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Single Idea 18251

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

Full Idea

Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers.

Gist of Idea

Irrational numbers are the limits of Cauchy sequences of rational numbers

Source

report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2

Book Ref

Lavine,Shaughan: 'Understanding the Infinite' [Harvard 1994], p.92

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]