What is the rank of an integer? Given an integer, e.g. $-4$, I can implement it as an equivalence class of ordered pairs of naturals as follows:
$$(-4)_{\mathbb{Z}} := \{(1_{\mathbb{N}},5_{\mathbb{N}}),(2_{\mathbb{N}},6_{\mathbb{N}}), \dots \}$$
where $(a,b)$ (with $a,b \in \mathbb{N}$) is a representative of the integer $a-b \in \mathbb{Z}$. 
What is the rank of $-4$ as an integer? Is it $\omega$, or $\omega+1$? 
(I don't really know much about this subject area, so apologies if I haven't phrased my question very well)
 A: It depends how you implement the ordered pair, and how you implement the natural number.
The usual implementations are:


*

*the Kuratowski definition of the ordered pair: $(a,b) = \{ \{a\}, \{a, b\}\}$

*the definition of a natural as an ordinal: $0 = \emptyset, 1 = \{0\}, 2 = \{0, 1\}, \dots$


The definition of the rank of a set is $$\mathrm{rk}(x) = \sup \{ \mathrm{rk}(y) + 1 : y \in x \}$$
with $\mathrm{rk}(\emptyset) = 0$.
Therefore the rank of $0 \in \mathbb{N}$ is $0 \in \mathrm{Ord}$; the rank of $1 \in \mathbb{N}$ is $1 \in \mathrm{Ord}$; and generally the rank of the natural $n$ is the ordinal $n$.
The rank of the ordered pair $(m, n)$ of naturals is therefore the rank of $\{ \{m\}, \{m,n\}\}$, which is the sup of $\mathrm{rk}(\{m\})+1$ and $\mathrm{rk}(\{m,n\})+1$, which is itself the sup of $m+2$ and $\max(m,n)+2$, which is $\max(m,n)+2$.
Therefore the rank of the equivalence class $[(m,n)]$ is the sup of $\max(m',n')+3$ over all $m',n'$ with $m+n' = m'+n$.
That's $\omega$, of course.

The intuition is that each ordered pair $(m,n)$ has finite rank, and we've just collected them all together, so the resulting set has rank $\omega$ for the same reason that the set $\mathbb{N}$ has rank $\omega$.
I'm told by a reliable source that the question of "what rank does this object have?" is not one that most mathematicians will find interesting, mainly because it's in general highly dependent on the implementations of the objects. Most mathematicians don't really care how the objects are implemented; only that they can be implemented.
