Cardinality of polynomials with real coefficients

Question

What is the cardinality of the set of all polynomials with real coefficients? I know the power set of $\mathbb{R}$ is "more infinite" than $\mathbb{R}$, so to speak, but I'm unsure of how to prove that there does or does not exist a surjection onto $\mathbb{R}[X]$ from $\mathbb{R}$. Is it equinumerous with the power set of $\mathbb{R}$, or something else more exotic?

Answer 1

$\mathbb R[X]$ has the same cardinality as $\mathbb R$ itself.

One fairly simple way to see this is to know that there are bijections $f: \mathbb R \to \mathcal P(\mathbb N)$ and $g: \mathbb N\times\mathbb N \to \mathbb N$. Then

$$h(a_0+a_1X+\cdots a_n X^n) = \{g(p,q)\mid p\in f(a_q)\}$$ defines an injection $h:\mathbb R[X]\to \mathcal P(\mathbb N)$, and since there are obviously at least as many polynomials as there are real numbers, the Cantor-Bernstein theorem takes care of the rest.