Now, for the next statement : The number of squares equals the number of all the natural numbers. This seems odd, because as we continue in the series of all the natural numbers ..See Fig. 3 below
the squares (underlined) occur less and less often. Still if we use Cantor’s matching principle of set theory we find :that the ‘infiniteness’ of the ‘square numbers’ corresponds one-to-one to the degree of infinity of the natural numbers which is the ‘countable infinity’ or א0. In the same way it can be shown that the set of all third powers, or any higher power, also tends to the same ‘countable infinity’. Thus if we can find a unique way to match any set to the natural numbers one-to-one the degree of infinity in the end remains the same viz. א0.
This helped Cantor resolve the Galileo’s paradox stated above – the number of points on the center, the smaller circle or any bigger circle all belong to ‘countable infinite’ rays.
For this purpose, Cantor used a line segment as a measure for ‘real numbers’. Whereas ‘natural numbers’ belong to the sequence 0, 1, 2, 3, 4, and so on, the ‘real numbers’ are those which are integers with a decimal placed between them somewhere e.g. 1.291 or 0.3125 or 365.89 and so on and so forth. These are also known as decimal numbers. In real life, we use ‘real numbers’ for measurements, weights, scientific calculations, cash transactions etc.
René Descartes (1596 – 1650) developed the idea of graphical representation, whereby, he represented ‘real numbers’ by the length of a line and accordingly defined a co-ordinate system. Thus a horizontal line can represent ‘real numbers’ from zero onwards. See Fig.5 below :Next let us look at ‘fractions’. By a ‘fraction’ we mean ‘the ratio of two natural numbers’ e.g. 1:4, 2:7, 1:125, 23:2365, ……. and so on. Fractions are also called ‘rational numbers’.
By contrast , there are irrational numbers, like P, √2, √3 etc. When we express these decimals they are infinitely long.In other words the decimal expansion of ‘irrational numbers’ neither terminates nor is periodic; it extends interminably e.g. P = 3.14159…. computers have checked it to a million decimal places without it ending or repeating.
‘Rational numbers’ always terminate or give a periodically recurring decimal. Conversely, a recurring or exact decimal will always be a fraction.
Now we make a very surprising statement : There is the same number of fractions as natural numbers ! Cantor made a grid as follows:-
we can again use the correspondence principle used earlier. Number each arrow-stop using the natural numbers :-
Each fraction has a ‘unique’ natural number associated with it and the entire grid can thus be covered exhaustively. Both sets therefore have the same ‘infiniteness’, that is they are both ‘countably infinite’ and have א0 elements. These concepts and others are explained in greater mathematical detail by Rudy Rucker and Paul Davies.
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.Now, 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’.
Proving this seems difficult at first, but Cantor’s reasoning was so simple and original that it is worth understanding it. His is a ‘proof by contradiction’. Here we start with a statement that is opposite of what we want to prove and then show that this leads to an inconsistency.
The essence of Cantor’s proof is that we assume that the decimals (irrational & rational numbers both put together) are as numerous as the natural numbers ( i.e. integers, and we have shown above that the rational numbers have the same infinity א0, as the integers themselves) then it should be possible to label them one by one.