How to Catch a Tortoise

‘As you’ve probably begun to see,’ David Foster Wallace writes in Everything and More, ‘Aristotle manages to be sort of grandly and breathtakingly wrong, always and everywhere, when it comes to infinity.’ A much milder version of this antagonism towards Aristotle appears in both Brian Clegg’s Brief History of Infinity and Robert and Ellen Kaplan’s The Art of the Infinite. Clegg writes that Aristotle ‘made a distinction on the matter of infinity that was to prove useful, but also was a fudge that made it possible to avoid the real issue for a couple of thousand years’. And the Kaplans describe how Georg Cantor won certain ‘striking insights . . . by going against the authority’ of Aristotle. All three books express what has become, nowadays, received wisdom: that Aristotle (‘the bad guy’) promulgated certain views about infinity that had the status of orthodoxy until the late 19th century, when Cantor (‘the good guy’) finally showed what was wrong with these views and how we should really think about the infinite. I do not myself subscribe to this standard account, and in due course shall explain why not.

First, however, I want to go back to our ordinary intuitive conception of the infinite – as that which is endless, unlimited, unsurveyable, immeasurable. I don’t mean to suggest that this conception is stable: all sorts of paradoxes beset it, and all sorts of refinements of it are designed either to circumvent or to accommodate these paradoxes. There are technical definitions of both endlessness and unlimitedness, for instance, whereby not only can they come apart from each other, but each can come apart from the infinite in its own appropriately technical sense. Even so, we should not allow any of this to make us lose sight of our starting point among these basic ideas of endlessness, unlimitedness, unsurveyability and immeasurability.

From the time of the early Greeks, the infinite, so conceived, has aroused suspicion, largely because of the paradoxes such as the famous one of Achilles and the tortoise, formulated 2500 years ago by Zeno of Elea. Achilles, who runs much faster than the tortoise, lets it start a certain distance ahead of him in a race. The paradox is that Achilles seems never to be able to overtake the tortoise, no matter how great the difference in their speeds. For in order to do so, he must first reach the point at which the tortoise starts, by which time the tortoise will have advanced a fraction of the distance initially separating them. Achilles must then make up this distance, by which time the tortoise will have advanced again. He must then make up this new distance, by which time the tortoise will have advanced yet again. And so on ad infinitum.

There is also a family of paradoxes, known since medieval times, based on the principle that if it is possible to pair off all the members of one set with all the members of another, then the two sets must be exactly the same size. (We can be certain that the set of older twins has just as many members as the set of younger twins, even if we have no idea how many there are of each.) This ‘same-size principle’ is certainly very compelling. You might even think that it is incontestable. But when it is applied to infinite sets, it seems to flout Euclid’s notion that the whole is greater than the part. It is possible, for example, to pair off all the members of the set of positive integers {1, 2, 3, 4, . . . } with all the members of the set of square positive integers {1, 4, 9, 16, . . . }. We can pair 1 off with itself, 2 with 4, 3 with 9, 4 with 16, and so on.

Aristotle, well aware of the problems that afflict the infinite, but also reluctant to abandon the concept completely, responded to this dilemma by drawing a distinction between the ‘actual infinite’ and the ‘potential infinite’. The infinitude of the actual infinite exists at some point in time. The infinitude of the potential infinite exists over time. (Imagine a clock endlessly ticking. Its ticking is potentially, but never actually, infinite.) All the objections to the infinite, Aristotle insisted, are objections to the actual infinite. They are objections to the idea of an infinitude that exists all at once. The potential infinite, by contrast, is a fundamental feature of reality. It is there to be acknowledged in any process that can never end: in the process of counting, for example; or in the process of dividing the racecourse on which Achilles is trying to overtake the tortoise; or in the passage of time itself. Paradoxes such as Zeno’s arise because we fail to pay due heed to this distinction. Having seen that there can be no end to the process of dividing the racecourse, we imagine that all those possible future divisions are somehow already there. We come to view the racecourse as already divided into infinitely many parts, and it is easy then for Zeno’s paradox to take hold. If we consider only the actual divisions that we have effected at any given time, then we can say that Achilles will overtake the tortoise once he has traversed a certain finite number of them.

