Re: Is continuum completely filled up?

OK, another ignorant question.

A real number can be defined by a Cauchy sequence - as I understand it this is an infinite sequence with a limit, with the difference between each successive term less than the previous (so that the sum of the series of the differences between terms is convergent). And this is just a fancy way of saying that a real number can be represented by an infinite e.g. decimal expansion, with the Cauchy series members for e.g. pi going {3,3.1,3.14,3.141, ..}

But if a real number can be represented by a sequence of rationals, why do we need any other numbers than rationals? Or is it that the actually infinite set of a sequence of rationals is logically something other?


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Andy Smith
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