Difficulties with real numbers as infinite decimals ( I) | Real numbers + limits Math Foundations 91

There are three quite different approaches to the idea of a real number as an infinite decimal. In this lecture we look carefully at the first and most popular idea: that an infinite decimal can be defined in terms of an infinite sequence of digits appearing to the right of a decimal point, each digit chosen arbitrarily and independently.

There are several very attractive reasons for taking this position, which we outline. Primary among these is that the definition allows us to turn a process into a number (ostensibly!).

But ultimately the idea founders on the rocks of reality: the impossibility of specifiying competely a general such number, the impossibility of defining the addition (and multiplication) of such numbers via finite algorithms, and the resulting problematic aspects of the laws of arithmetic.

We look also briefly at the role of the Axiom of Choice in trying to provide an axiomatic framework for real numbers as such `infinite choice decimals'.

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Видео Difficulties with real numbers as infinite decimals ( I) | Real numbers + limits Math Foundations 91 канала Insights into Mathematics