I want to talk about the infinite and the finite. What does it mean to say that something is infinite, that a set is infinite, or that a set is finite? Perhaps it seems obvious, but I think it is not so clear. Throughout history, we have had many different proposed definitions of the infinite and the finite, and so let us get into it.
If we want to understand the infinite, perhaps we think that concept is too difficult, and so we might ask instead: what does it mean to say that a set is finite? Perhaps you reply that this is absurd — of course we know what it means for a set to be finite. But do we really?
In fact, I claim that these two concepts, the finite and the infinite, are equally hard, because they carve exactly the same joint. To define the finite is to define the infinite, since infinite sets are simply those that are not finite. And to define the infinite is to define the finite, since finite sets are simply those that are not infinite. Defining one is defining the other, and so it is just as hard to define the finite as it is to define the infinite.
Aristotle defined the infinite as that which proceeds without end. As discussed in another lecture on potentialism, he held a potentialist understanding of the infinite: something is infinite, or there are infinitely many numbers, in the sense that one can always have more and more, as many as one likes, but one never has all of them at once. This is precisely the sense of proceeding without end, and it is closely connected with the broader ideas of potentialism.
We had also mentioned that Galileo was an early and prominent critic of this potentialist view.
Almost everyone was a potentialist up until about the end of the 19th century, when a sea change occurred. Galileo, however, had already raised an objection, which we discussed in the lecture on Galileo's paradox: the observation that the number of numbers is the same as the number of perfect squares. This is based on the idea that a set can be equinumerous with a proper part of itself. The natural numbers, zero, one, two, three, and so on, can be placed into one-to-one correspondence with the perfect squares, zero, one, four, nine, sixteen, twenty-five, and so on.
That situation is paradoxical because it exposes a tension between Euclid's principle and the Cantor–Schröder–Bernstein principle that sets have the same number of elements when they can be placed into one-to-one correspondence. Richard Dedekind, in the late 19th century, proposed that we simply take this phenomenon as a definition of the infinite: a set is Dedekind infinite exactly when it is equinumerous with a proper part of itself. The natural numbers are therefore Dedekind infinite because they are equinumerous with the perfect squares, which form a proper part of the whole.
Of course, the perfect squares are not the only example. We could equally have used the even numbers, since any natural number can be doubled to produce an even number, establishing a one-to-one correspondence between all the natural numbers and just the even ones. There are many proper parts that are equinumerous with the whole of the natural numbers, and any one of them suffices to confirm that the set of natural numbers is Dedekind infinite.
The deeper question is whether we should accept this as the correct definition of infinity in general, that is, whether every infinite set must contain a proper part equinumerous with itself. Finite sets certainly never exhibit this property; no equinumerosity between a finite set and a proper part of it can ever be found. But one might conceive of a strange, amorphous set that is infinite in the sense of failing to have finitely many elements, and yet for which no equinumerosity with any proper part can be established. Whether such sets are possible is a genuine question.
Let me turn to some alternative order-theoretic conceptions of the infinite. Perhaps being infinite has to do with how we can place a set into an order. A naive initial proposal is that a set is finite if and only if it can be placed into a discrete linear order with endpoints, that is, a linear order possessing a first element and a last element, where every point in between has an immediate successor and an immediate predecessor. One might think this captures exactly what it means to be finite.
The trouble is that this proposal does not actually work. There are clear counterexamples, situations that satisfy the condition and yet are obviously infinite. Consider the following linear order: take the positive integers 0, 1, 2, 3, 4, 5, … and place the negative integers −1, −2, −3, −4, … above them, in descending order. This gives a linear order in which 0 is the first element and −1 is the last element.
Every element drawn from the positive part has an immediate successor and an immediate predecessor within that part, and the same holds for every element in the negative part. So this is a discrete linear order with endpoints. And yet it is obviously not a finite set: it contains all the integers. The naive proposal therefore fails. Merely admitting a discrete linear order with endpoints does not force a set to be finite in the intuitive sense, and so it cannot serve as a definition of finiteness.
Let us consider some other proposals. One problem with the order we have been examining is that it can be rearranged. If we take the dividing point and interchange the two pieces, placing one part on the right going upward, perhaps getting smaller and smaller, and the other part on the left going downward, we obtain something that looks just like the integers: zero, one, two, three, and so on going up, and −1, −2, −3, −4 going down. The key feature to identify is that this set has a rearrangement with a linear order that has no largest element, and so according to Aristotle's conception that something is infinite when it proceeds without end, here is a linear order that proceeds without end.
