What is the Infinite?
What is the Infinite?
By Joel David Hamkins, O’Hara Professor of Logic, University of Notre Dame, USA
What is the infinite? What does it mean to say that a given set is infinite? How are we to describe this sublime, elusive notion? We should want a crisp, clear account, one suitable for mathematics.
Suppose I turn the tables and ask the dual question: What is the finite? You might think, how absurd, obviously we know all about the finite. Right? To my way of thinking, however, the two questions are equally hard, for they carve out the same boundary between the finite and the infinite. To define the finite is to define the infinite by complement; to have a clear criterion for one is to have a clear criterion for the other.
Meanwhile, it is not so easy and clear after all how to define either the finite or the infinite. In mathematical history there have been many proposals, and so let us explore a little the spectrum of concepts on offer.
Aristotle (384—322 BC) defines the infinite as that which proceeds without end, but he emphasizes that we can never achieve an actual infinity. We always find ourselves rather with only a finite quantity at hand, even while the infinite potentiality looms before us, tempting us. That perspective persisted in mathematics for millennia, although there was a sea change at the dawn of the 20th century and views have been completely overturned—we’re almost all actualists now.
Galileo (1564–1642) was a rare, early critic of potentialism and undertook to investigate the true nature of actual infinity. He observes in the Two New Sciences (1638) that the natural numbers can be placed into one-to-one correspondence with the perfect squares by associating each number n with its square n2 like this:
The situation is deeply paradoxical—do you see why? On the one hand, there seem to be many more numbers than there are perfect squares, as the gaps between the squares grow increasingly large. Most natural numbers, it would seem, are not perfect squares. And yet, the correspondence suggests that there should be exactly the same number of numbers as squares—they are equinumerous. Thus we observe the tension between the classical Euclid’s principle, according to which the whole is strictly greater than any proper part, and the Cantor-Hume principle, asserting that sets placed into a one-to-one correspondence will have the same size. The characters in Galileo’s dialogue ultimately throw up their hands in confusion.
The mathematician Richard Dedekind (1831–1916) proposes that we could try to take this feature as a definition. Namely, a set is finite—let us call it Dedekind finite—when it is not equinumerous with any proper subset, and otherwise Dedekind infinite, when it is. So the natural numbers come out as Dedekind infinite, of course, just as Galileo observes, and this is what we want. But will the criterion give the right result in all cases? After all, do we really expect that every infinite set will have this paradoxical property of being equinumerous with a proper part? Perhaps there could be some strange inhomogeneous infinite set that did not admit such kinds of self-directed correspondences, but we wouldn’t want to say therefore that it should count as finite.
Meanwhile, other approaches to defining the dichotomy between the finite and the infinite are grounded in elementary concepts of ordering. One naively attractive idea may be that a finite set is one for which we can place the elements in a discrete linear order, with a first point, and a next point, and a next, up to the very last point. Every intermediate point should have a next point and a previous point, like this:
The idea, of course, is that there should be only finitely many points between, but it would be circular to make that requirement part of the definition because we are attempting to define in the first place what it means to be finite. But there is a regrettable problem here, you see, in that merely having a discrete order with a first and last element, unfortunately, does not prevent an infinity sneaking into the middle like this:
Although infinite, this red-yellow set is nevertheless placed into a linear order with a first and last point, and every point between them has a definite next point and previous point, so it is a discrete order. The naive approach therefore doesn’t quite work.
To improve it, one thing to notice about this red-yellow order is that we can rearrange the two halves and produce an order with no first or last element, like this:
Harnessing this observation, let us define that a set is order infinite if it admits a linear ordering without any largest element. This proposal would seem to fit well with Aristotle’s definition, after all, since if the order has no largest element, then indeed it proceeds without end. But what if a set had no linear ordering at all? If indeed such a strange set is possible, a set that cannot be placed into a linear order, then it would not be realized as order infinite, but we wouldn’t want therefore to regard it as finite, since don’t we rather expect that every finite set can be placed into a linear order? Simply lay down the elements in a line until you run out.
In light of this, let us therefore define instead that a set is linearly finite, if it admits a linear order and all linear orders of the set have a final element. It follows that all linear orders of the set will have both a first and last element, since if there were no first element, then the inverse order would have no last element. The red-yellow set above is not linearly finite, since the rearranged order has no final element.
Another observation would be that the original red-yellow set above admits a rearrangement that is not discrete, by moving the right-most yellow ball to the center, like this:
The middle point now has neither an immediate successor nor an immediate predecessor, and so this order is not discrete. This suggests that we define that a set is discretely finite, if it admits a linear order, but all linear orders of it are discrete. It follows from this that the orders will also have a first and last element, since if not we could place one of the points from the middle to the extreme, resulting in a non-discrete order. So every discretely finite set is also linearly finite. The converse also holds, since if there is a non-discrete linear order, then we can rearrange it to place the failure of discreteness at the beginning or end and thereby violate the first/last element condition. So linearly finite and discretely finite are actually the same concept. This is welcome news, since it shows that our finiteness concepts are coming into alignment.
Perhaps a little simpler than rearrangements is to consider whether there might be a bad subset of the order, a nonempty subset that has no first or no last element. For example, we rule out the order above by finding a disqualifying subset like this:
The mathematician Paul Stäckel (1862–1919) similarly defines that a set is finite— let us call it Stäckel finite—if it can be placed into a linear order, such that every nonempty subset of the set admits a least and greatest element. This is equivalent to saying that the set has a well-order that is also conversely well-ordered. This implies that the set is discretely ordered—every intermediate point has a next point and a previous point, since if not, there would be a disqualifying subset, and the converse also is true.
