The Wikipedia article on Real Numbers says every "a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits continue in some way" I don't think this is right. Doesn't there have to be some real numbers that cannot be named, even if we allow that name to be letters from an infinite string of digits?(adsbygoogle = window.adsbygoogle || []).push({});

You can write out a countably infinite list of numbers in [0,1], and each of those numbers can have an infinite number of digits to the right of the decimal point. For any n, you can be certain you've written down the entire list of numbers with n decimal places. According to the Wikipedia article, the infinite list has to contain every real number. But doesn't Cantor's Diagonalization argument say that the list must be incomplete, that there are points which were not on that list?

So either 1) there is no method of knowing you have written all of the decimal representations for any number of digits m > n, at least as n approaches infinity. Not only that, but the decimal representation exists but you must necessarily have missed writing it. Or else 2) there are reals that don't have decimal representation, which I think is to say reals which cannot be named in and of themselves.

Frankly I dislike the first, for this reason. The generating method can include recursively applying the diagonalization procedure and adding any such determined omission onto the list. We can do that countably infinitely often but still Cantor's argument holds, doesn't it?

Am I going wrong, and if so, where?

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# Decimal representation of reals

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