This motivates the following definition. A set is order infinite if it has a linear order with no largest element. The original ordering of our set was no good because it did have a largest element, but the set admitted a rearrangement with no largest element, and so the set counts as order infinite. Of course, any set possessing such a linear order would be infinite according to our intuitive idea, since there is no largest element and the order proceeds without end. But should we expect every infinite set to have such a linear order?
Consider whether every infinite set has a linear order at all. If there existed a strange set that could not be given any linear order whatsoever, it would vacuously fail to be order infinite, since being order infinite requires having a linear order with no largest element, and yet we would not want to call it finite. We also have the intuition that every finite set should have a linear order; one ought to be able to count off the elements of a finite set, and that counting would itself impose an order. So the order infinite concept runs into difficulty if there are sets that admit no linear order at all.
Whether such sets exist depends on the set-theoretic principles we are working under, on the nature of set-theoretic ontology. These foundational commitments turn out to have real consequences for the definitions of finite and infinite. In light of this objection, however, we can attempt a repair. A set is linearly finite if it has at least one linear order, but every linear order on it has a final element. Here a linear order means an ordering in which for any two points, one appears before the other.
To see how this applies, consider our example set. It does have a linear order, but not every linear order on it has a final element: we exhibited a linear order on the very same set with no final element. Therefore this set is not linearly finite, and so it counts as infinite according to this account of finiteness.
Here is a slightly different definition. A set is discretely finite if it has a linear order, but every linear order on it is discrete: it will never admit an order that fails to be discrete. The two orders considered previously were both discrete, but the set of integers, I claim, is not discretely finite, because one can construct a linear order on it that is not discrete. Let me show how.
Begin with the standard discrete linear order of the integers. Now take one of the points and place it in the middle, specifically, insert it into the gap between the negative integers, which converge upward, and the positive integers, which proceed downward from above. The result is an order in which that chosen point sits in the gap between the two halves. All we have done is lift that point out of its original position and place it into that gap.
This gives a set that has a linear order, but it is not the case that every linear order on it is discrete, because this new order is not discrete. Most points behave well: each has an immediate successor and an immediate predecessor. The exception is the point placed in the gap. The points in the lower half increase without bound, so there is no largest element among them, meaning the inserted point has no immediate predecessor. Similarly, the points in the upper half decrease without bound, so there is no smallest element among them, meaning the inserted point has no immediate successor. The absence of either one of these would be enough to disqualify the order from being discrete.
Therefore, the integers are not discretely finite: the set admits linear orders, but not every linear order on it is a discrete order.
Here is another definition, proposed by Paul Stäckel in the late nineteenth century. A set is Stäckel finite if it admits a linear order with the property that every non-empty subset has both a least element and a greatest element. The condition that every non-empty subset has a least element is precisely what is known as a well order. Stäckel's definition therefore requires an ordering that is a well order and whose inverse is also a well order, so that every non-empty subset has both a least and a greatest element.
One might reasonably ask why it should be so difficult to say what we mean by finite. Why do we need such subtle and complicated conditions involving different kinds of orders just to capture this apparently simple notion?
Tarski, writing in the early twentieth century, addressed this question directly, producing a paper offering five different order-theoretic accounts of finiteness. His definitions were concerned with collections of subsets of a given set. For example, a set is Tarski finite if every non-empty family of subsets of the set has a minimal member, that is, the family is well-founded with respect to inclusion. Tarski proposed several further alternatives along similar lines.
Many of these order-theoretic notions turn out to be equivalent to one another, but they are not all provably equivalent within standard Zermelo–Fraenkel set theory alone. If one also assumes the axiom of choice, however, they become equivalent across the board. The axiom of choice is thus needed to establish these equivalences, and we will discuss it as a fundamental principle of set theory in a later lecture.
Let me turn now to another definition of finiteness: numerically finite. A set is numerically finite if you can count off its elements with numbers, that is, if you can label each element with a finite number. This fits the intuition of a shepherd counting sheep: one, two, three, four, and so on up to 117, or however many sheep one happens to have. A set is numerically finite if its elements can be counted off with the numbers from one up to some n, in which case we say it has n elements. (I habitually start at zero, but when counting sheep one would naturally start at one.)
The obvious objection, of course, is that this definition is hopelessly circular. I have said that a set is numerically finite if you can count off its elements using the numbers up to a finite number, but that word "finite" is doing all the work. We would need a completely independent account of what it means for a number to be finite in order for this definition to succeed. We certainly cannot say that a number is finite if it is numerically finite, since that is entirely circular and inadequate.
