Different sizes of infinity

Did you know there is infinity, and then there’s a bigger infinity?

This concept is quite bizarre – one would think that infinity has only one size – that is, infinite size. But this is not true. Georg Cantor, a German mathematician, proved this idea by way of contradiction. Take two infinite sizes – the natural numbers, and real numbers. Both systems are unbounded and infinite in size. It can be proved that one infinity is larger than the other. The opposite is assumed: suppose both sets of numbers have the same size of infinity. By constructing a mapping, we can take an arbitrary mapping and show correspondence. Suppose every natural number has a real partner. We pair them off. In the diagonal construction, it is shown that there exists numbers that can’t be mapped because as your mapping get arbitrarily precise, the real numbers increase still. So there will always be number left unmapped. QED.