symbol (א or ‘aleph’ is from the Hebrew alphabet; the symbol is read as
‘aleph nought’). So, in mathematical language, we can write card(N) = card(O) =
card(E) =.
Any set which can be put into a one-to-one correspondence with N is called a
‘countably infinite’ set. Being countably infinite means we can write the elements
of the set down in a list. For example, the list of odd numbers is simply 1, 3, 5,
7, 9,... and we know which element is first, which is second, and so on.
Are the fractions countably infinite?
The set of fractions Q is a larger set than N in the sense that N can be thought
of as a subset of Q. Can we write all the elements of Q down in a list? Can we
devise a list so that every fraction (including negative ones) is somewhere in it?
The idea that such a big set could be put in a one-to-one correspondence with N
seems impossible. Nevertheless it can be done.
The way to begin is to think in two-dimensional terms. To start, we write
down a row of all the whole numbers, positive and negative alternately. Beneath
that we write all the fractions with 2 as denominator but we omit those which
appear in the row above (like 6/ 2 = 3). Below this row we write those fractions
which have 3 as denominator, again omitting those which have already been
recorded. We continue in this fashion, of course never ending, but knowing
exactly where every fraction appears in the diagram. For example, 209/67 is in
the 67th row, around 200 places to the right of 1/67.