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Single Idea 15903

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy ]

Full Idea

Cantor's theory of Cauchy sequences defines a real number to be associated with an infinite set of infinite sequences of rational numbers.

Gist of Idea

A real is associated with an infinite set of infinite Cauchy sequences of rationals

Source

report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6

Book Ref

Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.38

A Reaction

This sounds remarkably like the endless decimals we use when we try to write down an actual real number.

The 5 ideas with the same theme [defining real numbers using Cauchy sequences]:

15903 A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine]
18251 Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine]
18247 Brouwer saw reals as potential, not actual, and produced by a rule, or a choice [Brouwer, by Shapiro]
18249 Cauchy gave a formal definition of a converging sequence. [Shapiro]
18250 Cauchy gave a necessary condition for the convergence of a sequence [Lavine]