polynomial is a finite list of monomials with rational coefficients, so $P_n$ is a countable union of countable sets.

$X_1$ is the set of real algebraic numbers and is countable because each nonzero polynomial has finitely many roots. For $n\ge 2$, $X_n$ is uncountable: the polynomial $x_1$ vanishes on $\{0\}\times {\mathbb{R}}^{n-1}$.

Start with $S_0=\mathbb{Q}$. Given countable $S_n$, let $S_{n+1}$ be the field generated by $S_n\cup \{\sqrt{|x|}:x\in S_n\}$. It is countable because its elements are values of rational expressions in finitely many members of a countable set. Then $S={\bigcup }_n{S}_n$ is countable and has (i)–(iii). Any other set with those properties contains every $S_n$ by induction, proving minimality and uniqueness.