Related to a more historically inclined question I'd like to ask, if the language and axioms of Hilbert's geometry do suffice to first state and then prove that in the standard model (Euclidean space) every circle can be (continuously) collapsed to a point, i.e. that Euclidean space has genus $0$. Or is this beyond Euclidean geometry?
(If it cannot be proved there must be models of genus $\neq 0$?)