Georg Cantor

Are there more irrational numbers than rational numbers, or more
rational numbers than irrational numbers? Well, there are infinitely
many of both, so the question doesn't make sense. It turns out,
however, that the set of rational numbers is infinite in a very
different way from the set of irrational numbers.

As we saw here, the rational numbers (those that can be written as fractions) can be lined up one by one and labelled 1, 2, 3, 4, etc. They form what mathematicians call a *countable infinity*.
The same isn't true of the irrational numbers (those that cannot be written as fractions): they form an
uncountably infinite set. In 1873 the mathematician
Georg Cantor
came up with a beautiful and elegant proof of this fact. First notice
that when we put the rational numbers and the irrational numbers
together we get all the real numbers: each number on the line is
either rational or irrational. If the irrational numbers were
countable, just as the rationals are, then the real numbers would be
countable too — it's not too hard to convince yourself of that.

So let’s suppose the real numbers are countable, so that we can make a list of them, for example

1.

2.

3.

4.

5.

and so on, with every real number occurring somewhere in the infinite list. Now take the first digit after the decimal point of the first number, the second digit after the decimal point of the second number, the third digit after the decimal point of the third number, and so on, to get a new number .

Now change each digit of this new number, for example by adding . This gives the new number . This new number is not the same as the first number on the list, because their first decimal digits are different. Neither is it the same as the second number on the list, because their second decimal digits are different. Carrying on like this shows that the new number is different from every single number on the list, and so it cannot appear anywhere in the list.

But we started with the assumption that every real number was on the
list! The only way to avoid this contradiction is to admit that the
assumption that the real numbers are countable is false. And this then
also implies that the irrational numbers are uncountable.

It's easy to see that an uncountable infinity is "bigger" than a
countable one. An uncountable infinity can form a continuum, such as
the number line, in a way that a countable infinity can't. Cantor went
on to define all sorts of other infinities too, one bigger than the
other, with the countable infinity at the bottom of the
hierarchy. When he first published these ideas, Cantor faced strong
opposition from some of his colleagues. One of them,
Henri Poincaré,
described Cantor's ideas as a "grave disease" and another,
Leopold Kronecker,
went so far as to denounce Cantor as a "scientific charlatan" and
"corrupter of youth". Cantor suffered severe mental health problems
which may have resulted in part from the rejection his work had met
with. But we now know that his work had simply come too soon: 150
years on, Cantor's ideas form a central pillar of mathematics and many
of his results can be found in standard textbooks.

See our infinity page to find out more about this and other things to do with infinity.