Birch and Swinnerton-Dyer Conjecture

To cite from the Clay Mathematics Institute description of millennium problems [claymath.org],

"this amazing conjecture asserts that if zeta(1) is equal to 0, then there are an infinite number of
rational points (solutions), and conversely, if zeta(1) is not equal to 0,
then there is only a finite number of such points.''

If \zeta(1) is not 0, then there is a finite number S of such points from 0<S<infinity. But the
S is not limited: 1,2,3,4,....,infinity; hence, it is infinity as maximum. But an infinite
number of points must have zeta(1)=0, i.e., even if zeta(1) is not 0,
there is zeta(1)=0.

I came to a contradiction.
More in: https://www.researchgate.net/publication/360014214_INFINITIES_IN_RIEMANN_AND_ABC_CONJECTURES