mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

, the Cantor–Dedekind axiom is the thesis that the

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s are order-

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

to the linear continuum of

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

. 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 In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

proved by

Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...

in his book '' Geometric Algebra''. More precisely,

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

s defined over the field of

s satisfy the

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s of

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, and, from the axioms of Euclidean geometry, one can construct a field that is

to the real numbers. Analytic geometry was developed from the

Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...

introduced by

René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...

. It implicitly assumed this axiom by blending the distinct concepts of real numbers and points on a line, sometimes referred to as the real number line. 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. Another consequence is that Alfred Tarski's proof of the decidability of first-order theories of the real numbers could be seen as an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

to solve any first-order problem in

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) * B.E. Meserve (1955) Real numbers Mathematical axioms {{mathlogic-stub

See also

References