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.

• 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 א₀!

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.