Infinity

6. Modern mathematics of the infinite

Despite Kant’s influence on Hegel, and despite his own commitment to infinite reason (as well as to infinite space), Kant certainly helped to propagate the Aristotelian tradition of treating the actual infinite with hostility and suspicion. As this tradition prevailed, the actual infinite came increasingly to be understood in the more general, non-temporal sense indicated in §2 above. Eventually, exception was being taken to any categorematic use of the word ‘infinite’. The most serious challenge to this tradition, at least in a mathematical context, was not mounted until the nineteenth century, by Cantor, whose mathematical contribution to this topic is unsurpassed (see Cantor, G.).

Objections to the actual infinite had tended to be of two kinds. The first kind we have already seen: objections based on the fact that we can never encounter the actual infinite in experience. Objections of the second kind were based on the paradoxes to which the actual infinite gives rise. These paradoxes fall into two groups. The first group consists of Zeno’s paradoxes and their variants. By the time Cantor was writing, however, the calculus (which had then reached full maturity) had done a great deal to mitigate these. Of more concern by then were the paradoxes in the second group, which had been known since medieval times. These were paradoxes of equinumerosity. They derive from the following principle: if (and only if) it is possible to pair off all the members of one set with all those of another, then the two sets must have just as many members as each other. For example, in a non-polygamous society, there must be just as many husbands as wives. This principle looks incontestable. However, if it is applied to infinite sets, it seems to flout Euclid’s notion that the whole is greater than the part. For instance, it is possible to pair off all the positive integers with those that are even: 1 with 2, 2 with 4, 3 with 6 and so on.

Cantor accepted this principle. And, consistently with that, he accepted that there are just as many even positive integers as there are positive integers altogether. Far from being worried by this, he defined precisely what is going on in such cases, and then incorporated his definitions into a coherent, systematic and rigorous theory of the actual infinite, ready to be laid before any sceptical gaze.

It might be expected that, on this understanding, all infinite sets are the same size. (If they are, that is not unduly paradoxical.) But much of the revolutionary impact of Cantor’s work came in his demonstration that they are not. There are different infinite sizes. This is a consequence of what is known as Cantor’s theorem: no set, and in particular no infinite set, has as many members as it has subsets. In other words, no set is as big as the set of its subsets. If it were, then it would be possible to pair off all its members with all its subsets. But this is not possible. Suppose there were such a pairing and consider the set of members paired off with subsets not containing them. Whichever member was paired off with this subset would belong to it if and only if it did not belong to it (see Cantor’s theorem).

In the course of developing these ideas, Cantor laid down some of the basic principles of the set theory which underlay them; he devised precise methods for measuring how big infinite sets are; and he formulated ways of calculating with these measures. In short, he established transfinite arithmetic.

Even so, there are many who remain suspicious of his work and who continue to think of the infinite in broadly Aristotelian terms. Cantor himself was forced to admit that there are some collections, including the collection of all things, which are so big that they cannot be assigned any determinate magnitude (their members cannot be given ‘all at once’). Concerning such collections, he even sometimes said that they are ‘truly’ infinite. There is in fact a real irony here: Cantor’s work can in many ways be regarded as corroborating Aristotelian orthodoxy.

Brouwer believed that Cantor had gone wrong in not showing sufficient respect for the first kind of objection to the actual infinite: that we cannot encounter it in experience. All Cantor had done, in Brouwer’s view, was to demonstrate certain tricks that can be played with (finite) symbols, without addressing the question of how these tricks answer to experience. The relevant experience here – the experience to which any meaningful mathematical statement must answer, according to Brouwer and other members of his intuitionistic school – is our experience of time. It is by recognizing the possibility of separating time into parts, and then indefinitely repeating that operation over time, that we arrive at our idea of the infinite. And such infinitude is potential, not actual – in the most literal sense (see Intuitionism §1).

There was a very different critique of Cantor’s ideas in the work of Wittgenstein, though it led to similar results (see Wittgenstein §14). Wittgenstein believed that insufficient attention had been paid (at least by those interpreting Cantor’s work, if not by Cantor himself) to what he called the ‘grammar’ of the infinite, that is to certain fundamental constraints on what could count as a meaningful use of the vocabulary associated with infinity. In effect, Wittgenstein argued that the word ‘infinite’ could not be used categorematically.