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

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.

- As we said earlier, Cantor pushed his mathematical reflections on set theory and the ‘infinitely many’ further. He showed that there exist sets with an ‘infiniteness’ which is ‘beyond countability’ i.e. these sets have a degree of infinity that is higher than
**א**!_{0}

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’.**[8]**

By contrast , there are irrational numbers, like P, √2, √3 etc.**[9]** 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:-

If we follow the arrows in Fig.6 as shown (and skip the common fractions like 1:1 is the same as 2:2 or 3:3 and so on ; 2:1 is the same as 4:2 or 6:3 and so on and so forth)

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 **א _{0} **elements.

- If we think of the rational numbers, say just between ‘0’ and ‘1’ (we can also write this as [0,1] ) , and arrange them by size, it seems this interval is completely filled by these fractions. But this is not so, between two rational numbers, however close, there will always be another rational number. To appreciate this take (here the 6 recurs endlessly and we can write instead 0.41‾6) 12

and we take the fraction (here the entire sequence 428571 cycles)

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 **א _{1} **and demonstrated that it is ‘bigger’ than the ‘countable infinity’,

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 **א _{0} **infinity of isolated points will always remain countable. If they are shrunk down together without limit,

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.

- Let us take the entire set of decimal numbers i.e.
*all*the numbers between [0,1]. This means that the rational and irrational numbers are in this together. And we know that the fractions or the rational numbers are ‘countably infinite’.

*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** א _{0}**, as the integers themselves)

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