Consider lists all of whose elements are \({ ∅ }\). One of these is the list of length 0, which is just \({ ∅ }\). If \({ x }\) is such a list then we can get another such list by prepending another \({ ∅ }\). In our representation of lists this is \({ x ∪ \{ ( ∅ , x ) \} }\) but in keeping with our policy of not relying on the particular representation we can introduce the notation \({ x ' }\) for the result of prepending a \({ ∅ }\) to \({ x }\). Similarly, concatenating two such lists gives us another such list. Rather than writing down an expression for this in our particular representation of lists we can simply denote the concatenation of \({ x }\) and \({ y }\) by \({ x + y }\). Rather than relying on the empty list being represented by the empty set we can write \({ 0 }\) for the empty list.
We the notation as above one can prove a number of elementary properties of these lists, such as the following.
\({ 0 }\) exists.
For all \({ x }\), \({ x ' }\) exists.
For all \({ x }\) we have \({ x ' ≠ 0 }\).
For all \({ x }\) we have \({ x + 0 = x }\).
The second and fourth of these are proved by induction on sets. These may look familiar. They are in fact our first four axioms for elementary arithmetic. To get the remaining axioms we need to define more operations and relations. \({ x - y }\), for example, can be defined to be the list, if there is one, which when concatenated with \({ y }\) gives \({ x }\). \({ x \le y }\) will mean that there is a list which, when concatenated with \({ x }\) gives \({ y }\) and \({ x > y }\) will mean that there isn’t one. The only definition which is somewhat tricky is the definition of \({ x · y }\). To get \({ x · y }\) we start with a pair of lists where the first is initially \({ 0 }\) and the second is initially \({ y }\). We then successively remove elements from the second list and concatenate copies of the \({ x }\) with the first. Once we’ve removed all elements of the second list the first list will be \({ x · y }\).
Once we have definitions for all the operations and relations it’s not terribly difficult to show that the axioms of elementary arithmetic are all satisfied. We can also show that the three rules of inference from elementary arithmetic also apply to lists of \({ ∅ }\)’s. Not surprisingly, the rule of induction for natural numbers follows from the induction theorem for finite sets. Note that induction for sets is a theorem though, rather than an axiom.
So we have a copy of elementary arithmetic sitting inside of set theory. There are other ways to embed arithmetic in set theory. The method above is not the most commonly used one. A more traditional approach is in terms of what are called the von Neumann ordinals. It’s better not to think of the natural numbers as some particular representation. We shouldn’t therefore ask questions like whether \({ x ⊂ x '' }\). This happens to be true in the representation we’ve chosen, and also happens to be true in the representation in terms of von Neumann ordinals, but isn’t true in some other representations of the natural numbers within set theory. In keeping with our policy of separating interface and implementation we should avoid applying operations or relations to natural numbers other than the ones from elementary arithmetic.
The particular construction chosen is somewhat arbitrary, but it’s not randomly chosen either. Intuitively, lists have a length. Prepending and element increments the length. Concatenating lists adds their lengths. Of course many different lists will generally have the same length but if we want to choose a particular one for each length then we can do that by choosing some item and using only lists where all elements are equal to that item. The simplest choice for that item is \({ ∅ }\) because it’s the only set whose existence is directly guaranteed by an axiom of set theory.