Infinity in Dedekind

NOTE: The Dedekind quotes are from Dedekind’s essay “The Nature and Meaning of Numbers”, or in the original German, “Was sind und was sollen die Zahlen?”. The paragraph numbers seem to be consistent across editions so I am not going to provide page references below but I am using the Dover edition, in case it makes a difference.

Introduction: Dedekind’s Influence on the Concept of Infinity

Richard Dedekind, a 19th-century mathematician, revolutionized how we think about infinity by redefining it as a property of sets, independent of numbers, instead of the other way around. His precise definition of infinite sets and his proof of their existence paved the way for the axioms of the modern set theory, offering mathematicians a systematic way to work with the infinite.

Historical Context: From Zeno to Galileo’s Paradox

Zeno may have been the earliest philosopher who thought deeply about infinity without ever directly addressing the concept itself. He has many paradoxes dealing with different counter-intuitive aspects of infinity.¹ In Aristotle’s rephrasing, he says “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.”² The paradoxes of Zeno and others revealed fundamental tensions in our understanding of infinity, challenging philosophers and mathematicians to bring clarity to the concept. Dedekind’s work was a direct response to these challenges, offering a formal and systematic way to define and work with the infinite.

Another particularly confusing paradox comes from Galileo, which bears directly on our discussion. He noted that while there are natural numbers that aren’t squares of other natural numbers—suggesting there are more natural numbers than square numbers—there is a one-to-one correspondence between natural numbers and their squares. This implies that the set of natural numbers and the set of square numbers are the same size, which is counterintuitive. His conclusion was that we simply cannot compare infinite quantities in the same way we compare finite ones. This tension between our intuition about size and the behavior of infinite sets would later prove crucial to how we think about infinity.³

Building upon these foundational puzzles, Richard Dedekind offered a novel way to define and understand infinity mathematically. His approach doesn’t exactly “solve” these paradoxes; after all, they’re paradoxical precisely because they challenge our intuitions, and describing them differently doesn’t make them more intuitive. However, his framework gives us powerful tools for thinking about them systematically, which I will explore in the rest of the post.

Dedekind’s Foundational Approach to Infinity

Before diving into Dedekind’s definition, it’s worth understanding the mathematical context of his time. The late 19th century saw mathematicians struggling to put familiar concepts – like numbers themselves – on rigorous foundations. Dedekind’s essay “The Nature and Meaning of Numbers” (1888) emerged from this intellectual environment. His goal was ambitious: to construct a rigorous foundation for natural numbers using only sets and functions, which required defining infinity without relying on numbers themselves. What makes his approach particularly fascinating is how he proved that infinite sets exist – by looking not at numbers or physical quantities, but by examining the realm of human thought itself.

I will focus on two ideas from the essay: his definition of an infinite set and his proof that infinite sets exist. By exploring these concepts, we’ll uncover how Dedekind resolved long-standing paradoxes of infinity and established a rigorous foundation for understanding infinite sets. The essay provides many other interesting definitions and constructions related to these developments, and I hope to come back to some of these in other posts.

Dedekind’s definition of infinity brings several key innovations. Interestingly, his definition of numbers builds upon his concept of infinity, reversing the typical order of dependence. Moreover, his definition offers a concrete, functional criterion for identifying infinite sets, making it remarkably accessible compared to earlier, more abstract treatments. By creating a clear, functional test for infinite sets, Dedekind transformed an abstract philosophical concept into something mathematically tractable.

His proof that infinite sets exist is even more interesting. He wants to construct the natural numbers using an infinite set, so he needs there to be an infinite set. Where does he just find such a set? Believe it or not, he finds it in his mind. He argues that his thoughts are infinite and provides a proof for it.

Without further ado, I’ll dive into his definition of an infinite set, connect it to what we mean by infinite today, and illustrate it in a few examples. Then, I’ll turn to his proof of the existence of infinite sets, based on his thoughts. I have many questions I’d love to ask Dedekind but instead, I’ll settle for asking them here.⁴ Finally, in the appendix, I’ll share some commentary on this proof from different books on philosophy of mathematics.

