= Countable closure under real polynomial roots
{title2=$X=\bigcup_{n\geq0}B_n$}
Every <countable set> $B$ of <real numbers> is contained in a <countable set> closed under taking all real <roots of a polynomial> with coefficients in that set. Let $\phi(A)$ consist of these <roots of a polynomial>, excluding the zero <polynomial>. There are countably many finite coefficient lists over a <countable set>, and each nonzero <polynomial> has finitely many <roots of a polynomial>; hence $\phi(A)$ is countable. Define $B_0=B$ and $B_{n+1}=B_n\cup\phi(B_n)$. The <countable union of countable sets> theorem shows that $X=\bigcup_nB_n$ is countable. Every finite coefficient list in $X$ belongs to one stage $B_N$, so its real <roots of a polynomial> belong to $B_{N+1}\subseteq X$. Retaining $B_n$ at the next stage matters: $\phi(A)$ need not contain $A$.
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