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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy

[defining real numbers using Cauchy sequences]

5 ideas
A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine]
     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.
     From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite II.6
     A reaction: This sounds remarkably like the endless decimals we use when we try to write down an actual real number.
Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine]
     Full Idea: Cantor introduced irrationals to play the role of limits of Cauchy sequences of rational numbers.
     From: report of George Cantor (works [1880]) by Shaughan Lavine - Understanding the Infinite 4.2
Brouwer saw reals as potential, not actual, and produced by a rule, or a choice [Brouwer, by Shapiro]
     Full Idea: In his early writing, Brouwer took a real number to be a Cauchy sequence determined by a rule. Later he augmented rule-governed sequences with free-choice sequences, but even then the attitude is that Cauchy sequences are potential, not actual infinities.
     From: report of Luitzen E.J. Brouwer (works [1930]) by Stewart Shapiro - Philosophy of Mathematics 6.6
     A reaction: This is the 'constructivist' view of numbers, as espoused by intuitionists like Brouwer.
Cauchy gave a formal definition of a converging sequence. [Shapiro]
     Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε.
     From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4)
     A reaction: The sequence is 'Cauchy' if N exists.
Cauchy gave a necessary condition for the convergence of a sequence [Lavine]
     Full Idea: The Cauchy convergence criterion for a sequence: the sequence S0,S1,... has a limit if |S(n+r) - S(n)| is less than any given quantity for every value of r and sufficiently large values of n. He proved this necessary, but not sufficient.
     From: Shaughan Lavine (Understanding the Infinite [1994], 2.5)