Defining Dedekind-Infinite Sets

Let’s delve into his precise definition and see how it aligns with our modern understanding of infinity.

64. Definition. A system $S$ is said to be infinite when it is similar to a proper part of itself; in the contrary case $S$ is said to be a finite system.

Dedekind defines a “system” as what we now call a “set.” When he says two systems are “similar,” he means there exists a bijection between them. His definition of similarity is actually more subtle—he defines similar transformations as what we know today as injective (one-to-one) functions. However, when defining the similarity of two sets, he focuses on the image rather than the codomain. Since functions are necessarily onto their images, whenever he says two sets are similar, we can think of this as meaning there exists a bijection between them. I’ll include his definition of similarity for completeness, if you want to check it out yourself.

Definitions of similar transforms and systems (Click to Expand)

As I noted above, he calls sets “systems”, and he also calls functions “transformations”.

26. Definition. A transformation $\phi$ of a system $S$ is said to be similar or distinct, when to different elements $a, b$ of the system $S$ there always correspond different transforms $a’=\phi(a)$, $b’=\phi(b)$. […]

30. Theorem. The identical transformation of a system is always a similar transformation.

32. Definition. The systems $R, S$ are said to be similar when there exists such a similar transformation $\phi$ of $S$ that $\phi(S)=R$, and therefore $\phi^{-1}(R)=S$. Obviously by (30) every system is similar to itself.

In the entire essay, vast majority of paragraphs are labeled as theorem, a handful as definitions, and a few are broad commentaries without any such label. 32 is a bit interesting because after giving an actual definition, he states a result within the same paragraph. So many lemmata, corollaries, and even remarks are labeled as theorem so I found it kind of surprising that this last sentence didn’t get its own theorem.

From 32, it is clear that the similar systems are indeed sets with bijections between them, as the definition uses the image, and not the codomain.

To understand this definition concretely, let’s visualize it using the set of natural numbers, $N$, and the function $\phi(n) = n + 1$ (which of course is a bijection):

This definition is remarkable both for its creativity and its practicality. In principle, it gives us a concrete condition we can check to determine whether a set is infinite. But how does it compare to our intuitive sense of infinity? Is it too permissive, allowing too many sets to be called infinite, or too restrictive? To explore this, let’s be precise and call sets that are infinite according to Dedekind’s definition “Dedekind-infinite” and those that aren’t “Dedekind-finite.”

One direction is clear: if a set is finite in our usual sense (has $n < \infty$ elements), it must be Dedekind-finite. We can prove this by showing that a bijection cannot map a finite set to a proper subset of itself.⁵

The more challenging question is whether infinite sets are necessarily Dedekind-infinite. This turns out to depend on our axioms – specifically, whether we accept the axiom of choice. Without delving too deeply into set theory, it’s worth noting that without the axiom of choice (but with the other standard axioms of set theory), we can construct infinite sets that are Dedekind-finite.⁶ This is another instance where axiom of choice makes our intuitions work.

Overall, this definition provides a way to think about some paradoxes like Galileo’s Paradox that we visited already and the Infinite Hotel Paradox. According to this definition of infinity, there is no paradox but Cantor will have more to say on how to compare different infinities so I won’t go into that here.

Examples

While Dedekind couldn’t use numbers to prove the existence of infinite sets (since he was trying to construct numbers using infinity), we can use familiar number systems to build our intuition about his definition. For the same reason, he provides almost no concrete examples in his essay, though he must have had some in mind while developing these ideas. Let’s explore some illuminating cases, if you are metaphysically and epistemologically comfortable with using numbers for this sort of thing:

Example 1: Natural Numbers

The natural numbers provide perhaps the most intuitive example of a Dedekind-infinite set. Let’s say $N = \lbrace 1, 2, 3, \ldots \rbrace $ and $\phi(n) = n + 1$ for $n \in N$. $\phi$ is clearly one-to-one and $\phi(N) = \lbrace 2, 3, 4, \ldots \rbrace := N’$. (You can refer to the Figure 1 above for a visualization.)

