This means that all the decimals are listed out exhaustively in an infinite column (the labeling can be random ; one such ordering is shown in Fig.10). We next construct a new decimal number ‘b’ as Cantor pointed out – take the first digit of a1 and change it to obtain the first decimal of ‘b’ ; then take the second digit of a2 and alter it to get the second decimal of ‘b’ and so on move diagonally down the entire infinite column.[For mathematical precision the changed digits should not be either 0 or 9].
A possible ‘b’ from Fig.10 could be :-
But, if our original list is complete then ‘b’ should be in it somewhere. The way Cantor constructed ‘b’ he ensured that it has at least one decimal digit different from each one of the ai’s . For example, if we say that an (‘ai’ with the nth index) is the same as ‘b’ – it cannot be so since the nth decimal of ‘an’ and ‘b’ are different by Cantor’s process of inception. Thus, a contradiction has arisen in our initial assumption. Therefore, the decimal numbers are ‘uncountably infinite’ and this infinity, א1 (Aleph – one) is higher than the ‘countable infinity’, א0 (Aleph – nought).