Infinity

The mathematician George Cantor (d. 1918) uses the word infinite to refer to a number defined as being greater than any finite number. In this sense of the word, the number of whole integers and the number of rational fractions are both “infinite” in the same degree. This is because for every fraction, no matter how many there may be of them, a new integer can always be assigned to it without ever running out of integers, and vice versa. In other words, you can use whole integers to number each item in a series of fractions.

The irrational numbers are rather different. Both integers and rational fractions of integers possess an inherent “graininess” because they are essentially definite, i.e. discontinuous with each other. Irrational numbers, on the other hand, occupy the spaces between each of the rationals, and fill them up continuously. The number of irrationals always exceeds that of the rationals, and therefore, according to Cantor, the “infinity” of the irrationals is of a different order.

The discovery of orders of infinity is highly significant for us. In fact Cantor’s set theory proves that there is an infinite series of infinities, each of a higher order than the last, right up to an “absolute” infinite, which he seems to have identified with God. As he wrote: “The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.”