The Shāntī Pāth & Cantor’s Infinities

We can make a new fraction in between these two rational numbers by adding both the numerator and denominator together as follows :-

It looks as if the rational numbers fill the interval [0,1] completely. However, this is not so. There are irrational numbers between the fractions everywhere. Oddly there are “many more” irrational numbers than fractions. Cantor’s great discovery was that the set of all decimals (i.e. all rational and irrational numbers) is ‘uncountably infinite’. This ‘uncountable infinity’ he called א1 and demonstrated that it is ‘bigger’ than the ‘countable infinity’, א0.Trying-to-breakNow, let us go back to the continuous line of Fig.5 and imagine attempting to dismember it by taking the points, one by one, from the left end and arranging them in a row individually as shown in Fig. 9:-

Even after removing an infinity of points, the line has not shrunk by the smallest of distance, for these isolated points, each of “zero” length, can never accumulate to measure any length at all. (Mathematically a point is defined   as “zero dimension” i.e a natural number however large, will give a non-existent product when multiplied by its “zero” length.) Similarly, an א0 infinity of isolated points will always remain countable. If they are shrunk down together without limit, they can never fill out even the shortest part of a continuous line. The infinity associated with ‘continuity’ is of a ‘higher nature’ ! It is ‘uncountably infinite’.