Are the reals countable?

John Gabriel (a non-mathematician) claims in his blog that the real numbers are countable. The author tries to enumerate the real numbers in the interval [0, 1) by writing out all those whose decimal representation has one digit after the dot (0.1, 0.2, …, 0.9), followed by those with two digits after the dot, then those with three digits, and so on.

He goes on further to say that this establishes a procedure for writing
down an ordered sequence of numbers in which every real number of the
source interval will appear eventually (although, it should be clear
that any number with an infinite decimal representation will never
occur in his sequence of numbers).

This argument shows that the subset of real numbers between 0
and 1 that have a finite decimal representation is countable. Although,
it fails to work for the interval [0, 1).

It’s rather entertaining to watch fellow mathematicians humor him. Either way, things like this happen all the time :-D.

** Note: Some posts on Math-Fail are user-submitted and NOT verified by the admin of the site before publication. If you find this post to be distasteful, non-math related, ?or something worse?, then definitely leave a comment letting me know. Thanks very much! Mike **

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• I would say that most of the time I don’t need to insult them – they do it to themselves by trying to make fun of my ideas which they do not understand or don’t like.

If you had any relevance, I would be concerned. Everyone knows that the site math-fail.com is a joke that lacks any credibility. So, it’s quite funny that my actions should bother you so much. As for me searching for other sites, me thinks you project yourself onto others too much!

Surely a fool (you) deserves an answer? Or perhaps not…

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4. In the following comment, I prove conclusively that “real” numbers do not exist:

http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html

My final proof that 0.999… is not equal to 1 can be found here:

http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-372.html#post23252

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