For well over two thousand years this view had the status of orthodoxy, though it did not persist in exactly the form Aristotle presented it. Later thinkers construed the references to time in the actual/potential distinction metaphorically. Existing ‘all at once’ came to mean something altogether more abstract than it had for Aristotle. Eventually, exception to the actual infinite became exception to the idea that the infinite could be a legitimate object of mathematical study at all.

Hence today’s received wisdom. Late in the 19th century, Cantor presented a coherent, rigorous, systematic mathematical theory of the infinite, by taking the paradoxes in his stride. He accepted the same-size principle, for instance, and what follows from it: that there are as many square positive integers as there are positive integers altogether. He didn’t flinch at the idea that the part can be as great as the whole. Neither, however, did he go to the other extreme of urging that all infinite sets are the same size (a conclusion which, in its own way, would not have been all that repugnant to commonsense). On the contrary, much of the revolutionary impact of his work lay in his proof that, granted the converse of the same-size principle – that two sets are the same size only if it is possible to pair off all the members of one with all the members of the other – then not all infinite sets are the same size. No set, and in particular no infinite set, has as many members as it has subsets. (Here is Cantor’s proof. Suppose that all the members of a given set S were paired off with subsets of S. Then some members of S would belong to their corresponding subsets, while others would not. All those that did not would themselves constitute a subset of S – call it T. But no member of S would be paired off with T. For if a member m were paired off with T, then m would belong to T if and only if m did not belong to T, which is of course impossible.) Cantor then devised infinite cardinals: numbers that can be used to measure the various different infinite sizes. He presented an arithmetic for these cardinals: carefully defining his terms, he indicated what happens when one infinite cardinal is added to another, or multiplied by another, or raised to the power of another. His work showed mathematical craftsmanship of the highest calibre. It was astonishing in both its depth and its beauty. No longer, it seemed, was there any reason to reject the actual infinite. The Aristotelian orthodoxy seemed finally to have been subverted.

But was it really so? Cantor needed to proceed cautiously. His work made indispensable use of the concept of a set (as glimpsed above). But what is a set? According to one conception of what a set is, often referred to as the ‘naive’ conception, a set is something that is constituted by all the things that have some given property. Moreover, for any given property, there is, on the naive conception, a corresponding set, constituted by all the things that have it: corresponding to the property of being a planet, there is the set of planets; corresponding to the property of being an older twin, there is the set of older twins; corresponding to the property of being a square positive integer, there is the set of square positive integers; and corresponding to the property of being a set, there is the set of sets. The naive conception can be shown to be incoherent, however. For suppose there is a set corresponding to any given property. Now consider the fact – I am still presupposing the naive conception – that some sets belong to themselves, and some do not. The set of sets would be of the former kind, because it is itself a set, while the set of planets would be of the latter kind, because it is not itself a planet. But what about the set corresponding to the property of being a set that does not belong to itself – call it R? Then R belongs to itself if and only if it does not, which is of course impossible. So the naive conception is incoherent. (This is Russell’s famous paradox. There is a striking resemblance between the reasoning involved in this paradox and the reasoning involved in Cantor’s proof sketched above. The connections between the two are very deep.)

In order to safeguard his theory from this kind of incoherence, Cantor needed to operate with a somewhat more sophisticated conception of a set, often referred to as the ‘iterative’ conception. According to the iterative conception, a set is something whose existence is parasitic on that of its members: the members exist ‘first’. Thus there are, to begin with, all those things that are not sets (planets, twins, positive integers etc). Then there are sets of these things. Then there are sets of these things. And so on, without end. Each thing, and in particular each set, belongs to countless further sets. But there never comes a set to which every set belongs. There is no set of all sets. How does this escape the incoherence in the naive conception? Well, on the iterative conception, no set belongs to itself. Hence R, if it existed, would be the set of all sets. But there is no set of all sets: there is no such thing as R.

