I'm Joel David Hamkins, and I'd like to tell you about Galileo's Paradox and the notion of equinumerosity. What does it mean to say that two sets, perhaps infinite sets, have the same size? Along the way, we'll discover some tension between certain ideas about equinumerosity and Euclid's Principle, which holds that the whole is greater than any proper part.
Let me begin with a story about the young Gottlob Frege. He was hosting a dinner party, and walking past the dining room, which had already been laid out for the guests, he could tell at a glance that the number of plates on the table was the same as the number of knives, without counting either. How could he know this?
The answer is simple: he observed that at every place setting there was exactly one plate and one knife. This allows us to establish a one-to-one correspondence between the plates and the knives, and thereby determine that the two sets have the same size, even without counting them.
This is the principle underlying the definition of equinumerosity. We say that two sets, call them A and B, are equinumerous, written with this symbol, if there is a one-to-one correspondence between the elements of A and the elements of B. This means a matching of elements of A with elements of B such that every individual in A has a partner in B and vice versa, and no two different individuals are ever matched to the same one. That is what it means to have a one-to-one correspondence between two sets.
Frege had observed that the plates were equinumerous with the knives. A similar observation arises in an everyday setting: when I look out at a lecture hall, I can recognize that the number of people in the classroom equals the number of noses in the classroom, simply because every person has a nose, every nose is attached to a person, and nobody has an extra nose in their pocket. Without counting the students at all, I can come to know that the number of people is the same as the number of noses.
This brings us to a principle I call the Cantor-Hume Principle, also known simply as Hume's Principle, which states that two sets have the same number of elements if and only if they can be put into one-to-one correspondence, that is, if and only if they are equinumerous. The principle is named after Hume because Hume wrote about it in 1739, in the sentence: "When two numbers are so combined as that one has always a unit answering to every unit of the other, we pronounce them equal." Frege cited this sentence in his work at the end of the nineteenth century, and it came to be known as Hume's Principle throughout much of the philosophical literature.
I prefer to call it the Cantor-Hume Principle, however, in light of Cantor's work with the same idea. It is absolutely central to Cantor's thinking, particularly in the infinite case. Frege worked with Hume's Principle primarily in the finite case, using it to found his theory of arithmetic, whereas Cantor carried out his spectacular work with equinumerosity in the infinite and even the uncountable case, territory we will explore in future lectures.
The concept of equinumerosity also figures in Galileo's work, particularly in his dialogues between Salviati and Simplicio in the Dialogue Concerning Two New Sciences of 1638. Before we get to that later work, however, I want to go much further back and discuss the paradox of Aristotle's wheel.
Consider a wheel with a center, an axle, and a certain radius and circumference. We set it rolling along the ground through exactly one full revolution. Labeling the relevant points: the center begins at A, the point on the axle at B, and the bottom contact point at C. After one complete revolution, these points have traveled to D, E, and F respectively. As the wheel turns, the circumference of the outer circle is laid off along the segment CF, so the length of CF equals the circumference of the outer circle.
Now Aristotle observed that we should also consider the path traced by the axle, the inner, smaller circle. As the wheel turns, that inner circle also traces a path, from B to E, and its points are laid off in a one-to-one manner along the segment BE. But BE has the same length as CF. The paradox is this: the inner circle has a smaller circumference than the outer circle, yet both circles appear to be placed in one-to-one correspondence with segments of equal length. By the reasoning of what we would later call Hume's principle, the two circles should therefore be the same size, which is absurd.
One possible explanation is that the inner circle is not truly rolling along the middle segment at all. If we imagine two shelves dusted with powder and the wheel rolling freely along the lower shelf, the inner circle would be carried along faster than its own circumference warrants, and would scrape rather than roll cleanly against the upper shelf. Conversely, if the wheel were driven by contact with the upper shelf, the outer circle would scrape along the lower one. The rolling and the laying-off are not the same thing.
But this mechanical observation does not dissolve the deeper puzzle. Even granting the scraping, there remains a genuine one-to-one correspondence between the points of the smaller circle and the points of the segment BE. The paradox therefore persists from the standpoint of the Cantor–Hume principle, and it is a genuine conceptual difficulty about the relationship between size and one-to-one correspondence.
