One reason I suspect that is that polynomials alone don't really distinguish between the set of all algebraics and transcendentals, because for the former at least one polynomial will give zero as a result, and for the latter, all polynomials will give nonzero values, but there seemed to be no way to "average" all the possible values of all polynomials (with rational coeffcients) which there are $\aleph_1$ many of them when acting on the reals, which there are also $\aleph_1$ many of them, without false positive results like cancellations such as 1-1=0