The iterative conception, like the naive conception, has great intuitive appeal. But is it not also strikingly Aristotelian? Notice the temporal metaphor that sustains it. Sets are depicted as coming into being ‘after’ their members, in such a way that there are ‘always’ more to come. Their collective infinitude, as opposed to the infinitude of any one of them, is potential, not actual. Moreover, it is this collective infinitude that has best claim to the title. For the basic ideas characteristic of our ordinary intuitive conception of the infinite – endlessness, unlimitedness, unsurveyability, immeasurability – more properly apply to the entire hierarchy than to anything in it. This is not least because of the success that Cantor enjoyed in subjecting the sets within the hierarchy to careful mathematical scrutiny. He showed, for example, that the set of positive integers is not unlimited (in size): it has fewer members than it has subsets. He also showed that it is not immeasurable: we can give a precise mathematical measure to how big it is. There is a sense, then, in which he established that what is ‘really’ infinite is something of an altogether different kind. (He was not himself averse to talking in these terms.) In a curious way, his work served, in the end, to corroborate the Aristotelian orthodoxy that ‘real’ infinitude can never be actual.

The quarrel that I have just picked with these books should not be allowed to detract from what each of them has to offer. Wallace and Clegg both provide a historical account of the concept of infinity, tracing a story from early Greek discomfort with the infinite, through the development of the calculus, to the advent of transfinite arithmetic in the work of Cantor. The Kaplans give a more general introduction to various mathematical delights, which, although it makes the reader continually aware of the infinite framework within which mathematical practice proceeds, does not always have the infinite itself in focus.

Clegg’s book is engagingly written and includes several interesting biographical sketches, which animate the mathematics and give a good sense of the very human concerns that have helped to propagate the story of this (very inhuman?) concept. There are, however, some slips and inaccuracies. In his presentation of Russell’s paradox, for example, Clegg gets into an uncharacteristic muddle concerning the difference (which is as crucial in this context as it is anywhere) between the members of a set and its subsets. He calls the set of humans ‘humanity’, and comments that each one of us, as well as being a member of this set, is also a member of a nation: humanity has us as its members and includes nations among its subsets. But he also says, immediately afterwards, that nations are members of humanity. He exacerbates the confusion by talking indiscriminately about what humanity ‘contains’: this can mean either what humanity has as members or what it has as subsets. Still, there is much to be learned from this book and much pleasure to be derived from it.

The same is true of the Kaplans’ book, which guides the reader through some extremely difficult mathematical ideas in ways that are both imaginative and diverting. Mathematics is often said to be the science of the infinite; the Kaplans want us to appreciate mathematics as the art of the infinite, an art which involves invention, narrative and an inexhaustible pursuit of variations on themes. They would have done better, however, to have resisted indulging in slightly irritating, and indeed bemusing, metamathematical-cum-philosophical asides, such as ‘there is no problem that cannot be solved,’ or, we can ‘be sure that problems will have solutions’. Not only is this optimism controversial and expressed without adequate justification, it is also inconsistent with the equally controversial and equally unjustified pessimism expressed on the final page of the main text, where we are told that a problem with which Cantor wrestled throughout his life – whether any set is intermediate in size between the set of positive integers and the set of sets of positive integers – is ‘for ever undecidable’.

As for Wallace’s book, the less said, the better. It’s a sloppy production, including neither an index nor a table of contents, and after a while his breezy style grates. No one who is unfamiliar with the ideas behind his dense, user-unfriendly mathematical expositions could work their way through them to gain any insight into what he is talking about. Worse, anyone who is already familiar with these ideas will see that his expositions are often riddled with mistakes. The sections on set theory, in particular, are a disaster. When he lists the standard axioms of set theory from which mathematicians derive theorems about the iterative conception of a set, he gets the very first one wrong. (It is not, as Wallace says, that if two sets have the same members, then they are the same size. It is that two sets never do have the same members.) From there it is pretty much downhill. He goes on to discuss Cantor’s unsolved problem, which I mentioned at the end of the previous paragraph. There are many different, equivalent ways of formulating the problem; Wallace gives four. The first and fourth are fine. The second, about whether the real numbers ‘constitute’ the set of sets of rational numbers, does not, as it stands, make sense. And the third, about whether the cardinal that measures the size of the set of real numbers can be obtained by raising 2 to the power of the smallest infinite cardinal, is simply wrong: we know it can. Any reader keen to gain insights into the infinite would do better to go back to Aristotle.