Suppose you did not believe Cantor. You know that each number between 0
and 1 can be expressed as an extending decimal, for example, ½ =
0.500000000000000000... and 1/π = 0.31830988618379067153... and you
would have to say to Cantor, ‘here is my list of all the numbers between 0 and 1’,
which we’ll call r 1 , r 2 , r 3 , r 4 , r 5 ,. .. If you could not produce one then Cantor
would be correct.
Imagine Cantor looks at your list and he marks in bold the numbers on the
diagonal:
r 1 : 0.a 1 a 2 a 3 a 4 a 5...r 2 : 0.b 1 b 2 b 3 b 4 b 5...r 3 : 0.c 1 c 2 c 3 c 4 c 5...r 4 : 0.d 1 d 2 d 3 d 4 d 5...
Cantor would have said, ‘OK, but where is the number x = x 1 x 2 x 3 x 4 x 5...where x 1 differs from a 1 , x 2 differs from b 2 , x 3 differs from c 3 working our way
down the diagonal?’ His x differs from every number in your list in one decimal
place and so it cannot be there. Cantor is right.
In fact, no list is possible for the set of real numbers R, and so it is a ‘larger’
infinite set, one with a ‘higher order of infinity’, than the infinity of the set of
fractions Q. Big just got bigger.