So the question becomes: do we have a concept of finite number that stands independently of our concept of finite set? One might propose that a finite number is simply one that counts the elements of a finite set, but that, too, is circular. The definition of numerically finite therefore faces a genuine foundational difficulty: it presupposes the very concept it is meant to explain.
Frege grappled deeply with the question of what it means to say that a number is finite, and he embarked on his logicist program, the attempt to reduce all of mathematics to logic. He wanted to ground the entire foundation of mathematics in basic logical principles, on which the rest of mathematics could then be built.
He began with the concept of the number zero, which he defined as the number of elements in the empty set. If you have a contradictory property, the extension of that concept has no instances, giving you the empty set, and the number of elements belonging to that concept is what Frege called zero. He also introduced the concept of the successor relation: if a set has a certain number of elements, the successor of that number is the number of elements in the set obtained by adding one more element.
Taking zero and the successor relation as primitive notions, Frege then defined a finite number as a number that possesses every property which zero has and which is always transferred from any number to its successor. To put it precisely: a number is finite if it has every property that zero has and that, whenever it holds of any number N, also holds of the successor N + 1.
This definition fits our intuitions perfectly. Zero is the smallest finite number, and if N is finite then N + 1 is finite. Frege is identifying finiteness with the most minimal such property, the smallest collection of numbers closed under zero and successor. When you examine this carefully, you realize that Frege's definition amounts to what we would call induction: to be a finite number is precisely to satisfy the inductive property, so that whenever a property holds at zero and is always transferred from N to N + 1, it holds of all the finite numbers.
Dedekind is remarkable for having given us not one but two different concepts of finiteness. There is the concept of Dedekind finite and Dedekind infinite, where a set is Dedekind infinite when it is equinumerous with a proper part of itself, and Dedekind finite when it is not. But Dedekind also gave us a concept of finite number through his theory of Dedekind arithmetic.
Dedekind arithmetic concerns the structure of the natural numbers equipped with a constant for zero and the successor operation, the plus-one operation. Dedekind wrote down the most fundamental axioms governing this successor relation. The first axiom states that zero is never the successor of anything: there is no number smaller than zero, no predecessor to zero. The second axiom states that the successor function is one-to-one. If the successor of x equals the successor of y, then x equals y, so two numbers that share the same successor must themselves have been equal.
The third axiom captures the idea that every number is obtained from zero by repeatedly applying the successor operation. It is stated as follows: if a set of numbers contains zero, and whenever n belongs to the set its successor also belongs to the set, then every number belongs to the set. Expressed in terms of properties rather than sets, this says that if zero has a given property, and whenever a number has that property its successor also has it, then all numbers have the property. This is precisely the principle of induction.
This theory is what we call Dedekind arithmetic, and it is certainly what we expect to hold of our conception of the natural numbers under the successor operation.
Dedekind proved a remarkable theorem about this theory: the theory is what is called categorical. Suppose you have two different structures, my conception of the natural numbers with my zero and my successor operation, and your conception with your zero and your successor operation, and both satisfy the axioms. Dedekind showed that our two number systems are simply copies of one another; they are isomorphic. There is a correspondence from my number system to yours, or the other way, that preserves all structural features. A categorical theory, in this sense, has only one model up to isomorphism: the axioms completely capture the structural situation of the intended model.
The way Dedekind proved this is by showing, on the basis of his axioms, that one can undertake definition by recursion. A recursive definition defines a function by reference to earlier instances of that same function. Let me show how this concept leads to the categoricity result. Suppose my number system is M with zero and successor S, and your number system is N̄ with zero-bar and successor S̄. The individual numbers in your system may be constituted by entirely different objects; your zero may be a different object from mine; the nature of your successor operation may differ from mine. The question is precisely whether two such systems, both satisfying Dedekind's axioms, must share the same fundamental structure.
Dedekind defines an isomorphism, call it π, mapping my numbers to yours. The definition is straightforward: π must send my zero to your zero, the successor of my zero to the successor of your zero, and so on. More precisely, if π maps some number n to π(n), then it must map the successor of n to the successor of π(n) on your side, that is, π(Sn) = S̄(π(n)). This is a recursive definition, since π at each new instance is defined in terms of π at the previous instance. Dedekind proved that such definitions by recursion are legitimate within his theory, and so this definition of π succeeds.