Let me now turn to Galileo's analysis of the paradox of Aristotle's wheel. Galileo changed the setup slightly by replacing the circular wheel with a hexagonal one. He drew a hexagon with a smaller hexagonal axle at its center, and then considered what happens when this hexagonal wheel is rolled.
Of course, a hexagonal wheel does not roll smoothly. When it is tipped forward, it falls with a ker-thump onto the next flat side, and this process repeats, again and again, until all six sides have each had a turn resting on the ground. The wheel then comes to rest in a position corresponding to one full revolution.
The crucial observation concerns what happens to the inner hexagonal axle during this motion. Each time the outer hexagon tips forward and a new side falls flat, the inner hexagon is lifted slightly and then comes to rest again. As a result, the inner hexagon does not trace a continuous path along the ground; instead, it lands in a series of separated positions, with gaps between them.
The total length of those resting positions corresponds exactly to the circumference of the inner hexagon, which is less than the total length traced by the outer hexagon. The gaps between the inner hexagon's contact points account precisely for the discrepancy: they explain why the inner figure corresponds less densely to the line traced below it.
Galileo then extended this reasoning by imagining polygons with ever more sides: 20, 40, 100, and so on. As the number of sides increases, the gaps become smaller and smaller, the ker-thump becomes less pronounced, and the motion becomes increasingly smooth, approaching that of a true circle. In the limit, the polygon becomes Aristotle's wheel. And yet, for every polygon in this sequence there are gaps; the question is what becomes of those gaps in the limiting case of the actual circle, where they seem to vanish entirely. This is the paradox of Aristotle's wheel, and this is Galileo's answer to it.
Let me now turn to Bolzano's observations about the equinumerosity of geometric figures. These appear in Paradoxes of the Infinite, published in 1851. What Bolzano observed is the following: if you have two circular arcs, one from a larger circle and one from a smaller, the larger arc appears to have greater length. And yet we can establish a one-to-one correspondence between their points by drawing rays from a common center, matching each point on the inner arc to exactly one point on the outer arc.
Provided the two arcs subtend the same angle, this ray construction yields a bijection between the points of the smaller arc and the points of the larger arc, even though those arcs have different lengths. Bolzano took this as evidence against the part-whole principle, the principle that the whole must always be strictly greater than any of its proper parts, since here a smaller geometric object corresponds perfectly with a larger one.
Another example he discussed is that of any two circles. If we place them concentrically, we immediately obtain a one-to-one correspondence between their points: given any point on one circle, the point on the other circle at the same angle is its unique correspondent. Again, two circles of different sizes are placed in perfect one-to-one correspondence, further challenging the intuition that a larger figure must contain strictly more points than a smaller one.
Galileo observed a similar phenomenon with line segments. Suppose you have a short line segment and a longer line segment. One can imagine a foliation of lines between them, a fanning out, that specifies a natural correspondence. The point halfway along the smaller segment corresponds to the point halfway along the larger segment; the point 10% of the way along the smaller segment corresponds to the point 10% of the way along the larger segment; and so on. Every point on the shorter segment gets mapped to a point on the longer segment in a one-to-one way, giving us a one-to-one correspondence between a shorter segment and a longer one.
This may seem puzzling if one holds that two sets which are equinumerous should have the same size. What it suggests, perhaps, is that the length of a segment is not quite the same thing as the size of the segment when we think of it as a collection of individual points. One might say that these two segments have different lengths, but that the number of points they contain is the same, as witnessed by the one-to-one correspondence. That is precisely what one would have to say in order to hold on to the Hume principle.
Galileo gives another example involving a finite segment. He argues that if you have an open line segment, we can build a correspondence between it and the entire infinite line. The construction proceeds as follows: inscribe a semicircle within the segment, noting its center point. Given any point on the segment, drop it perpendicularly down to the semicircle, and then project it outward from the center of the semicircle onto the infinite line.
To see how this works, consider a point on the segment that lies very near one of its edges. When it is dropped down to the semicircle, it lands near the base of the arc, and when projected outward from the center, it travels quite far along the line. Conversely, every point on the infinite line can be projected back through the center of the semicircle to a unique point on the arc, and from there lifted to a unique point on the original segment.
This gives a one-to-one correspondence between the finite open segment and the infinite line: given any point on the segment we obtain a point on the line, and given any point on the line we can reverse the process and recover exactly where it came from on the segment. The conclusion is that a finite segment is equinumerous with the infinite line.
