Is the set of reals, R, a countable set?

Discussion in 'General Math' started by emile.seaburn, Sep 8, 2005.

  1. emile.seaburn

    emile.seaburn Guest

    Hi all! Is the set of reals, R, a countable set? If yes, how do I go about proving this? Thanks!

     
    emile.seaburn, Sep 8, 2005
    #1

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  2. emile.seaburn

    Jim Spriggs Guest

    No, it isn't. If it were countable then the reals could be listed as
    follows

    N_1 . a_1 a_2 a_3 ...
    N_2 . b_1 b_2 b_3 ...
    N_3 . c_1 c_2 c_3 ...
    ... . ...

    where the N's are the integral parts, and the small letters are the
    digits after the decimal points.

    We now choose a digit a that is different from a_1 and is neither 0
    nor 9 (to avoid problems with equalities like 0.999... = 1.000...); a
    digit b that is different from b_2 and is neither 0 nor 9; a digit
    c that is different from c_3 and is neither 0 nor 9; and so on. Now
    consider this number

    0.abc...

    it differs from each number in the list because it differs from the n-th
    number in the n-th place after the decimal point.

    So our assumption that we could list _all_ real number is shown to be
    false.
     
    Jim Spriggs, Sep 8, 2005
    #2

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