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Natural numbers can be represented as sets, however there are more than one representation of natural numbers in set theory (for example von Neumann's and Zermelo's). But all the representations of number 2, for example, should have some properties in common, namely the properties which we intuitively attribute to number 2. And since there is only one number 2 (as opposed to many of its representations), my question is: could we identify number 2 with the properties that all set-theoretical representations of number 2 have in common, that is, with a well-formed, pure-set-theoretical formula that all these representations have to satisfy?

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  • $\begingroup$ Could you give an example of a property that all set-theoretical representation of the number 2 has to have? Because I can't think of one offhand. And wouldn't any possible set-theoretical representation of the number 2 also be a possible set-theoretical representation of the number 1, and vice versa? It wouldn't do to identify the numbers 1 and 2 with the same collection of properties, would it? $\endgroup$ – bof May 13 '16 at 7:24
  • $\begingroup$ What is a number? $\endgroup$ – Asaf Karagila May 13 '16 at 7:31
  • $\begingroup$ @AsafKaragila And can you name it? $\endgroup$ – Stefan Mesken May 13 '16 at 7:34
  • $\begingroup$ @Stefan: I'd name it Bob. So yes, I can name it. :P $\endgroup$ – Asaf Karagila May 13 '16 at 7:35
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There are two issues with the question which hint at a negative answer.

  1. The number $2$ does not live in vacuum. Namely, in order for something to be a number, it usually has to be a part of some arithmetic structure. And while $2$ can usually be found in every arithmetic structure (the natural numbers, the integers, the rationals, the reals, the complex, the ordinals, the cardinals, and so on and so forth), in each structure it might have different properties.

    Not to mention that "properties" does not include any description of these properties. For example in $\Bbb R$ with first-order logic the number $2$ does not have the property of being "a natural number", because $\Bbb N$ is not definable via a first-order formula over $\Bbb R$. But it is definable with a second-order logic formula, so with second-order logic $2$ does satisfy the property of being a natural number.

    Another example is that in $\Bbb N$ we have that $2$ is a prime number, but in the ring $\Bbb R$ it isn't a prime number since it can be divided by any non-zero number to result in another real number.

  2. There is no "special" representation of $2$. Sure, we can argue that the set $\{\varnothing,\{\varnothing\}\}$ is a pretty damn special snowflake of a set, and it is the number $2$. But we'd be wrong. It will be the von Neumann ordinal $2$.

    As far as interpreting the natural numbers, or the real numbers, or the complex numbers, or anything really, every set of the right cardinality could do, and every element in that set can be anything in the interpretation. There is nowhere in the interpretation of mathematical structure inside $\sf ZFC$ that we require that the underlying set will have some properties that agree with the intended interpretation.

    Given any interpretation of $\Bbb N$, for example, we can take any permutation of that set and come up with an isomorphic but different interpretation. Since given a set $A$, any two elements can be swapped via a permutation, this means that if $A$ is the underlying set of the interpretation of $\Bbb N$, via permutations we can come up with interpretation where $2$ is any element of $A$. And if we want something which is not in $A$ to be $2$, then we can just remove the object designated as $2$, and add the new object instead.

    The point is that $\sf ZFC$ is very agnostic when it comes to interpreting mathematical objects as sets. It really doesn't care what are the sets we've used. So evrey set in the mathematical universe has equal shot of being $2$ under some interpretation of whatever mathematical structure you have in mind for the number $2$.

Now, in the comment the OP has asked about a formula which characterizes all the interpretations of the natural numbers. That we can do in a reasonable way. Because we can describe the natural numbers in a fairly robust way. In fact, in several ways:

  • $\Bbb N$ is the unique model of $\sf PA$ which is well-founded.
  • $\Bbb N$ is the unique well-ordered set which is infinite, but every proper initial segment is finite.
  • $\Bbb N$ is the unique linearly-ordered set which has no maximum, but every element has finitely many predecessors.

And so on and so forth. So first we need to decide what does it mean for a set to be the natural numbers. The natural numbers are not just a countable set. They have $\leq$ and they have $+,\cdot$ and $0$ and $1$. Of course we can define the order from $+$ and we can define $0$ and $1$ from the order, and using second-order logic we can also define $+$ and $\cdot$ from the order.

But for the sake of argument we can opt for maximality and take all those symbols as part of our language. So really what we are after is a formula which tells us when a first-order interpretation to the language of arithmetic is isomorphic to the natural numbers.

So you need to write this complicated formula which states what it means for something to be an interpretation for the language of arithmetic, and what does it mean for a sentence in the language of arithmetic to be true in that interpretation, and then you can say that you have an interpretation for the language of arithmetic which satisfies the axioms of $\sf PA$, and the order is well-founded (this is an external property, of course).

If you want to think about the natural numbers in other ways, that's also possible. But as long as you are thinking about them as a structured set, and you're looking for all the interpretations of that structure, this is definitely doable. But it would probably take a few pages to write down this formula, and it is a horrible exercise in formalization of things in the language of set theory.

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  • $\begingroup$ Your point no. 1 seems very important. It is probably not representations of an individual natural number that need to have some common property but the representations of the whole SERIES of natural numbers. For example the von Neumann representation of the series of natural numbers and the Zermelo representation of the series of natural numbers need to have some common properties and thus satisfy some formula. But what would such a formula look like? $\endgroup$ – Tom May 13 '16 at 8:31
  • $\begingroup$ But there is no "common formula". Every countable set can be used to represent the natural numbers, that is exactly the second point in my answer. $\endgroup$ – Asaf Karagila May 13 '16 at 8:54
  • $\begingroup$ Well, then I would say that the formula that every representation of natural numbers must satisfy is that the representation is an infinite countable set. $\endgroup$ – Tom May 13 '16 at 10:52
  • $\begingroup$ Let me add something to my answer. $\endgroup$ – Asaf Karagila May 13 '16 at 10:53
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There is little reason why the number $2$ - when viewed as a set - should have any property that we intuitively assign to the number $2$. We define numbers as sets in order to make them available to our theory - how exactly this is done is rarely of interest and mostly besides the point.

That doesn't say, however, that there are no 'natural' candidates to code natural numbers. In fact, finite von Neumann ordinals seem like an excellent choice (and provide some technical advantages). However, everything works (almost) equally well if we'd decide to use another coding, e.g. we could replace any ordinal $\alpha$ with $(\alpha,\alpha)$.

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    $\begingroup$ This sort of question comes up quite regularly, albeit in slightly different flavors. The common answer to all of them is that in set theory, we mostly don't care how we represent a certain object as sets. We mainly construct them as sets, simply to show their existence, i.e. to show that a set with desired properties exists which may be regarded as a representation of the object that we were looking for. $\endgroup$ – Stefan Mesken May 13 '16 at 7:25

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