In mathematical logic, the Cantor–Dedekind axiom is the thesis that the real numbers are order-isomorphic to the linear continuum of geometry.
In other words, the axiom states that there is a one-to-one correspondence between real numbers and points on a line.
This axiom became a theorem proved by Emil Artin in his book Geometric Algebra.
Analytic geometry was developed from the Cartesian coordinate system introduced by René Descartes.
Artin's proof, not only makes this blend explicitly, but also that analytic geometry is strictly equivalent with the traditional synthetic geometry, in the sense that exactly the same theorems can be proved in the two frameworks.