It's amazing what people will nodeshell.
used the diagonal argument as a way of explaining how the cardinality of the real number
s is somehow greater than that of the natural numbers
, that is, that there are "more" real numbers than natural numbers.
It goes like this:
Assume that you can find a function
that puts the real numbers into a one-to-one correspondence
with the positive integers. Each real number has a decimal expansion
. For example:
f(1) = 0.1231243412331212...
f(2) = 0.4359348729384792...
f(3) = 0.3304923870398473...
f(4) = ....
and so on.
Now, consider the real number whose decimal expansion whose first digit is obtained by taking the first digit
of f(1) and adding 1, whose second digit is f(2)+1
, and so on, always taking the n
th digit of f(n)
and adding 1. If a digit is 9, use 0 instead of adding 1.
This number is different from all the real numbers that f
maps to, yet in order for f
to be a one-to-one correspondence, it must exhaust the real numbers! This means that f
cannot be a one-to-one correspondence. Since we placed no conditions on what f
might be, this means no function between the integers and the real numbers can be a one-to-one correspondence.
Please note that this argument is Cantor's way of explaining his proof that for every set there exists a larger set
in layman's terms. Kalmakka's little parlor trick is briefly interesting but harmless. The general proof isn't that hard to understand, but that's for another node.