Once we have this map, we must verify that it is a bijection, that it hits every one of your numbers. Consider the set of numbers in your system that actually appear in the range of π. This set contains zero-bar, and it is closed under your successor operation, because the successor of anything in the range of π is itself in the range of π, by the recursive clause. A subset of your numbers that contains zero-bar and is closed under S̄ must, by the induction axiom, be all of your numbers. Therefore every one of your numbers is π(n) for some n in my system, and the map is onto.
What we have proved, then, is that any two number systems satisfying Dedekind's axioms admit a mapping between them that witnesses their isomorphism: they are structurally identical, perfect copies of one another. That is precisely what it means for the theory to be categorical.
The significance of this for the notion of the finite is that the categoricity result shows that Dedekind's theory identifies the concept of finite number. To be a finite number means to be one of the numbers in a model of this theory, that is exactly what it means to be a finite number. Crucially, the theory never uses the word "finite," so the definition is not circular. We have an independently standing concept: a finite number is a number that occurs in a model of Dedekind arithmetic.
Once you have a model of Dedekind arithmetic with the successor operation, you obtain from it the entirety of number theory. Addition can be defined by recursion from the successor relation, multiplication by recursion on addition, and from there one can define exponentiation, primes, the ordering, and all of the familiar number-theoretic concepts that appear in the development of number theory. All of them follow deductively from Dedekind's theory.
Peano famously wrote a treatise doing exactly that, explicitly acknowledging Dedekind's theory. He wrote down these axioms and showed, in a quite elegant and beautiful manner, how to develop the entirety of elementary number theory on their basis. This is, in fact, how a great deal of number theory is introduced at an elementary level: proving things by induction, establishing associativity of addition, commutativity of multiplication, and all the other familiar basic facts. Every one of them can be proved within Dedekind's theory.
People sometimes think of induction as a curiosity, useful mainly for proving recursive formulas, the way it is often treated in high school. But this view is completely wrong. The perspective that grows out of Dedekind and Peano's work shows that induction is not some curiosity; it is a core principle at the very foundation of our ideas about number. Essentially all of the most fundamental facts of number theory are proved using induction.
Consider the concept of fractions. Three-sixths is the same as one-half, and five-fifteenths is the same as one-third; we can cancel and reduce to find the lowest terms. But can every fraction be placed in lowest terms? In elementary school we always succeed when we try, but why should we expect to always succeed? Is it a theorem? It is, and we prove it by induction.
Before giving that proof, it is worth stepping back and discussing the different forms induction can take. The form appearing in Axiom Three of Dedekind's theory is what might be called common induction. Common induction says: if you want to prove that every number has a certain property, prove that it holds of zero, and then prove that whenever it holds of a number n it also holds of the successor n + 1. From those two things you may conclude it holds of all numbers. The zero case is called the anchor case, and the implication from n to n + 1 is called the induction step.
There is another form called strong induction. Here you prove just one thing: that if the property holds of every number below n, then it holds at n. No separate anchor case is needed, because the case of zero follows automatically. There are no numbers below zero, so the hypothesis that the property holds of all numbers below zero is vacuously true, and the conclusion therefore applies to zero. Once it holds at zero, it holds at every number below one, so it holds at one; and so on for every number in turn.
The difference between the two principles lies in the strength of the assumption available during the induction step. In common induction you may assume only that the property holds at n when proving it for n + 1. In strong induction you may assume it holds at all earlier numbers, a stronger assumption, which is why the principle bears that name. Despite this difference, the two principles are equivalent, and Dedekind arithmetic could equally well have been formulated using strong induction in place of common induction.
There is a third equivalent formulation called the least number principle, which states that every non-empty set of natural numbers contains a smallest element. Its connection to strong induction is illuminated by the phrase minimal criminal. When proving by strong induction that every number has some property, suppose for contradiction that some number fails to have it. Among all such failures there would have to be a least one, the minimal criminal, the least counterexample. But because the property holds for every number below this minimal criminal, the strong induction hypothesis forces it to hold at the criminal itself, contradicting the assumption that it was a counterexample. Thus the set of counterexamples can have no least member, which by the least number principle means it must be empty.
This shows that the strong induction principle is equivalent to the least number principle, and both are equivalent to common induction. All these different ways of talking about induction turn out to say exactly the same thing.
Let us return to the lowest terms claim: every fraction can be put in lowest terms. Suppose we have a fraction p/q that is not yet in lowest terms. To show we can reduce it, consider the set of all possible numerators that could appear in an equivalent representation of p/q. That is, we look at all values p′ such that p′/q′ equals p/q for some q′. By the least number principle, this set of numerators must contain a smallest element, so there is a smallest possible numerator p′ that can be used to write a fraction equivalent to p/q.
