Let $S \subseteq \mathbb{R}^d$ be a $d$-dimensional convex set (i.e. $\exists d+1$ affinely independent points in $S$). Let the origin of the coordinate system lie in the interior of $S$ and let: $$S^* = \underset{a\in S}\bigcap K(a) $$ where $K(a) = a \cdot x \le 1$
Then, $S^*$ is the polar of $S$. Also, if $S$ is a $d$-polytope in $\mathbb{R}^d$, then $S^*$ is a dual of $S$.
Since a $d$-polytope can be represented by a countably infinite set of points, this implies that the dual (which is also a polytope) is formed from the intersection of a countably infinite number of halfspaces --- which clashes with the definition of a polytope as the intersection of finitely many closed halfspaces.
What am I doing wrong here?
If the polytope can be represented by a countably infinite set of points, the dual can be represented by the intersection of countably infinite number of halfspaces.
Which is true and not in contradiction with the existence of a finite representation.