$N’$ is of course a proper subset of $N$ and $N$ and $N’$ are similar, given that $\phi$ is a one-to-one transform. Thus, $N$ is Dedekind-infinite.

This example leads us naturally to one of the most famous thought experiments about infinity: Hilbert’s Infinite Hotel.

The Infinite Hotel Paradox

Imagine a hotel with infinitely many rooms, one for each natural number. Even when the hotel is completely full, it can still accommodate new guests. How? Simply ask each guest to move to the room numbered one higher than their current room. If $k$ new guests arrive, repeat this process $k$ times. Everyone retains a room, yet we’ve created $k$ empty rooms for our new guests.⁷ A similar process can also accommodate infinitely many new guests; we can send the guest in room $i$ to room $2\times i$, and now we have infinitely many empty rooms. Easy peasy.

Notice that, under the definition of infinity, Hilbert’s Infinite Hotel is not much of a paradox at all. It is almost the definition of the infinity. Of course it is counterintuitive when you first hear about it but that is only because our mathematical intuition is mostly built on finite quantities. You can imagine how a similar argument might explain (explain away) Galileo’s paradox as well.

Example 2: $[0, 1]$

This one is perhaps even simpler. Let’s say we have a set of real numbers this time, namely, $I = [0, 2]$ and the function $\phi(x) = x/2 $ for each $x \in I$. Again, $\phi$ is clearly invertible and one-to-one. So, $I’ := \phi(I) = [0, 1]$ and $I$ are similar and $I’$ is a proper subset of $I$. Thus, $[0, 2]$ is also Dedekind-infinite.

Example 3: $\lbrace 1 \rbrace$

For contrast, let’s examine a finite case.

Consider $\lbrace 1 \rbrace$, which has only one proper subset: the empty set. No function can map a non-empty set to the empty set – for any function $\phi$, we must have $\phi(\lbrace 1 \rbrace) = \lbrace \phi(1) \rbrace$.⁸ Therefore, $\lbrace 1 \rbrace$ cannot be similar to any of its proper subsets, making it Dedekind-finite.

Of course, there is nothing special about 1. The same argument would hold for any singleton set. In fact, by an inductive proof, we can extend the argument to cover sets of the form $\lbrace 1, 2, 3, \ldots, n \rbrace$ for any $n < \infty $. Once again, note that we are not using any property of the elements (and we wouldn’t use them in the induction either). So, the argument would work for any set with $n < \infty$ elements.⁹

These examples illustrate how Dedekind’s definition captures our intuitive understanding of infinite sets, while also providing a rigorous mathematical framework to identify and work with them.

Proving the Existence of Infinite Sets: Dedekind’s Thought Experiment

Having understood what Dedekind means by infinity, we arrive at the crucial question: How do we know infinite sets exist at all? Dedekind’s proof of this fact is one of the most fascinating—and perhaps controversial—arguments you can find in a mathematical text. Rather than constructing an infinite mathematical object, he turns to something far more immediate: the contents of his own mind. Here’s his proof, which I’ll break down step by step:

66. Theorem. There exist infinite systems.
    Proof. My own realm of thoughts, i.e., the totality $S$ of all things, which can be objects of my thought, is infinite. For if $s$ signifies an element of $S$, then is the thought $s’$, that $s$ can be object of my thought, itself an element of $S$. If we regard this as transform $\phi(s)$ of the element $s$ then has the transformation $\phi$ of $S$, thus determined, the property that the transform $S’$ [$S’ = \phi(S)$] is part of $S$; and $S’$ is certainly proper part of $S$, because there are elements in $S$ (e.g., my own ego) which are different from such thought $s’$ and therefore are not contained in $S’$. Finally it is clear that if $a, b$ are different elements of $S$, their transforms $a’, b’$ are also different, that therefore the transformation $\phi$ is a distinct (similar) transformation (26). Hence $S$ is infinite, which was to be proved.

