Cantor–Dedekind axiom

In mathematical logic, the phrase Cantor–Dedekind axiom has been used to describe 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 is the cornerstone of analytic geometry. The Cartesian coordinate system developed by René Descartes implicitly assumes this axiom by blending the distinct concepts of real number system with the geometric line or plane into a conceptual metaphor. This is sometimes referred to as the real number line blend:^[1]

A consequence of this axiom is that Alfred Tarski's proof of the decidability of the ordered real field could be seen as an algorithm to solve any problem in Euclidean geometry.

Notes

1. George Lakoff and Rafael E. Núñez (2000). Where Mathematics Comes From: How the embodied mind brings mathematics into being. Basic Books. ISBN 0-465-03770-4.

References

• Ehrlich, P. (1994). "General introduction". Real Numbers, Generalizations of the Reals, and Theories of Continua, vi–xxxii. Edited by P. Ehrlich, Kluwer Academic Publishers, Dordrecht
• Bruce E. Meserve (1953) Fundamental Concepts of Algebra, p. 32, at Google Books
• B.E. Meserve (1955) Fundamental Concepts of Geometry, p. 86, at Google Books