 The most visionary “arithmetizer” of all was Georg  Cantor[31]  (1845–1918). Cantor’s analysis of the continuum in terms of infinite point sets led to his theory of transfinite numbers and to the eventual freeing of the concept of set from its geometric origins as a collection of points, so paving the way for the emergence of the concept of general abstract set central to today’s mathematics. Like Weierstrass and Dedekind, Cantor aimed to formulate an adequate definition of the real numbers which avoided the presupposition of their prior existence, and he follows them in basing his definition on the rational numbers. Following Cauchy, he calls a sequence \(a_1,\) \(a_2 ,\)…, \(a_n,\)… of rational numbers a fundamental sequence if there exists an integer \(N\) such that, for any positive rational \(\varepsilon\), there exists an integer \(N\) such that \(\lvert a_{n+m} - a_n \rvert \lt \varepsilon\) for all \(m\) and all \(n \gt N\). Any sequence \(\langle a_n\rangle\) satisfying this condition is said to have a definite limit b. Dedekind had taken irrational numbers to be mental objects associated with cuts, so, analogously, Cantor regards these definite limits, as nothing more than formal symbols associated with fundamental sequences (Dauben 1979: 38). The domain \(B\) of such symbols may be considered an enlargement of the domain \(A\) of rational numbers. After imposing an arithmetical structure on the domain \(B\), Cantor is emboldened to refer to its elements as (real) numbers. Nevertheless, he still insists that these “numbers” have no existence except as representatives of fundamental sequences. Cantor then shows that each point on the line corresponds to a definite element of \(B\). Conversely, each element of \(B\) should determine a definite point on the line. Realizing that the intuitive nature of the linear continuum precludes a rigorous proof of this property, Cantor simply assumes it as an axiom, just as Dedekind had done in regard to his principle of continuity.