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For some context I did a course on set theory where I was taught about ZFC, and the construction of the natural numbers, integers etc.

I think I was far too young to take the course because it’s left me really confused and questioning about everything.

You guys might not be qualified for this kind of thing but I don’t feel real anymore (as strange as it sounds) and I feel dissociated from everything.

The construction of the integers and rationals felt easy and intuitive after the natural numbers were defined. However, when I think about what a natural number is, I just don’t know anymore.

I know in set theory they are symbols defined recursively in terms of empty sets, but I just don’t understand how they are used in such broad contexts and take so many forms when they are just sets. The bijection (counting) elements of a set feels strange to me, since I just feel uncomfortable now saying a set has 3 elements.

Mikhail Katz
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Frazer
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    The truth is, first courses in set theory often also teach you one particular kind of mathematical philosophy. Though the maths they teach is indisputable, there are valid philosophical standpoints totally different from what they teach. – Trebor Dec 14 '23 at 12:58
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    The usage of the natural numbers and all the properties you know and love came first. The definition involving set theory and emptysets and such came after and was specifically constructed in such a way so as to make sure that natural numbers had all the properties we wanted and were used to, but so that it was consistent and able to be unambiguously defined "from the ground up" using set theory since we want to be able to say mathematics is consistent. – JMoravitz Dec 14 '23 at 13:00
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    It is perfectly fine in your normal life to think of numbers as you always have. We only defined it with emptysets and the like to prove the point that we could define it that way if asked, nothing more. – JMoravitz Dec 14 '23 at 13:01
  • See Natural number: "the natural numbers are the numbers 1, 2, 3, etc., possibly including 0 as well." Various approaches: the axiomatic one (see Peano axioms) and the "construction" one (see set-theory). – Mauro ALLEGRANZA Dec 14 '23 at 13:01
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    I think the question would be far more at home over at philosophy.stackexchange – tomasz Dec 14 '23 at 13:05
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    You are in good company: Natural numbers were created by God, everything else is the work of men — Kronecker – Lee Mosher Dec 14 '23 at 13:41
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    Your question would be better suited to the philosophy stack exchange, where similar questions have been asked before. See this question for instance. Regarding you feeling derealization and discomfort, try not to sweat it :) There is no obvious answer for what mathematical objects are, nor even a universally accepted answer in research. So you're not the first to feel that way. And remember that an object is not necessarily how we represent it. The number "two" is not just its encoding in ZFC. – Jam Dec 14 '23 at 15:06
  • Math consists of establishing definitions and then studying their logical consequences - it's not about what things "really are". The concerns you've expressed in this post are too vague to be answerable with math. Do you understand the definition of natural numbers? Is there a property they have that confuses you? – Karl Dec 15 '23 at 02:52
  • @MikhailKatz I have come to answer I am satisfied with. It baisically says numbers are just part of the English language and I accept that I can understand and interpret English. 2 doesn’t really have meaning by itself it’s just a symbol of the language we speak, but when used in context my brain just “understands” it. Like I don’t question why I can understand the meaning of everything I am writing, I just accept. It’s taken centuries of evolution. See my answer here: https://philosophy.stackexchange.com/a/106397/70540 – Frazer Dec 15 '23 at 12:23
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    Since the question has been closed, I can't give this as an answer, but I have always found the logicist account (in the vein of Frege and Russell-Whitehead) of the natural numbers compelling: Cardinal numbers are equinumerosity classes and the natural numbers are the finite cardinal numbers (i.e., those in the smallest inductive class of cardinal numbers). This is in some sense equivalent to the categorical specification, but I find it far more philosophically satisfying because it explains why the natural numbers are a natural object and why they're related to counting things. – James E Hanson Dec 16 '23 at 15:07
  • @FraserPye, what you described above are the metalanguage integers, which are not the same as the formal-language (or "object-language") integers. Formal constructions, such as the set-theoretic one starting from the empty set, give a larger collection than the metalanguage integers. In any formal construction, there are going to be "ideal" elements that may not have been originally intended. – Mikhail Katz Dec 17 '23 at 08:44
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    @LeeMosher, Kronecker is routinely misquoted in this context; see https://hsm.stackexchange.com/questions/3770/did-kronecker-attribute-immutable-origin-to-the-integers – Mikhail Katz Dec 24 '23 at 12:06
  • @MikhailKatz: Benecerraf's very insightful paper What numbers coujld not be is highly relevant here and I can't see a reference to it in any of the comments. – Rob Arthan May 13 '24 at 19:10
  • @RobArthan, Why do you feel Benacerraf's paper is relevant here? – Mikhail Katz May 14 '24 at 09:37
  • @MikhailKatz: because it contrasts the "constructions of the natural numbers" with what they "really are". I thought that was relevant to the OP's concerns. – Rob Arthan May 14 '24 at 19:19

3 Answers3

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You shouldn't think of an object by how it is implemented but by how it behaves. (By how it is specified.)

A better definition of the natural numbers would be "a thing" up to isomorphism we will call $\mathbb N$ together with "a succesor function" $S : \mathbb N → \mathbb N$ and "a zero" $0:1\rightarrow\mathbb N$ that satisfies for any $X$, any $f : X → X$ and $x_0 : 1 → X$, that there is a unique map $\mathbb N \rightarrow X$ (up to ---unique--- isomorphism) that makes the following diagram commute $\require{AMScd}$ \begin{CD} 1+\mathbb N @>{(0\ \ \ S)}>> \mathbb N\\ @VVV @VVV\\ 1+X @>>(x_0\ \ \ f)> X \end{CD}

This says you should think of natural numbers as a zero followed by iterations of the successor function: $0$, $S(0)$, $S(S(0))$, $S(S(S(0)))$ and so on.

Together with a way to transform this structure; so $0$ will be mapped to $x_0$, $S(0)$ will be mapped to $f(x_0)$, $S(S(0))$ will be mapped to $f(f(x_0))$, effectively giving you the strength of a for-loop, letting you iterate a function a fixed finite number of times on a desired point of $X$.

What you do in set theory is give a particular implementation of this, a codification ---you may want to check that the natural numbers as you have them defined actually satisfies the univeral property.

The natural numbers are not their description in set theory. No mathematical object is its description in any particular theory. We just need a way to talk about these objects and we arbitrarily choose one such codification. But what really matters is how they behave, here specified by a universal property.

Ask further questions if any of this is too heavy. Cheers!

Julián
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    +1 for the philosophy. Related: https://math.stackexchange.com/questions/2185587/what-actually-is-a-polynomial/2185648#2185648 – Ethan Bolker Dec 14 '23 at 18:35
  • @Julián so when we define addition, we look at the “properties” we are used to? What are these “properties”? Also, where did the properties of the natural numbers come from? – Frazer Dec 14 '23 at 21:50
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    The properties of the natural numbers from which you can construct the usual arithmetic operations are precisely the ones here: a successor function and the axiom that allows for induction. Mathematicians invented (some would say discovered) this formalism by thinking about the observed properties of the ordinary everyday integers we use in life - addition and multiplication are commutative and the latter distributes over the former. – Ethan Bolker Dec 14 '23 at 22:17
  • @mr_e_man fixed thanks – Ethan Bolker Dec 14 '23 at 22:20
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    I like the philosophy here but I think appealing to a commutative diagram, an arrow theoretic conception of an object, is not appropriate for the OP. It may be worth adding some words which say the same thing but actually say something, rather than draw. Still +1 as it’s a nice answer for a broader audience than just the OP – FShrike Dec 16 '23 at 11:59
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Set theory such as ZF is a customary framework in which one expects all higher-level objects to be developed, but the particular construction such as von Neumann's shouldn't be taken as "the natural numbers" but merely one way of constructing them. Von Neumann's construction is convenient because the order relation among natural numbers becomes simply the inclusion at the set-theoretic level.

Thus, if $0,1,2$ are represented by $\{\}$, $\{\{\}\}$, $\{ \{\}, \{\{\}\} \}$ (hope there are no misprints here :-), then the inequality $1<2$ corresponds to the membership $\{\{\}\} \; \in \; \{ \{\}, \{\{\}\} \}$, etc.

Again you shouldn't be discouraged if this seems overly complicated (it is!). It is merely one of several possible constructions. The relevant properties of the natural numbers as far as your stage of learning is concerned is given by the Peano Axioms involving the successor function, induction, etc.

Mikhail Katz
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0, 1, 2, 3, 4, 5, etc.

That is it. It is what you get when you count things, starting from zero.

All of those mathematical abstractions are attempts to capture what this is. And there is going to be more than one way to do it, especially the etc part.

But it all goes back to one sheep, two sheep, three sheep.

One way of doing it is the successor function. You identify a natural number to a specific set that has the natural number's number of elements in it, and you make a way to build them.

There is one one set with no elements, so that is what you identify as zero.

If you define succ( S ) to be S U {S} something really neat happens. {S}, the singleton that contains nothing but another set, can't be in that set. Ie, {apple, orange} is a set. {{apple,orange}} is the set containing the set containing apple and orange. It has an extra layer of {}s. Anyhow, {apple,orange} is not an element of {apple, orange} - it isn't apple, nor is it orange, it is the set containing both. So S U {S} is guaranteed to have exactly 1 more element than S, as S cannot contain itself. (S is a subset of itself, but not an element of itself).

To count, you need to go from zero, to the next number. And then the next number. In this construction we use succ as our way to get the next number.

0, succ(0), succ(succ(0)), etc.

Now, succ(0) is succ({}) which is {} U {{}}, which is just {{}}. We call the number after 0 to be 1 - so in this construction, 1 is {{}}. 1 is the set containing zero.

Lets look at 2 - succ(1), or {{}} U {{{}}}, or {0} U {{0}} or {0} U {1}, or {0,1}. 2 is the set containing 0 and 1.

This ends up holding true -- the number n is the set {0, 1, 2, ..., n-1} - the set containing all of the natural numbers before n. It has n elements, which is actually useful - the cardinality of the natural number n is n.

Now, all we did here was we constructed a example of natural numbers. From here we can go and define + and * and we end up checking that they correspond to our intuition of how counting numbers add and multiply.

You can instead construct the natural numbers a pile of other ways. You could define them in base 10, where natural numbers are a finite sequence of such base 10 symbols, and define addition and multiplication based on the rules you learnt in elementary school. (finite, you say? Yes, you can define finite before you define the natural numbers; a set is finite if no strict subset of itself is at least as big as it is. And you can define "at least as big as" as "there exists a surjection".)

This is because natural numbers are "0, 1, 2, 3, 4, 5, and so on". We invent abstractions of that describe them, and those abstractions in turn let us prove things about them. But the base concept is literally counting things in the real world.

If we went and discovered something extremely strange about our natural number abstraction, what would happen is we'd change the abstraction. Like, imagine it turns out we prove the existence of a natural number between 3 and 4 in a given abstraction: rather than accept this, we'd be far more likely to reject the abstraction.

Even ZF shouldn't be treated as foundational. ZF itself is an attempt to abstract the naive concept of sets: when or if it misbehaves, we consider inventing a new axiom system. On the other hand, math has a long history of finding misbehaving abstractions and discovering new, interesting constructs, like non-Euclidean geometry.

Yakk
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