From the article: "nearly all real numbers are transcendental."
I'll put forward the facetious argument that only half of the real numbers are transcendental. Between every pair of non-equal rationals exists a transcendental. Between every pair of non-equal transcendentals exists a rational. Therefore, only half of the real numbers are transcendental.
Of course, what the author meant is that the set of non-transcendental numbers has measure zero. This implies, among other things, that a number chosen randomly from, say, [0,1] is transcendental with probability 1.
I'll put forward the facetious argument that only half of the real numbers are transcendental. Between every pair of non-equal rationals exists a transcendental. Between every pair of non-equal transcendentals exists a rational. Therefore, only half of the real numbers are transcendental.