Are the reals countable?

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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.

7 thoughts on “Are the reals countable?”

  1. I find your entry rather amusing. Accusations against my credentials and confused arguments don’t make you appear any more credible.
    Is it not strange my knols have such high ratings? Hmmm, I wonder, could this be because the knols make sense or could it be because the mathematics academia are so dumb that anything making even a bit of sense seems to be more credible than what they teach?

  2. seriously, do you search for people who link to you in order to insult them personally? this is at least the second place I’ve seen someone make fun of your ideas, and both times you went out of your way to answer.

    1. 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 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…

  3. The set of Real numbers is not countable because it is mostly made up of irrationals. As a result, it has no injective function that maps perfectly to the set of Natural numbers.

    1. That’s baloney. The “set” (it does not exist) of “real” numbers is uncountable because “irrational” numbers do not exist.

      As for an injective function, that’s also nonsense because Cantor knew nothing about injective or bijective functions. His so-called Diagonal argument (a bowel movement) is based on “listing” all the real numbers.

      Now, if as Cantor falsely believed that any real number can be represented in decimal, then the “real” numbers are indeed countable. Of course, this is untrue because any radix system represents only rational numbers. Non-terminating representations are ill-defined. For example, as a result of Cantor’s malaise, you have mainstream mathematicians thinking that 0.999… = 1 which is FALSE.

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