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I recently learned about the von-Neumann construction of the naturals, where $0:=\emptyset$ and $n:=\{0,1,2,...,n-1\}$. I get the idea - if we define $\mathbb N$ as the smallest inductive set containing the empty set, then it satisfies the Peano axioms and the von-Neuman construction seems natural.

However this just doesn't sit right with me. If we do analysis, linear algebra etc. we treat $\emptyset$ and $0$ as fundamentally different concepts. Where does this change in thought happen? For example if we just do analysis and don't think about ZFC, something like "Let $x\in 0$" would be to say the least weird (I don't want to say wrong, but if wouldn't want to write that in an exam). Or no one would write "Let $x=\emptyset+1$". So at the very least there is some kind of difference.

Edit: The comments aren't really trying to answer my question (marked in bold, and complemented by examples like $x\in0$. Yes I know everything in ZFC is a set, which includes functions, numbers, cartesian products and whatnot. And as I've written it makes sense (for me too at first) to just define $0:=\emptyset$, as perfectly fulfills the the Peano axioms. That's everything I've written. tl;dr I would be thankful if you could focus on my question specifically and not talk around the bush

bochner.martinelli
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    It doesn't matter, its just a construction – Jakobian Apr 23 '25 at 16:37
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    You are confused by the intuitive idea of a "type." In set theory, unfortunately, "set" is the only thing we have for a type. We never use $\emptyset=0$ when studying actual numbers, and, we could have defined the natural numbers otherwise in set theorem, but the nice thing about this encoding is that the cardinality of the set encoding the natural number $n$ is, intuitively, $n.$ – Thomas Andrews Apr 23 '25 at 16:40
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    It is not only very convenient to take $0$ to be the empty set, but also follows from the intention behind von Neumann's definition of ordinals; We want to identify finite ordinals with natural numbers, and an ordinal with the set of its proper initial segments. $0$ has no proper initial segment, so it must correspond to the empty set. – Hanul Jeon Apr 23 '25 at 16:40
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    Other examples are that set theory defines "functions" as sets, when, intuitively, they are a different "type" of thing. Ordered pairs are defined as sets, even though we treat them as different types. – Thomas Andrews Apr 23 '25 at 16:41
  • @TankutBeygu I don't think this question duplicates with what you mentioned because the OP asks about the ordinals, and your linked question is about cardinals. – Hanul Jeon Apr 23 '25 at 16:44
  • @HanulJeon That brings to mind a question I've had: Who was the first mathematician to include $0$ as an ordinal? Originally, ordinals were like "first, second, third." It was precisely because of this that people for millennia didn't see zero as a number: Counting a set involved ordering the set and realize "this last bean is the tenth bean, so I have $10$ beans." There is no zeroth place in a race. Was it Von Neumann who first codified this change? – Thomas Andrews Apr 23 '25 at 16:45
  • @ThomasAndrews I have no idea; My knowledge is that human being took around 2000 years to realize zero is a natural number. We can say zero is an ordinal once we realize zero is a natural number. – Hanul Jeon Apr 23 '25 at 16:47
  • @HanulJeon Technically, it took a long time to realize $0$ was a number. Took a couple centuries longer to call it a natural number, but that is just a choice. You can due Peano starting with $0$ or $1,$ and whether $0$ is a natural number is just a definition. Peano himself started with $1.$ – Thomas Andrews Apr 23 '25 at 16:49
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    @ThomasAndrews I strongly disagree with that "whether zero is a natural number is just a definition." It is not just a definition, but the most natural definition we can take. In set theoretic perspective, we want to identify finite cardinals, finite ordinals and natural numbers, otherwise we must exclude the least number (zero) from the natural numbers that is weird. I do not see any reason to exclude a single element otherwise can form an initial segment over the class of ordinals. (I also think your question is off-topic.) – Hanul Jeon Apr 23 '25 at 16:52
  • In regard to the original question, it is worth realizing that natural numbers are not, technically, integers, and integers are not technically rational numbers, and rational numbers are not real numbers, but we use $0$ for the "name" of elements in all of these sets. The key isn't that $0$ is the set-theoretic definition of some version of the natural numbers $0.$ We could use $0_{ord}$ rather than $0,$ but it really doesn't matter. – Thomas Andrews Apr 23 '25 at 16:58
  • I think you understand that people don't study linear algebra by building it up from ZFC axioms. We don't have the lifespan for that. I don't think anyone outside of logic is really walking around thinking about the number $0$ as the empty set per ZFC. At some point you see that construction, think "oh neat," and then you move on. I'm sure it happens at different moments for different people. In fact, it usually happens in reverse: people have a number-sense for $0$ decades before they ponder ZFC. – Randall Apr 23 '25 at 17:01
  • But the meaning of ordinal was for generations defined so there was no zero ordinal. It wasn't until modern times we refined ordinal. We could call $\emptyset$ equal to $1$ as a model for the natural numbers, but I agree, in the Von Neumann model, it is much more convenient to call $0$ the empty ordinal. But the relation between cardinal and ordinal was complicated, historically, and the fact that you can define cardinality as an ordinal was not at all clear. I agree, in set theory it might make sense to call $0$ and ordinal, but that is more a choice for convenience. – Thomas Andrews Apr 23 '25 at 17:04
  • In particular, the usual set theory definition of cardinal as ordinalsrequires either the axiom of choice or so,e equivalent, or else the cardinality of sets is not well-defined. The fact that ZFC lets us define a cardinal as a set is a nice little miracle, but , without choice, we can only define what it means for two sets to have the same cardinal. A cardinal is then not an object in set theory, it is a meta equivalence relation on all sets. @HanulJeon – Thomas Andrews Apr 23 '25 at 17:15
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    It might be more helpful to think of this construction and the accompanying properties, as proving that ZFC has a model of PA. It is perfectly fine if you don’t want to work in this model (who does?) but it is nice that ZFC sort of “contains” PA. – Malady Apr 23 '25 at 18:14
  • How can $+$ be addition in the real numbers, but also in a finite field? – Asaf Karagila Apr 23 '25 at 19:50
  • Mates, this discussion ends up at Robert Kaplan's The Nothing that is: A Natural History of Zero or something like that. – Tankut Beygu Apr 23 '25 at 20:08
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    This phenomenon is referred to by some as "junk theorems". In my opinion, the real way to resolve the issue is through a the philosophy of mathematics and what it means to do maths (in particular, that this has little to do with technicalities related to formal ZFC-proofs). You might like to read this, this, this. – Izaak van Dongen Apr 24 '25 at 17:25