There is another example of this kind of paradoxical situation with equinumerosity, known as Galileo's Paradox. Galileo considered the natural numbers, starting from zero: zero, one, two, three, and so on. Some of those numbers are perfect squares. Zero is zero squared, one is one squared, two squared is four, three squared is nine, four squared is sixteen, and so on. If you notice, the perfect squares grow farther and farther apart as the numbers increase.
To make this vivid, consider the natural numbers laid out in order: zero, one, two, three, four, five, six, seven, eight, nine, and so on. The squares among them, zero, one, four, nine, sixteen, and so on, are scattered with increasing gaps between them. It seems reasonable to conclude that there are strictly more natural numbers than perfect squares, since there are always non-square numbers in between, and the density of the squares decreases as we go further out.
This intuition appeals to what is called Euclid's Principle: the whole is strictly greater than any proper part. According to Euclid's Principle, the collection of all natural numbers must be greater than the proper subcollection consisting only of the perfect squares. And yet, Galileo observed that we can construct a one-to-one correspondence between the natural numbers and the perfect squares, by associating each number n with its square n², and each perfect square with its square root. This pairs zero with zero, one with one, two with four, three with nine, four with sixteen, five with twenty-five, and so on.
By the Cantor–Hume Principle, which holds that two sets have the same number of elements if and only if there is a one-to-one correspondence between them, the number of natural numbers should be exactly equal to the number of perfect squares. This reveals a direct tension between the two principles. The Cantor–Hume Principle says the sets are the same size; Euclid's Principle says the whole must be strictly greater than its proper part. When a collection can be placed in one-to-one correspondence with a proper part of itself, as is the case here, the two principles cannot both be maintained. That conflict, are the natural numbers and the perfect squares the same size, or not, is precisely what is at stake in Galileo's Paradox.
At this point, I want to read what Galileo himself wrote about this situation in his lovely dialogue.
The dialogues are between two characters, Salviati and Simplicio. Salviati is the clever one, and when reading these dialogues one gets the clear impression that it is really Galileo himself speaking through Salviati. Simplicio serves as the foil. What Salviati says about this situation is the following: "But if I inquire how many roots there are, it cannot be denied that there are as many as there are numbers, because every number is a root of some square. This being granted, we must say that there are as many squares as there are numbers, because they are just as numerous as their roots, and all the numbers are roots. Yet at the outset, we said there are many more numbers than squares, since the larger portion of them are not squares."
Salviati continues: "Not only so, but the proportionate number of squares diminishes as we pass to larger numbers. Up to 100 we have only 10 squares, that is, the squares constitute one-tenth of the numbers up to 100. But up to 10,000 we find only one one-hundredth part to be squares, and up to a million, only one one-thousandth part. On the other hand, in an infinite number, if one could conceive of such a thing, he would be forced to admit that there are as many squares as there are numbers taken all together." One can see that Galileo, through Salviati, is already committed to what we now call the Cantor–Hume principle, and this is, of course, earlier than Cantor.
He continues: "We can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite. Neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former. And finally, the attributes equal, greater, and less are not applicable to infinite, but only to finite quantities. When, therefore, Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones do not contain more points than the shorter, I answer him that one line does not contain more or less or just as many points as another, but that each line contains an infinite number."
That is Galileo in 1638, throwing up his hands at the paradox. He has no resolution to offer. He is saying that we cannot make comparisons between infinite quantities precisely because we seem forced into the contradictory position of asserting both that the number of squares is the same as the number of numbers, and that the number of squares is strictly less than the number of numbers. Finding this intolerable, he refuses to make any such comparative judgments about infinite sets at all.
Despite Galileo's conclusion, thinkers progressed on this question, and the contemporary mathematical attitude resolves the paradox by taking the Cantor-Hume principle as basic. Two sets are equinumerous, that is, they have the same number of elements, if and only if there is a one-to-one correspondence between them. On this view, mathematicians give up Euclid's principle that the whole is greater than any proper part. Instead, it is accepted as a feature of infinity that an infinite set can be the same size as a proper part of itself, and that this is not a contradiction but rather part of the very nature of infinite sets, just as the natural numbers are the same size as the proper subset of perfect squares.