I claim that this fraction p′/q′, with the smallest possible numerator, must already be in lowest terms. If it were not in lowest terms, there would be a common factor shared by p′ and q′, and we could divide both by that factor to obtain an equivalent fraction with a strictly smaller numerator. But this contradicts the assumption that p′ was already the smallest numerator available. Therefore no such common factor can exist, and p′/q′ is in lowest terms. Since there can be no minimal criminal, no fraction that resists being put in lowest terms, every fraction can indeed be reduced to lowest terms.
Let me do another example. The Fundamental Theorem of Arithmetic states that every positive integer has a factorization into primes, a unique factorization into primes, the prime factorization of the number. The theorem is really two claims: an existence claim and a uniqueness claim. We assert not only that a factorization exists, but that there is only one. All factorizations of a given number use the same primes, and while the order may vary, the multiplicity of each prime appearing is uniquely determined.
Let us prove the existence part of the Fundamental Theorem of Arithmetic by strong induction. We take a number n and suppose, as our strong induction hypothesis, that every positive integer less than n already has a prime factorization. We wish to show that n itself has a prime factorization. If n is prime, then n is already its own prime factorization, and we are done.
If n is not prime, then n can be written as a product a × b for some smaller positive integers a and b. Both a and b fall under the induction hypothesis, so each has a prime factorization: write a = p1 · p2 · … · pk and b = q1 · q2 · … · qm. Since n = a × b, we simply combine these two lists of primes to obtain a prime factorization for n.
The key point is that knowing all smaller numbers have prime factorizations is sufficient to guarantee that n itself has one. By the principle of strong induction, every positive integer therefore has a prime factorization. This completes the existence part of the Fundamental Theorem of Arithmetic, and I hope it illustrates how induction is a core tool for establishing these fundamental facts of number theory.
The final example I want to discuss is the pigeonhole principle. The pigeonhole principle asserts that if you have finitely many pigeons and fewer pigeonholes, and you place all the pigeons into the pigeonholes, then at least one pigeonhole must contain at least two pigeons. In other words, you cannot distribute a larger number of pigeons into a smaller number of holes without some doubling up.
This seems completely obvious, and yet it is precisely the kind of principle that expresses such a basic fact that it becomes difficult to see how one would actually prove it. Of course, everything ultimately requires proof from first principles. But before we prove it, I want to point out that we actually have a surprising amount of apparent evidence against the pigeonhole principle from ordinary experience.
Suppose there is an enormous heap of pennies on a table, and two people independently count them. Do they always arrive at the same number? In practice, two people counting the same collection of objects often come up with different totals. If both counts were correct, that would constitute a counterexample to the pigeonhole principle. Of course, whenever this happens we attribute it to human error: someone miscounted, or a penny slipped past unnoticed. But the situation does arise, and it could in principle be taken as evidence against the pigeonhole principle. Similarly, when an election is followed by a recount, the two tallies are almost never perfectly identical. That too could be read as evidence against it.
I am not, of course, seriously advocating that we deny the pigeonhole principle. It is a fundamental principle of number theory, and we can prove it. Let us do so now using the principle of induction. Specifically, we apply the least number principle: if there were any counterexample to the pigeonhole principle, there would have to be a smallest one, a minimal criminal, as it were.
Suppose, then, that we have a counterexample: some number of pigeons that fit, without any doubling up, into a strictly smaller number of pigeonholes. Now remove one pigeon together with the pigeonhole it occupies. We still have a valid counterexample, because we have subtracted one from both the number of pigeons and the number of pigeonholes, so the pigeons still outnumber the holes, and they still fit without doubling up. This means that from any counterexample we can produce a strictly smaller counterexample.
But this contradicts the least number principle, which guarantees that any smallest counterexample cannot have a smaller counterexample below it. Therefore, no counterexample can exist, and the pigeonhole principle is proved.
Let us return to the concept of Dedekind finite. Recall that Dedekind's proposal was not an account of the finite numbers as such, that role is played by what we call numerically finite, but rather the notion of Dedekind infinite: a set is Dedekind infinite if it is equinumerous with a proper part of itself. In the Galileo paradox situation, the natural numbers are equinumerous with the perfect squares, which is precisely why that set is Dedekind infinite. The question before us is whether the concept of Dedekind finite, the negation of Dedekind infinite, coincides with numerical finiteness.
