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 Mathematics, the real numbers may be described informally in several different ways In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, the word continuum has at least two distinct meanings outlined in the sections below Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In other words the axiom states that there is a one to one correspondence between real numbers and points on a line. It is not an axiom in the ordinary mathematical sense. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject
This axiom is the cornerstone of analytic geometry. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry The Cartesian coordinate system developed by Rene Descartes explicity assumes this axiom by blending the distinct concepts of real number system with the geometric line or plane into a conceptual metaphor. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Cognitive linguistics, conceptual metaphor refers to the understanding of one idea or Conceptual domain in terms of another for example understanding Quantity This is sometimes known 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. Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California In Logic, the term decidable refers to the existence of an Effective method for determining membership in a set of formulas In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.
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References
- Erlich, P. . (1994). "General introduction". Real Numbers, Generalizations of the Reals, and Theories of Continua, vi-xxxii. Edited by P. Erlich, Kluwer Academic Publishers, Dordrecht
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