7. Counting infinity

‘Counting infinity’ returns to the mathematics of infinity, discussing Cantor’s remarkable theory of how to count infinite sets, and the discovery that there are different sizes of infinity. For example, the set of all integers is infinite, and the set of all real numbers (infinite decimals) is infinite, but these infinities are fundamentally different, and there are more real numbers than integers. The ‘numbers’ here are called transfinite cardinals. For comparison, another way to assign numbers to infinite sets is mentioned, by placing them in order, leading to transfinite ordinals. It ends by asking whether the old philosophical distinction between actual and potential infinity is still relevant to modern mathematics, and examining the meaning of mathematical existence.