This was such an unexpected turn of events for me that I thought it was an inside joke between him and Cantor or something like that but it really is his proof that infinite sets exist. I guess, it is not too different from cogito, ergo sum but given the time difference with Descartes, it is not what I expected from Dedekind. I have more questions than answers and some of my criticism is unfair because I have the benefit of over a century of development. Before getting into it, let me express the proof more compactly so I can refer back to its parts easily.

If we let $S$ be the set of Dedekind’s thoughts, and define $\phi(s)$ as the (potential)¹⁰ thought “$s$ can be object of my thought” (let’s call this $s’$), the proof makes three key claims but I add what I consider to be a hidden claim:

0. (His) thoughts exist.
1. For any thought $s$, the corresponding thought $s’$ is also in $S$ (closure under $\phi$)
2. The set $S’$ of all such “second-order” thoughts is a proper subset of $S$.
3. The transformation $\phi$ is one-to-one.

Each of these claims deserves careful scrutiny.

Step 0. Do thoughts exist?

The answer to this question of course depends on our metaphysical inclinations and we cannot decide it in this post.¹¹ However, if Dedekind was going to use such an argument, isn’t there a more direct way of doing it? In what sense, if at all, this assumption is better than the axiom of infinity from ZF? Why did the objects of the proof have to be thoughts? Why did they have to be his thoughts, as opposed to just thoughts?

Step 1. Is it true that $s’ = \phi(s)\in S$ for each $s\in S$?

I don’t think we can answer this question within Dedekind’s mind but here are some of my own thoughts.¹²

• It implies a kind of logical omniscience. If I can have the thought $s$, I can have the thought ‘$s$ can be object of my thought’. Is this necessarily true? I could imagine having unconscious thoughts; thoughts that I can have without having any conception that it can be the object of my thought.
• $s’ = \phi(s)\in S$ for each $s\in S$ is representing a recursion. We are assuming that the set of thoughts is closed under recursion defined by $\phi$: Dedekind starts with a set of thoughts and generates new thoughts that express the idea that existing thoughts are potential thoughts of the mind. I am, once again, not convinced that this is a better (or cheaper) way to “gain” infinity than the axiom of infinity.
• There is also a Godelian intuition against this argument. If we model our minds as formal systems and our thoughts as the theorems (and axioms) of the system, this would imply that there are thoughts that cannot be derived from other thoughts within our minds, as long as our minds are sufficiently rich formal systems. So, we can have a thought $s$, without being able to obtain it from other thoughts. Notice that for this to work, we need to change the direction of theorem-generation. Just above, I mentioned how Dedekind essentially assumes that the mind is closed under the recursion of thoughts given by $\phi$. Here, we would need to have a theorem-generation rule that says something like “if ‘$s$ can be object of my thought’ is a thought, then $s$ is a thought”. This sounds a lot like $\phi^{-1}$. Of course, we need to define what “sufficiently rich” means in this context more carefully. I think there is a formalization that makes this intuition work out in such a way.

Step 2. Is $S’$ a proper subset of $S$?

My objection to this will mostly come from the next subsection so I won’t go into any details here. However, one thing about this problem that is not covered in the next subsection is the following possibility. Suppose I have the thought $s$, “There is a dog on my desk.” and assuming that the idea of the thought, $s’$, is itself a thought, “‘There is a dog on my desk.’ can be the object of my thought.” Do we know that $s’$ is different from the thought “delenda est Carthago”?

Step 3. Is $\phi$ a one-to-one function?

If we apply $\phi$ repeatedly—say, a million times—can we confidently assert that each application produces a distinct new thought? Or does the process eventually collapse into repetition, producing “higher-order” thoughts that are indistinguishable from previous iterations?