For those who feel a strong pull toward Euclid's principle and wish to preserve as much of it as possible, there is a natural way to retain the core of the intuition. The difficulty lies specifically with the claim of strict excess; what the intuition most fundamentally demands is perhaps only that the whole of any object is at least as great as any proper part. That weaker formulation need not be abandoned, and in fact this slight revision to Euclid's principle is enough to make it fully compatible with the Cantor-Hume principle.
The prevailing attitude in mathematics today is therefore to define "same size" by the relation of one-to-one correspondence, the idea that Cantor emphasized so strongly in his work, while retaining the principle that the whole is at least as great as any proper part. It is this combination that naturally introduces the framework for comparing different sizes of infinity.
One thing to observe is that the Cantor-Hume Principle, the principle stating that two sets have the same number of elements if and only if they can be put into one-to-one correspondence, provides an identity criterion for when two sets have the same size, but it does not give us a criterion for when one set is at least as large as another. It concerns only sameness of size, not comparative size. The Reflexive Euclid Principle I proposed, by contrast, is about that comparative judgment.
I want to introduce what I call the Comparative Size Principle. We say that one set has at least as many elements as another if the smaller set is equinumerous with a subset of the larger. To make this precise: we say that X is less than or equal to Y in size, that is, Y is at least as large as X, if X can be placed into one-to-one correspondence with some part of Y, whether all of Y or merely a proper subset of it.
Notice that if X is equinumerous with a proper part of Y and there are points of Y left over, we do not immediately conclude that Y is strictly bigger, because Y itself may be equinumerous with that proper part, as is the case with the natural numbers and the perfect squares. This is therefore a reflexive notion: X is less than or equal to Y in size if X is equinumerous with some part of Y, possibly the whole of Y, possibly only a proper part.
Given this reflexive relation, we can define the corresponding strict order in the natural way. We say that X is strictly smaller than Y, equivalently, that Y is strictly larger than X, if and only if X is less than or equal to Y in size, but Y is not less than or equal to X in size. In other words, Y is at least as large as X, but X is not at least as large as Y. That is what it means for one set to be strictly larger than another.
A certain question arises naturally when one introduces these notions. If X is less than or equal to Y in size, meaning that X is equinumerous with a part of Y, and Y is less than or equal to X in size, meaning that Y is equinumerous with a part of X, must it follow that X and Y are equinumerous with each other? We can picture the situation: there is a one-to-one map sending points of X into Y, witnessing that X is at most as large as Y, and a one-to-one map going the other way, sending points of Y into a part of X, witnessing that Y is at most as large as X. Each set is equinumerous with a part of the other.
The question is whether one can always assemble these two injections into a single bijection, an exact one-to-one correspondence between X and Y with nothing missing on either side. If the answer were no, that would be evidence that this is not a robust notion of comparative size. But if the answer is yes, then the notion fulfills a rock-bottom principle we would want any concept of comparative size to satisfy: if Y is at least as large as X and X is at least as large as Y, then they are the same size.
The answer is indeed yes, and this result is known as the Cantor–Schröder–Bernstein theorem, proved at the end of the nineteenth century. It is, I think, a somewhat missing piece of the philosophical analysis, because there is an enormous literature on the Cantor–Hume principle, the principle that two concepts have the same number of instances if and only if those instances can be placed in one-to-one correspondence, yet there is comparatively little discussion in the philosophical literature of the Cantor–Schröder–Bernstein result, which is what makes the accompanying concept of comparative size robust.
The history of the theorem is intricate. Cantor mentioned it in 1887 without proof; Dedekind proved it in 1888 but did not publish his proof; Cantor proved it again in 1895, but only under the assumption that cardinal sizes are linearly ordered, a principle that is in fact equivalent to the axiom of choice. Schröder then proved it in 1896 as a corollary to a theorem of Jevons, and Bernstein proved it independently in 1897. Crucially, the theorem does not require the axiom of choice, and the argument that establishes it without that assumption is quite beautiful.
The proof works by taking the two injective maps and delicately assembling pieces of each to construct a single bijection between X and Y. It is a challenging argument, but not an overwhelmingly difficult one, and the correspondence it produces has an elegant structure. The conclusion is that equinumerosity does indeed satisfy the basic principle we demand: if X is less than or equal to Y in size and Y is less than or equal to X in size, then X and Y have the same size.