Aristotle gave the first systematic rebuttal of Zeno, in particular he wrote in Physics: "…a line cannot be composed of points, the line being continuous and the point indivisible". According to Aristotle, a line can be composed only of smaller, indefinitely divisible lines, and not of points without magnitude. This was the mainstream view until the "dissociation" of continuum by Cantor and Dedekind at the end of 19-th century. Under the new paradigm the paradox was resolved by the Lebesgue measure theory, which postulated that length is only countably additive, so the continuum wide summation of point lengths is invalid. This vindicated Aristotle's view: in terms of magnitude continuum can not be assembled from points.
The idea that continuum is a "set of points" leads to a number of conceptual problems, this being one of them. Others include its well-orderability, which strains credulity and implies existence of Lebesgue non-measurable sets, and the undecidability of the continuum hypothesis. A lot of effort was expended, by Zermelo, Lebesgue and others, to reconcile the intuition of continuum with "arithmetical" set theory. However, despite the objections of intuitionists like Hermann Weyl, in the end the benefits to analysis far outweighed the costs for most practitioners. Weyl's views echoed Aristotle's:"The notion that a set is a “gathering” brought together by infinitely many individual arbitrary acts of selection, assembled and then surveyed as a whole by consciousness, is nonsensical; “inexhaustibility” is essential to the infinite... Exact time- or space-points are not the ultimate, underlying atomic elements of the duration
or extension given to us in experience."