5 Answers5

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In general in mathematics we don't really care what something is, but rather what can be done with it. Maybe my "natural numbers" are not the same as yours. It doesn't matter, as long as every true statement about my natural numbers translates to a true statement about yours and vice versa.

The von Neumann construction produces a set and a "successor" function that satisfy the Peano axioms for the natural numbers. Whether the members of this set are the "actual" natural numbers is irrelevant. The point is simply that there is such a construction, so that arithmetic is in principle founded on set theory. Once it has been constructed, nobody actually uses this construction in practice.

Robert Israel
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  • I agree with everything you wrote. I have no problem with defining $0:=\emptyset$. But you can't deny that further in mathematics we treat those differently. Why? – bochner.martinelli Apr 23 '25 at 17:07
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    Because it is just an encoding. There are a lot of meanings for the symbol $0.$ This is one of many different things we call $0.$ If we had to write $0_{ord}$ for the ordinal $0,$ $0_{\mathbb N}$ for the natural number, $0_{\mathbb Z},$ $0_{\mathbb Q},$ $0_{\mathbb R},$ $0_{\mathbb C}.$ $0_{\mathbb R^n}$ for all the different meanings we give $0,$ we'd hate mathematics. They are all things we call $0,$ and we hope we distinguish which one we are talking about by context. But formally, we'd need different symbols for all of these. @bochner.martinelli – Thomas Andrews Apr 23 '25 at 17:23
  • In theory, you could define a "natural number set object" as a triple, $(N,0_N,s)$ with $0_N\in N$ and function $s:N\to N$ with certain properties. This is roughly how it is defined in category theory. But it has a problem: A function is not a first class thing in set theory. The only things we can discuss in set theory are sets. We can define what we mean by function in set theory, but we don't like axioms that depends on definitions, and there is no way to prove from the other axioms that there is such a triple. Even the idea of an ordered triple requires a definition. @bochner.martinel – Thomas Andrews Apr 23 '25 at 17:38
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Where does this change in thought happen?

This is backwards. Zero and the empty set are fundamentally different concepts. The "change in thought" is when we realize we can define things behaving like natural numbers in set theory.

There are many things we could take away from this insight.

  • Perhaps we can just view it as a curiosity.

  • Historically, many people thought it a good idea to unify all of mathematics under one foundational system, and this shows that set theory, as a foundational system, can "handle" the study of the natural numbers. Indeed, it can encode much more mathematics than just the natural numbers... practically anything, but it's somewhat of a controversial question how useful it is to view all of mathematics as founded in set theory, now that collective worry over foundational issues has waned.

