Скачать или смотреть Difficulties with Dedekind cuts | Real numbers and limits Math Foundations 116 | N J Wildberger

Richard Dedekind around 1870 introduced a new way of thinking about what a real number `was'. By analyzing the case of sqrt(2), he concluded that we could associated to a real number a partition of the rational numbers into two subsets A and B, where all the elements of A were less than all the elements of B, and where A had no greatest element. Such partitions are now called Dedekind cuts, and purport to give a logical and substantial foundation for the theory of real numbers.

Does this actually work? Can we really create an arithmetic of real numbers this way? No and no. It does not really work. In this video we raise the difficult issues that believers like to avoid.

Video Content:
00:00 Intro to Dedekind's approach to "real numbers"
5:08 "Cuts" of the rationals
7:13 Principles of Mathematical Analysis
12:38 Subsets of Q
24:02 Prior theory of 'infinite sets'
32:09 Arithmetic with 'Dedekind cuts'
35:05 Unwieldy, infinite sets

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