In the following quote, Frege makes fun of inattention, in the context of an empiricist approach but I think it applies here unironically.¹³

Inattention is a very strong lye; it must not be applied at too great a concentration, so that everything does not dissolve, and likewise not too dilute, so that it effects a sufficient change in the things. Thus it is a question of getting the right degree of dilution; this is difficult to manage, and I at any rate have never succeeded.

This is a completely out of context and has nothing to do with what he meant but I think it works very well here. Do we really pay attention to million and first recursion? Can we even do that, even if we wanted to?

Critiques and Philosophical Considerations

Here, I’ll share some quotes that illustrate how others interpret Dedekind’s proof, without much commentary from myself. I read these after writing the post before this section so they are more or less independent of my own thoughts. However, I noticed that many of these sources also had problems with similar aspects of the proof.

I don’t claim these are representative views on the proof; I restricted my search to books that I have in my physical library to make sure I would finish this post in finite time.

Introduction to Mathematical Philosophy by Bertrand Russell

The following is from pg.s 138-9 of my copy of the book (Routledge, 2004). His objections seem to be mostly related to what I call Step 1 above, and his arguments turned out to be parallel to what I stated above as well.

There is an argument employed by both Bolzano¹⁴ and Dedekind¹⁵ to prove the existence of reflexive classes. The argument, in brief, is this: An object is not identical with the idea of the object, but there is (at least in the realm of being) an idea of any object. The relation of an object to the idea of it is one-one, and ideas are only some among objects. Hence the relation “idea of” constitutes a reflexion of the whole class of objects into a part of itself, namely, into that part which consists of ideas. Accordingly, the class of objects and the class of ideas are both infinite. This argument is interesting, not only on its own account, but because the mistakes in it (or what I judge to be mistakes) are of a kind which it is instructive to note. The main error consists in assuming that there is an idea of every object. It is, of course, exceedingly difficult to decide what is meant by an “idea”; but let us assume that we know. We are then to suppose that, starting (say) with Socrates, there is the idea of Socrates, and then the idea of the idea of Socrates, and so on ad inf. Now it is plain that this is not the case in the sense that all these ideas have actual empirical existence in people’s minds. Beyond the third or fourth stage they become mythical. If the argument is to be upheld, the “ideas” intended must be Platonic ideas laid up in heaven, for certainly they are not on earth. But then it at once becomes doubtful whether there are such ideas. If we are to know that there are, it must be on the basis of some logical theory, proving that it is necessary to a thing that there should be an idea of it. We certainly cannot obtain this result empirically, or apply it, as Dedekind does, to “meine Gedankenwelt”—the world of my thoughts.

In essence, Russell questions the validity of Dedekind’s use of thoughts as mathematical objects. He argues that assuming every object has a corresponding idea leads to problematic conclusions, especially when considering the limitations of human cognition and the nature of abstract ideas.

The Logicism of Frege, Dedekind, and Russell by Demopoulos and Clark

This article appeared as a chapter in The Oxford Handbook of Philosophy of Mathematics and Logic (edited by Stewart Shapiro). There are many references to Dedekind’s work in the book but this chapter is the most relevant to our concerns. Here is what the authors say on the proof of the theorem 66 (pg.s 153-4 in my paperback edition of the handbook):

The success of Dedekind’s methodology hinges on the existence of an infinite system. As Dummett ([1991 a], pp. 49ff.) has observed, Dedekind sought to secure the existence of such a system by the provision of a concrete example, one from which the concept of number could be viewed as having been abstracted in the manner just reviewed. It would then follow by his categoricity theorem (theorem 132) that the peculiarities of the starting point of the construction can be discounted, and the generality of the construction of the infinite system secured. This is the goal of theorem 66, whose proof runs as follows: [they quote the proof above verbatim here.]