The logician Alfred Tarski (1901–1983) proposes several additional concepts of finite in the early twentieth century, several of which have to do with collections of subsets of the set. For example, a set is Tarski finite if every nonempty family of subsets of the set has a minimal such set, that is, a subset in the family that does not admit any proper subset in the family. So in the end we have many finiteness concepts.
Apart from these order-theoretic notions of finiteness, another natural approach to the problem is to try to use the finite numbers themselves to define the finite sets (and thus to define the infinite by complement). That is, we say that a set is numerically finite if it can be counted off with the finite numbers from 1 up to some finite number n, like so:
This certainly agrees with ordinary usage—the shepherd counts off finitely many sheep in the flock. But one worries whether it might be hopelessly circular? After all, we would seem to need to know already which exactly are the finite numbers in order to know which sets are finite, and vice versa.
Nevertheless, the logician Gottlob Frege (1848–1925) attempts in the late 19th century to solve this problem by providing a direct independent account of the finite numbers. Frege mounts an elaborate investigation of the nature of abstract objects and concepts, the logicism program, by which he aims to reduce all of mathematics to logic. In Frege’s account, we abstract from any set of objects to its number in such a way that fulfills the Cantor-Hume principle that sets placed into a one-to-one correspondence have the same size. Frege then defines that the successor of a number is simply the number arising from a set with one additional element. Using this, he provides his definition of finiteness:
The finite numbers are precisely those numbers that have every property of zero that is also transferred from every number to its successor.
Do you see how it works? If a property holds of zero and is transferred from every number to its successor, then it must also hold of one, and therefore also of two, of three, and so on successively by iterating the transfer property. Furthermore, finiteness itself is one of these magical properties, since zero fulfills the definition, and if a number is finite according to the definition, then its successor will also be finite. The finite numbers will therefore be the smallest collection of numbers with these properties. In a sense he is defining the finite numbers and the finite sets simultaneously, but not in a circular manner. Rather, it is a clever kind of recursive ramping-up, a form of mathematical induction. In this way, Frege provides us with a definition of the finite numbers, based in logic, perhaps a core success of his logicist project.
Dedekind mounts a similar idea before Frege, when he identifies the axioms that characterize the natural numbers up to isomorphism. Namely, the theory of Dedekind arithmetic posits a set of numbers, including the number 0, with an operation S called the successor operation, with the properties that (1) zero is not the successor of any number; (2) the successor operation is one-to-one; and (3) every number is generated by S from 0, in the sense that every set of numbers containing 0 and closed under S has all the numbers. This latter axiom is known as the induction axiom.
Dedekind proves that these axioms determine the structure up to isomorphism, meaning that if you have your number system ⟨ℕ,0,S⟩ with your zero 0 and your successor operation S and I have my alternative system ⟨ℕ*,0*,S*⟩ with my alternative zero 0* and alternative successor operation S*, but both of our systems fulfill the Dedekind theory, then in fact there is an isomorphism , a translation from your system to mine revealing them to be copies of one another. Dedekind furthermore proved that in his theory, one can define addition and multiplication and indeed develop all the key ideas of elementary number theory.
This was the beginning of the philosophy of structuralism in mathematics, by which we do not say nor care to say what the mathematical objects are in themselves, but rather only how they relate to each other in a mathematical system. All that matters are the structural relations, rather than their individual identity or essence, since other objects playing the same structural roles would serve just as well for mathematics.
The fact that the structure of natural numbers is uniquely determined by Dedekind’s theory shows that our concept of finite number is well grounded in set theory. We thus obtain a rigidly definite concept of finite number, from which we can define our concept of numerically finite and consequently also the accompanying conception of the infinite.
Altogether we have thus seen in this essay several distinct concepts of the finite—but are they equivalent? The answer turns out to be quite subtle and interesting. The first thing to say is that in the standard Zermelo-Fraenkel ZFC set theory, indeed we can prove that all the finiteness definitions I have mentioned are equivalent. In this sense, we’ve got it right—we know what it means to be finite and what it means to be infinite in a way that is fruitfully clear and precise for mathematics.
But meanwhile, some of the equivalences are not provable in the weaker theory ZF, that is, without the axiom of choice. Without the axiom of choice, it turns out, there can be the kind of badly behaved sets that we were worried about earlier. For example, we can prove that it is relatively consistent with ZF that there is a numerically infinite set that is Dedekind finite. That is, the set is infinite, in that one cannot count off its members using any natural natural number, but still it does not happen to be equinumerous with any proper subset, and so those two conceptions do not align perfectly without the axiom of choice. It is also relatively consistent with ZF that there is a numerically infinite set that admits of no linear order, and so this is an infinite set that is not order infinite, which thus pulls these concepts also apart.
Nevertheless, even without the axiom of choice one can prove that the concepts of numerically finite, linearly finite, discretely finite, Stäckel finite, and Tarski finite are all equivalent, and any set that is finite in these senses is also finite in all the other senses. That is, numerical finiteness is the strongest, most restrictive notion of finite that we have. I take this as evidence that indeed this is what we should mean by finite—a set is finite, full stop, if and only if it is numerically finite, which means that it can be counted off by numbers up to some particular natural number. This is not circular, since the Dedekind theory provides an independent categorical account of the natural numbers, and this is now almost universally accepted in mathematics as the definition of what it means to be finite. And thus by complement we get also an accompanying robust definition of the infinite.
Joel David Hamkins is a mathematician and philosopher who undertakes research on the mathematics and philosophy of the infinite, working on a broad spectrum of topics in logic and the philosophy of mathematics. He is serializing several new books on his substack Infinitely More, releasing the latest chapters and material each week from the books-in-progress The Book of Infinity and A Panorama of Logic. He is also a Visiting Research Fellow, Mathematical Institute, University of Oxford.
July 14, 2024