The pigeonhole principle tells us that if you have more than N pigeons, you cannot place them into N pigeonholes. In other words, no finite number N is equinumerous with a proper part of itself. It follows that every numerically finite set is Dedekind finite, giving us an implication in one direction. The harder question is whether the converse holds: if a set is not equinumerous with any proper part of itself, why should it be possible to count it off? Where would that counting come from?
To address this, suppose a set X contains a copy of the natural numbers as a subset, that is, it contains distinct points x0, x1, x2, x3, and so on, one for each natural number. Then we can construct an equinumerosity between all of X and a proper part of X. The idea is to shift that embedded copy of the natural numbers: associate each xn with xn+1, and leave every point outside this copy associated with itself. Under this correspondence, nothing maps to x0, so the image is the proper subset X ∖ {x0}. This shows that X is Dedekind infinite.
We therefore have the following equivalence: a set is Dedekind infinite if and only if it contains a countably infinite subset. Now suppose we have an infinite set and wish to build such a subset. We pick an element x0, any element at all. Since the set is infinite, it is not exhausted by this single point, so we pick another element x1 from what remains. Continuing in this way, we can always choose a further element, because at no finite stage will we have exhausted an infinite set. The result is a countably infinite subset, which by the argument above makes the set Dedekind infinite.
The critical question is what it means to say "pick any element you like" at each stage of this process. Whether that picking procedure is mathematically legitimate is precisely what the axiom of choice concerns, and we will devote a full lecture to it. When the axiom of choice holds, if it is part of one's conception of mathematical reality, then Dedekind finiteness and numerical finiteness are equivalent, and both coincide with all the other notions of finiteness we have discussed.
Nevertheless, set theorists have proved by sophisticated arguments that there exist models of the Zermelo–Fraenkel axioms of set theory in which the axiom of choice fails and in which there are infinite sets that are Dedekind finite, sets with no countably infinite subset whatsoever. The two notions are therefore not provably equivalent: Dedekind finite and numerically finite are genuinely distinct concepts in the absence of the axiom of choice. It is worth noting that the full axiom of choice is not required to establish their equivalence; the weaker principle known as countable choice suffices.
Returning to the main question: what does it mean to say that a set is finite? We had many different concepts on offer: Dedekind finite, linearly finite, discretely finite, Tarski finite, numerically finite, and so on. In certain situations we can prove equivalences between these various notions, but one thing we can prove in general is that numerically finite implies all the others.
A set is numerically finite if you can count it off with a finite number, where we have an independently standing concept of finite number derived from Dedekind arithmetic. Precisely because numerical finiteness implies all the other notions of finiteness, and is itself clearly acceptable as a notion of finiteness, there is strong reason to take it as the primary notion, and that is what mathematicians generally do today. If you ask a mathematician what it means for a set to be finite, they will say it means being equinumerous with the set of predecessors of some natural number, which is exactly numerical finiteness.
We want the strongest notion that still has the property that the conventional numbers all count as finite, and that is precisely the notion of numerical finiteness. The concept of numerical finiteness has, in this sense, won the contest over how to define finiteness. What is quite striking, however, is how these different concepts of finiteness come apart from one another once you begin giving up certain foundational principles, such as the axiom of choice.
This third axiom, the induction axiom, if you notice carefully, was not an axiom involving quantifiers only over the numbers themselves. Rather, we were quantifying over sets of numbers. We said: for every set A, if it contains zero and is closed under the successor operation, then A contains all the numbers. This is what is called a second-order assertion. Dedekind arithmetic is a theory expressible in second-order logic, and it is not, in fact, expressible in first-order logic; that is, it cannot be captured by principles that quantify only over individuals. We genuinely need to quantify over sets of individuals in order to attain this categoricity result.
This is a consequence of developments that arose much later in the twentieth century, in model theory and in the study of first-order logic. The Löwenheim–Skolem theorem, for example, tells us that no first-order theory, one expressed by quantifying only over individuals, can properly define the notion of finiteness, because every theory that has a model also has models of arbitrarily large, indeed uncountable, size.
Similarly, the Skolem paradox lies at the core of these difficulties. It shows that if the axioms of set theory are consistent, then there is a countable model of set theory, and that other models of set theory can be mistaken about, and disagree with, the very notion of finiteness. All of these intricate counterexamples reveal just how delicate and difficult it is to express the concept of finiteness in a formal theory.
It is therefore no surprise that our account of the finite numbers in Dedekind's theory required a second-order axiom. Indeed, this is not merely an artifact of Dedekind's particular approach; it is provably necessary for any successful formal definition of numerical finiteness.