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I've always argued that this fact was irrelevant, because although the set of 'all reals' are uncountable, the set of computable numbers IS countable. The fact
by MarkPNeyer 17y ago
I've always argued that this fact was irrelevant, because although the set of 'all reals' are uncountable, the set of computable numbers IS countable. The fact that there are numbers which cannot be represented by computers is interesting but has 0 practical applications.
- sophacles 17y agoSo let us look at just what you can fit into a 32 bit floating number. I know you can do reals in other ways, with arbitrary precision libraries, but the point is still the same. Anyway, the 32bit float. The fact that I have a limited amount of precision, means that errors exist as a result of truncation and rounding. For details search google (the main reason for my float example). The cumulative effects of these can be large enough be represented within the given amount of precision. Since we now have a proof that those numbers are innumerably many, we should develop techniques to deal with it, rather than just add precision.
- gabrielroth 17y agoFrom a pure math perspective, the limitations of computers are irrelevant.