here's my try:
Let X be a countable set of lines in the plane. the cardinality of the set of all lines in the plane with a slope between 0 and 2π is ℵ so there must be some line in the plane, Γ, with a slope α that is not in X. so now, every line in X must intersect Γ at only one point. because X is a countable set then there is only a countable number of points on Γ that the lines of X cover so there is an uncountable set of points on Γ that the lines of X don't cover (because |Γ|=ℵ). Hence, the union of lines in X don't cover the plane.
first of all, is this proof correct? and second, can anyone give me another proof, maybe an easier one?
