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

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

1. emile.seaburnGuest

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

2. Jim SpriggsGuest

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