Once we allow some principle of recursive reasoning, we are landed with the infinite. There are an infinite number of natural numbers: for 0 is a natural number, and if anything is a natural number, its successor is a natural number. So there are natural numbers, and for any finite number, n, it is demonstrably false that there are exactly n natural numbers. We feel impelled to allow the question ‘How many natural numbers are there?’, and the only possible answer seems to be ‘an infinite number’. But we have qualms. Infinity seems out of this earth. It smacks of Platonism, mysticism and theology. The word ‘infinite’ is a negative concept, contrasting not only with ‘finite’, in a strict mathematical sense, but with ‘definite’ and with ‘comprehensible’. Often, especially in theology and ancient philosophy, the Infinite is the Whole,
(to pan), the Universe, the Absolute, whose logic is difficult and fraught with inconsistencies. We are wise to be wary.