- probably something like ‘half the whole numbers are odd and half are even’.
Cantor would agree with the answer, but would give a different reason. He would
say that every time we have an odd number, we have an even ‘mate’ next to it.
The idea that both sets O and E have the same number of elements is based on
the pairing of each odd number with an even number:
If we were to ask the further question ‘is there the same number of whole
numbers as even numbers?’ the answer might be ‘no’, the argument being that
the set N has twice as many numbers as the set of even numbers on its own.
The notion of ‘more’ though, is rather hazy when we are dealing with sets with
an indefinite number of elements. We could do better with the one-to-one
correspondence idea. Surprisingly, there is a one-to-one correspondence between
N and the set of even numbers E:
We make the startling conclusion that there is the ‘same number’ of whole
numbers as even numbers! This flies right in the face of the ‘common notion’
declared by the ancient Greeks; the beginning of Euclid of Alexandria’s Elements
text says that ‘the whole is greater than the part’.
Cardinality
The number of elements in a set is called its ‘cardinality’. In the case of the
sheep, the cardinality recorded by the farmer’s accountants is 42. The cardinality
of the set {a, b, c, d, e} is 5 and this is written as card{a, b, c, d, e} = 5. So
cardinality is a measure of the ‘size’ of a set. For the cardinality of the whole
numbers N, and any set in a one-to-one correspondence with N, Cantor used the