  • In the study of set theory for its own sake, the Von-Neumann construction of the natural numbers (and more generally the ordinals) is incredibly important, as it forms a sort of "backbone" to the set theoretical universe, and for many other reasons.

None of this reflects any tension with the fact that mathematicians ordinarily think of zero and the empty set as different things. They are different things, even if there are some conceptual similarities (the empty set contains zero things), and we find it useful to define a version of the natural numbers in set theory that makes this identification.

spaceisdarkgreen
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One key point: ZFC (or any other set theory) isn’t necessarily “the foundation” of mathematics (see this excellent answer).

We’re not required to worry about what natural numbers “really are.” They are an abstraction: some collection of things with a successor relation that obeys the Peano axioms. It doesn’t matter whether the underlying “things” being counted are marbles, or apples, or abstract “points” (e.g. in a game of rugby), or sets (e.g. von Neumann odinals)

When mathematics is done within a set-theoretic foundation, we are in effect positing a particular “representation” or “model” of mathematical entities (natural numbers are von Neumann ordinals, ordered pairs are sets of the form $\{\{x\}, \{x, y\}\}$ and so on). Conceiving of all mathematics as living within a single “grand unified theory” like ZFC has its uses (such as for thinking about independence results like the Continuum Hypothesis), but that doesn’t mean we have to do math that way. The results of elementary number theory don’t change depending on whether we’re counting marbles or sets.

  • Auxiliary point: Even when working within a set-theoretic foundation, one should (I think) take care not to “reach around the abstraction” (to borrow a phrase from computer science terminology). For example, when doing number theory it doesn’t really make sense to say $1 \in 3$, even though “under the hood” it’s true (assuming ZFC foundation) that $\{\emptyset\} \in \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}$
NikS
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    My favorite way of "reaching around the abstraction" is the statement that $7$ is a topology on $6$. – Misha Lavrov Apr 24 '25 at 01:24
  • @MishaLavrov : Ha ha, good one (“good” in the sense of “entertaining” but presumably not “recommended”…) – NikS Apr 24 '25 at 01:32
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The change of thought happens because, in fields of math other than set theory, the theorems are not theorems about set-theoretic constructions, but about what follows from the foundations of that field.

Consider the identity $2+2=4$. This is a theorem of number theory, but its content is not the statement that $$\{\varnothing, \{\varnothing\}\} + \{\varnothing, \{\varnothing\}\} = \{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\},\{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\}\}.$$ The content of the theorem is that, in any mathematical object that obeys the laws of the natural numbers, it is true that $2+2=4$.

In less elementary number theory, we may reason about both natural numbers and sets: sets of natural numbers, or sets of sets of ordered pairs of natural numbers, and so on. But even in that setting, $0=\varnothing$ will not be a theorem. It is not true in any mathematical object that obeys the laws of natural numbers that $0 = \varnothing$; it is simply true in one popular construction of the natural numbers.

The same goes for any theorem of analysis, linear algebra, and so on. Their statements may use numbers, ordered pairs, functions, and so on that have set-theoretic constructions. But the theorems are not making claims about those set-theoretic constructions. A theorem about functions from $\mathbb R$ to $\mathbb R$ is a theorem that is true whenever we have an "object that acts like a function" to and from a "set that acts like $\mathbb R$".

The role of the set-theoretic construction is to demonstrate that - assuming nothing more than the axioms of ZFC - none of these theorems are vacuous. It would be vacuously and contentlessly true that $2+2=4$ if it turned out that there is in fact no mathematical object that obeyed the laws of natural numbers. By constructing $\mathbb N$ in some way, we confirm that there's at least one possible object that those theorems could be talking about.

Misha Lavrov
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The change in thought happens when you move from the language of set theory to another language.

Most people don't write mathematics in the Language of Set Theory, they write in the language of Fields, or Groups, or whatever, or some Hybrid Language. If they are writing in a first order language at all.

That's why you find it weird. You are mishmashing different formal languages,or working informally, outside of any defined language of mathematics.

One can make an anology to binary, and more abstract programming languages.

Adding strings of 1's and 0's to get arithmetic, might seem very contrived.

And of course no human would solve a complicated integral using binary by hand.

So we use higher order labguages to do that, when we code.

Set Theory is the binary Code, and The Language of Fields, or Groups or Arithmetic is a higher Order Language.

Michael Carey
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