-1
$\begingroup$

My understanding is that in elementary set theory, the natural numbers are defined so that $0 = \emptyset$ and $n+1 = n \cup \{ n \}$. I understand that this gives us some very pleasant properties that let us prove the Peano Axioms and define the ordinals and cardinals. I feel I have a good grasp of how this definition works on a technical level.

However, I don't feel I understand the intuition or practical motivation for this definition. To put it crudely, I think, "I have three apples," makes sense, but "I have $\{ \emptyset, \{ \emptyset \}, \{ \emptyset, \{ \emptyset \} \} \}$ apples," is surely nonsense! What is the intuitive connection here?

To finesse my question a bit more, my current understanding is that the sets represented by $0, 1, \dots \omega$ do not actually have any concrete relationship to the natural numbers as used in number theory or conventional, everyday usage. Instead, it seems to me that the sets are indeed important to elementary set theory, and labeling them with the same symbols we use for the conventional natural numbers is a convenient notation.

But I can't help noticing that discussions of set theoretic natural numbers consistently phrase it like, "The natural numbers are defined as," not, "The natural numbers represent/correspond to the sets defined thusly." So it seems there is a deeper connection, and I'm worried I'm missing a fundamental piece of understanding.

$\endgroup$
9
  • $\begingroup$ Not very rigorously speaking, in ZFC, every object should be constructed "from" $\varnothing$, since we "can" define $\varnothing$: $\varnothing=\{x: x\ne x\}$, and therefore $\{\varnothing\}$, $\{\{\varnothing\}, \varnothing\}$ etc can de defined. But we can not define, for instance, what $1$ is unless we define $1=\{\varnothing\}$. $\endgroup$
    – Asigan
    Commented May 10 at 9:03
  • $\begingroup$ Related: math.stackexchange.com/questions/2854103 $\endgroup$
    – Asigan
    Commented May 10 at 9:08
  • 1
    $\begingroup$ Also related: math.stackexchange.com/questions/68659 The natural numbers can be defined this way and then proved to satisfy Peano's "axioms". You can then forget about their set-theoretic definition and only use Peano's axioms and their consequences "in number theory or conventional, everyday usage". $\endgroup$
    – Anne Bauval
    Commented May 10 at 9:15
  • 1
    $\begingroup$ For the record, in the von Neumann definition of natural numbers (which is the one you're referring to), $3=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$, whereas $\{\emptyset,\{\emptyset,\{\emptyset\}\}\}\notin\omega$. One interesting feature of this definition is that $n=\{x\in\Bbb \omega\,\:\, 0\le x<n\}$, therefore $n$ has $n$ elements. Of course, the unspoken rule is that the set-theoretic definition of natural numbers should not interfere with the scientific aspects of mathematics. $\endgroup$
    – Sassatelli Giulio
    Commented May 10 at 9:34
  • 2
    $\begingroup$ Your apples biject with the elements of $\emptyset^{+++}$, where $n^+:=n\cup\{n\}$. But natural numbers mean whatever Peano says, not what set theory says, because naturals aren't the sets representing them in a model of Peano. That way lies madness. $\endgroup$
    – J.G.
    Commented May 10 at 9:47

1 Answer 1

Reset to default
0
$\begingroup$

After reading the comments in response to my question and the linked answers, here's the understanding I have developed. I'm posting it as an answer because I think it's more detail than makes sense in a comment.

The understanding I gave in my question is largely correct: The sets $0, 1, \dots \omega$ are not the same thing as the natural numbers in a literal sense. Instead, they're sets of special interest to ZF set theory that are used often enough that it is convenient to label them with the natural numbers. "I have $\{ \emptyset, \{ \emptyset \}, \{ \emptyset, \{ \emptyset \} \} \}$ apples" is, indeed, nonsense.

But what I was missing is just how incredibly useful those sets are. In particular, they have the very compelling property that, for anything we can prove about the natural numbers, there is a corresponding proof that the inductive sets have an analogous property. As a result, if we care to, we can rerwite any proof that presumes the existence of the natural numbers so that it only assumes the ZF axioms.

Philosophically, this is wildly important. It means that there is a fundamental sense in which the natural numbers exist unto themselves and must behave in certain ways. Practically, this is largely useless. And that's ok! In most applied contexts, whether the natural numbers exist intrinsically, are a product of human invention, or something else isn't important - they can still be used to model and make predictions and whatever just fine. But boy howdy, it is existentially reassuring to have a solid philosophical foundation for mathematics.

Based on this view, here are a few ways of understanding the intuition behind, "I have three apples," in the context of set theory. Each of them is valid and useful, and which one I prefer to focus on depends on the context I am in.

  • There is not, in fact, a conventionally useful relationship between the three apples and $\{ \emptyset, \{ \emptyset \}, \{ \emptyset, \{ \emptyset \} \} \}$. It does not help in most practical contexts to try to think about the set, so don't bother!
  • There is a bijection between each of the three apples and the sets labeled $1, 2,$ and $3$. This bijection doesn't necessarily have a practical meaning, but it exists. And this existence is guaranteed by our construction of the sets in question.
  • The cardinality of the set labeled $3$ is the same as the cardinality of the set containing my apples. The ordinals corresponding to the sets $1, 2$ and $3$ correspond to the ordinalities of the three apples within the set containing my apples.
  • Due to the Peano axioms, the fact that there is a bijection between my apples and the natural number sets means that there is a fundamental reason why, for instance, adding two more apples would mean I have five, or that having three apples means I have more than two. And it's reassuring that there's a strong philosophical reason for those things to be true unto themselves (at least inasmuch as ZF can be assumed to be largely consistent).
  • The construction of the natural numbers allows us to reason sensibly and rigorously about infinite quantities. This isn't always practically useful, but when it is, it saves us from a whole lot of crackpottery, ill-defined notions, and results that seem correct at first glance but fall apart under scrutiny.
$\endgroup$
1
  • 1
    $\begingroup$ If you're familiar with programming, an analogy which I like (I got it from others here): the Peano axioms are like an abstract interface, and the object $\omega$ of ZFC is a particular implementation. And even if you decide to take $\omega$ of ZFC as your "reference implementation" in some situations, it's still good to keep the abstract interface in mind. $\endgroup$
    – Daniel Schepler
    Commented May 10 at 21:18

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .