Alternative formulations of the natural numbers

Question

Do there exist other ways to create the natural numbers, other than the definition given by $$0=\emptyset \\ x^+ = x \cup \{x\}$$ For example, one could also define the succesor operation as $$x^+ = \{x\}$$ Would both formulations lead to a set with the same basic properties? Its obvious that at least some properties would be lost, for example, the $\subset$ would no longer order the set, but its also clear that this set could still be ordered.

In essence, my question is this: Is the typical definition of the natural numbers the only valid way to define the natural numbers in ZF? And if not, why do we choose that particular formulation of the natural numbers (i.e. what unique properties does it have)?

Answer 1

Any countable set could become "a valid version of the natural numbers". Simply by virtue of transport of structure from whatever copy you consider canonical.

You can define the von Neumann $\omega$, use it to define the integers, the rationals, the reals, the complex numbers, and then decide that $\Bbb N$ is the canonical copy of the natural numbers in the set that is the complex numbers. Nobody is stopping you from doing so, and for the most practical part, it would not change a single thing in mathematics.

We have a good reason to choose the von Neumann ordinal $\omega$ as our canonical copy, though:

1. It is very easy to define it, and the order on the numbers is the simplest order conceivable in the world of set theory: $\in$. So, for example, proving that the order is a well-order is easier, since $\min A=\bigcap A$ for any non-empty set of von Neumann ordinals.
2. Unlike the definition of Zermelo, $x+1=\{x\}$, the ordinals extend very nicely to the realm of the transfinite.
3. The von Neumann ordinal $n$ is a set with $n$ elements, whereas the Zermelo ordinal is a singleton (or the empty set). This allows us to define the arithmetic on the finite von Neumann ordinals either as ordinal arithmetic (i.e. by recursion) or as cardinal arithmetic. This is another layer of simplification in the definition of the natural numbers which does not exist in other interpretations.

But ultimately, the main point is that we need a copy, and (1) already ensures that $\omega$ is the simplest one.

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Let me add a remark that this is not a bad question, but it is sometimes motivated from bad teaching. People sometimes have the impression that set theory is this very rigid framework: ordered pairs are $\{\{a\},\{a,b\}\}$; the reals are sets of rational numbers with certain properties; etc. But this is far from the truth. In $\sf ZF$ we have the Replacement schema which can be understood (and is equivalent to) "implementation agnostic foundation".

Since the language of set theory has just one extralogical symbol (that is, a symbol which is not equality, connectives, quantifiers, and variables) which is $\in$, everything else needs to be implemented in one way or another. But just like we can implement a search algorithm in many many different ways, and even the same algorithm in different languages, different operating systems, and different computing architectures, just like that we can implement mathematical objects in many many different ways into set theory. That is part of the strength of set theory as a foundational theory.

Unfortunately, because we often don't care about the implementation, and because we choose the simplest one when the opportunity presents itself, some people outside of set theory often get the impression that "that's just the way to do it". And they pass this impression on when they teach other subjects which begin with a few words on set theoretic foundation. But that is really not the right way to look at set theoretic foundation.