How do we know that the set of sets that each contain real coefficient polynomials of distinct degrees is countable? If I have the set $V=\left \{S_{0},S_{1},S_{2},...\right \}$ where $S_{0}$ is the set of real constant polynomials, $S_{1}$ is the set of real polynomials with degree 1 and so on. How do we know that $V$ is countable?
It isn't. Just the set of polynomials over $\mathbb{R}$ of degree 0 is equinumerous with $\mathbb{R}$, so isn't countable; that being a subset of $\mathbb{R}[x]$ means the later one can't be countable either.
It isn't. Just the set of polynomials over $\mathbb{R}$ of degree 0 is equinumerous with $\mathbb{R}$, so isn't countable; that being a subset of $\mathbb{R}[x]$ means the later one can't be countable either.