Cantor-Dedekind axiom

Cantor-Dedekind axiom

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. It is not an axiom in the ordinary mathematical sense.

This axiom is the cornerstone of analytic 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. This is sometimes known as the "real number line" blend [cite book | author = George Lakoff and Rafael E. Núñex | title = Where Mathematics Comes From: How the embodied mind brings mathematics into being | publisher=Basic Books|year=2000|id= ISBN 0-465-03770-4] :

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

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


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Axiom — This article is about logical propositions. For other uses, see Axiom (disambiguation). In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self evident or to define and… …   Wikipedia

  • Controversy over Cantor's theory — In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has found wide acceptance in the mathematics community, it has been criticized in several areas by mathematicians and philosophers. Cantor… …   Wikipedia

  • Georg Cantor — Infobox Scientist name = Georg Ferdinand Ludwig Cantor image width=225px caption = birth date = birth date|1845|3|3 birth place = Saint Petersburg, Russia death date = death date and age|1918|1|6|1845|3|3 death place = Halle, Germany residence =… …   Wikipedia

  • Richard Dedekind — Infobox Scientist name = PAGENAME box width = image size =180px caption =Richard Dedekind, c. 1850 birth date = October 6, 1831 birth place = Braunschweig death date = February 12, 1916 death place = Braunschweig residence = citizenship =… …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • Hilbert's program — Hilbert s program, formulated by German mathematician David Hilbert in the 1920s, was to formalize all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent.Hilbert proposed that the… …   Wikipedia

  • Analytic geometry — Cartesian coordinates. Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties. This article focuses on the classical and elementary meaning …   Wikipedia

  • logic, history of — Introduction       the history of the discipline from its origins among the ancient Greeks to the present time. Origins of logic in the West Precursors of ancient logic       There was a medieval tradition according to which the Greek philosopher …   Universalium

  • Dénombrabilité — Ensemble dénombrable En mathématiques, un ensemble est dit dénombrable, ou infini dénombrable, lorsque ses éléments peuvent être listés sans omission ni répétition dans une suite indexée par les entiers. Certains ensembles infinis, au contraire,… …   Wikipédia en Français

  • Ensemble Dénombrable — En mathématiques, un ensemble est dit dénombrable, ou infini dénombrable, lorsque ses éléments peuvent être listés sans omission ni répétition dans une suite indexée par les entiers. Certains ensembles infinis, au contraire, contiennent trop d… …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”