Cantor carried the concept of the ‘countable infinity’, which he termed א₀ (Aleph – nought) to the ‘indenumerable infinity’ which he named א₁ (Aleph-one). [A very readable account of Cantor’s work is given in ‘Fractals – Images of Chaos’ [7]]. Note here that the description ‘countable infinity’ also means ‘that which in principle can be counted’ and it can also be termed as ‘denumerable infinity’.The ‘indenumerable infinity’, א₁ is likened to ‘that which cannot be counted’ and is part of the continuum. Cantor developed ‘degrees of infinities’ between א₀ to א₁ by using one-to-one correspondence between the elements of two sets.

• e.g. What do we make of the statement : There are as many even (or odd) numbers as there are natural numbers ? There are infinitely many of each sort and at first it seems that the even(or odd) numbers appear alternatively and therefore should be less than the natural numbers…but it is not so…for we can match them as follows :

We see the one-to-one correspondence between the two sets and we conclude that since every number has its match – both the sets approach the same degree of infinity – in this case א₀.