Following Frege, we may say that for Dedekind the realm of thoughts is an objective one, one that exists independently of us. Our access to this realm is taken for granted, since thoughts are transparent to our reason. The realm of thoughts yields an exemplar of an infinite system because the relation, $x$ is an object of my thought $y$, is a one-one function which maps a subsystem of objects of my thought onto a proper part of itself. From this exemplar we abstract the concept of an infinite system, and thus of number. For Dedekind’s proof to succeed, it is essential that thoughts be proper objects—arguments to concepts of first level, in fact to the concept $x$ is an object of my thought, one that holds of thoughts and of objects which are not thoughts. Dedekind says very little about the nature of thoughts, so it can hardly be urged against him that his account rests on an inconsistent theory of thoughts. But were we to regard Dedekind’s tacit theory of thoughts as essentially Frege’s—together with the further condition that thoughts are proper objects—the concept/thought paradox would be an obstacle to the success of his account of our knowledge of the infinity of the numbers: if the proof of theorem 66 fails, we lack an exemplar from which the concept of number can be abstracted and cannot be said to have an account of how we come to know the numbers’ most salient property.

This is also consistent with the way I read the proof. It is an original approach but it may rest on shaky philosophical (and even shakier mathematical) ground. Without a consistent and paradox-free theory of thoughts as mathematical objects, the proof’s effectiveness is questionable.

Infinity and the Mind by Rudy Rucker

[…] Indeed, in 1887 Cantor’s friend, Richard Dedekind, published a proof that the Mindscape is infinite, where Dedekind’s word for Mindscape was Gedanken-welt, meaning thought-world.

Dedekind’s argument for the infinitude of the Mindscape was that if $s$ is a thought, then so is “$s$ is a possible thought,” so that if $s$ is some rational non-self-representative thought, then each member of the infinite sequence $$\lbrace s, s \text{ is a possible thought}, s \text{ is a possible thought is a possible thought }, \ldots \rbrace$$ will be in the Mindscape, which must, therefore, be infinite.

[…]

Dedekind modelled his argument after an argument that appears in Bernard Bolzano’s Paradoxes of the Infinite (ca. 1840): “The class of all true propositions is easily seen to be infinite. For if we fix our attention upon any truth taken at random $\ldots$, and label it $A$, we find that the proposition conveyed by the words ‘$A$ is true’ is distinct from the proposition A itself$\ldots$”

So we can see that the Mindscape, the class of all sets, and the class of all true propositions are all infinite. Does this guarantee that infinite objects exist? Not really. For a case can be made for the pluralist claim that the Mindscape, the class of all sets, and the class of all true propositions do not exist as objects, as unities, as finished things.

[…]

There is a highly relevant passage in a letter Cantor wrote to Dedekind in 1905: “A multiplicity can be such that the assumption that all its elements ‘are together’ leads to a contradiction, so that it is impossible to conceive of the multiplicity as a unity, as ‘one finished thing.’ Such multiplicities I call absolutely infinite or inconsistent multiplicities. As we can readily see, the ’totality of everything thinkable,’ for example, is such a multiplicity$\ldots$”

Again, the reason that it would be a contradiction if the collection of all rational thoughts were a rational thought Τ is that then Τ would be a member of itself, violating the rationality of Τ (where “rational” means non-self-representative).

Rucker suggests that Dedekind’s proof might inadvertently rely on constructing a set that is too “big” to exist without contradiction. The notion of the Mindscape as an infinite set is appealing but may not withstand rigorous logical scrutiny due to these inherent paradoxes. I didn’t notice how Russell’s Paradox might have something to do with this proof until reading this text, although, of course, it connects back to the Godelian arguments I made at some level.

Conclusion: Dedekind’s Lasting Legacy in Foundations of Mathematics

Dedekind’s formal definition of infinity was a turning point in how mathematicians conceptualize infinite sets. By grounding infinity in the existence of a set that can be put into a one-to-one correspondence with a proper subset of itself, Dedekind’s approach provided a rigorous framework that underpins modern set theory. While subsequent thinkers have critiqued and refined Dedekind’s ideas, his influence remains foundational to modern mathematics. His innovative approach to infinity continues to shape how we conceptualize and work with the infinite in both philosophy and set theory.