Are there any cumulative axiomatizations of the numbers?

Question

By "cumulative axiomatization" I mean an axiomatization of the numbers where each set in the hierarchy of number types is explicitly a subset of the previous set. That is, $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{R}\subset\mathbb{C}$. All of the axiomatizations and constructions that I've seen have made each set of numbers an unrelated set, which violates my intuition that the naturals are (for example) a proper subset of the integers rather than isomorphic to a proper subset of the integers.

I'm thinking of an axiomatization that starts with the naturals, and then the first axiom in the axiomatization of the integers is $\mathbb{N}\subset\mathbb{Z}$, and then so on for the other sets. Has there been any work in this area?

Answer 1

After a long time, I came back to this question because I found a way to axiomatize numbers in a cumulative manner and I really don't see anything wrong with it, it actually seems to me like a good way to axiomatize extensions.

Generally, we start from some set $A$ with some structure attached and we want to extend it to another set $B$ with a 'larger' structure. We do this via an embedding, $A \hookrightarrow B$, ie. an injective homomorphism from $A$ onto some subset $E$ of $B$ (which has a substructure inherited from $B$'s structure). Via this embedding we identify the elements of $E$ with the elements of $A$ and effectively consider $A$ as a subset of $B$. What this means is up to discussion in my opinion and one way to see it is how I put it in the comments. But now I want to show another way to see it:

Instead of 'rewriting history' and changing what we mean by the set $A$, why not change $B$? We are still constructing it, so, why not go through 2 versions of it?

Say we have found a set $B_{\beta}$ such that $A \overset{\varphi}{\longleftrightarrow} E \subset B_{\beta}$. We'll suppose $A \cap B_{\beta} = \varnothing$. We can just replace $E$ in $B_{\beta}$ with $A$. Define the following conversions: $\beta : A \cup B_{\beta} \to B_{\beta},\ \beta(x) = \varphi(x)$ if $x\in A$ and $\beta(x) = x$ if $x\in B_{\beta}$,
$\alpha : B_{\beta} \to A \cup B_{\beta},\ \alpha(x) = \varphi^{-1}(x)$ if $x \in E$ and $\alpha(x) = x$ otherwise.

Now we can 'upgrade' from the beta version to the alpha version: define $B = A \cup (B_{\beta}\setminus E)$. Using the conversions above, we can translate the structure of $B_{\beta}$ to the desired structure of $B$ that is compatible with the one of $A$. For example, if $B_{\beta}$ is a group with some operation $\circ$, define the operation $\circ$ on $B$ as $x \circ y = \alpha(\beta(x) \circ \beta(y))$. You can verify that $(B, \circ)$ is indeed a group and that $(A, \circ)$ is a subgroup of it.

This way, we can define the integers as "all the natural numbers with their usual rules and also an additional set of numbers $-n$ for every non-zero natural number $n$" after seeing that pairs of the form $(\pm, n)$ and $0$ hold for our purposes and we can embed the natural numbers into this structure.