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Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Constructive Cauchy sequences vs. Dedekind cuts, published by jessicata on March 15, 2024 on LessWrong.
In classical ZF and ZFC, there are two standard ways of defining reals: as Cauchy sequences and as Dedekind cuts. Classically, these are equivalent, but are inequivalent constructively. This makes a difference as to which real numbers are definable in constructive logic.
Cauchy sequences and Dedekind cuts in classical ZF
Classically, a Cauchy sequence is a sequence of reals x1,x2,…, such that for any ϵ>0, there is a natural N such that for any m,n>N, |xmxn|
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