tl;dr: We have axiomatic systems that tell us when a set of objects is a number. Sets of objects that comply with those axioms can be seen as numbers.
Axiomatically we can define the natural numbers in terms of set theory via Peano's axioms. Let $N$ be a set and a $S:N\rightarrow N$ a function. Then we say $N$ is a set of
P1. $\exists n_0$ such that $S(n)\neq n_0$, for all $n\in N$.
P2. $S$ is injective.
P3. If $K$ is a set such that:
$n_0\in K$, and
For all $n\in N$, $n\in K\Rightarrow S(n)\in K$,
then $K=N$.
In this way, we can associate "natural numbers" or "counting numbers" with the set $\mathbb{N}=\{1,2,\ldots\}$. Any such set of objects that adhere to these properties can be considered counting numbers. Thus the set $\{\{1\}, \{1,2\}, \{1,2,3\},\ldots\}$ can be seen as a set of counting numbers since it adheres to these axioms under the function $S(A)=A\cup\{\max{(A)}+1\}$.
From here, we can define $+$ on $N$ recursively by:
- $n+n_0 =: S(n)$
- $n+S(m) =: S(n+m)$
Furthermore, we can define the integers via an equivalence relation $\sim$ on $N\times N$:
$$(m,n)\sim (p,q) \equiv m+q=n+p.$$
It is fairly straightforward to verify this is an equivalence relation. We then define the integers by $$Z=: N\times N/\sim.$$
We also see that
$$\{[(n,n_0)]:n\in N\}$$
adheres to the Peano axioms with
$$S([(n,n_0)])=:[(S(n),n_0)].$$
So we identify $N\subseteq Z$.
For the integers, we may define + and $\cdot$ by:
- $[(m,n)]+[(p,q)]=[(m+p,n+q)]$
- $[(m,n)]\cdot [(p,q)]=[(mp+nq,np+mq)]$
We can show that these operations are well-defined and confidently use the notation $$\mathbb{Z}=\{0,1,-1,2,-2,\ldots\},$$
where
- $n=:[(n,0)]$
- $-n=:[(0,n)]$
- $0=:[(n,n)]$
So here we now see that any set of counting numbers (i.e., a set that adheres to the Peano axioms) can be extended to a set which behaves like the integers, where we may talk about the "negative" of a number.
But then we may wish to think of the rationals, which similarly, we can define an equivalence relation $\sim$ on $Z\times N$:
$$(n,m)\sim (p,q) \equiv nq=mp$$
As above, we now form the rationals via equivalence classes
$$Q=:Z\times N/\sim.$$
This matches our usual understanding of rationals as numbers of the form $\frac{p}{q}$, where $q\neq 0$. To see this, we associate
$$\frac{p}{q}=:[(p,q)]$$
So we can now write
$$\mathbb{Q}=\left\{\frac{p}{q}:p\in\mathbb{Z},q\in\mathbb{N}\right\}$$
We can define concepts like addition, subtraction, and absolute value for this set, though I won't do that here.
Thus far, the construction of these numbers have all been based on logic, set theory, and algebraic concepts. But what about the reals? The reals are different because they have a topological structure. For this we need the concept of Cauchy sequences.
Define a rational sequence as a function $q:N\rightarrow Q$, where $q(n)=:q_n$ and we often write $q=\{q_n\}$. We say a sequence $q$ is Cauchy if for every $\epsilon>0$, where $\epsilon\in Q$, there exists $n\in N$ such that:
$$|q_k - q_m|<\epsilon, \forall k,m\geq n$$
The intuition behind this is that elements of the sequence can be pushed arbitrarily close together eventually. We then define:
$$C_Q = \{q:N\rightarrow Q:q \text{ is Cauchy}\}$$
This is the set of all rational Cauchy sequences. On this set we define $\sim$ an equivalence relation:
$$q\sim p \equiv q_n-p_n\rightarrow 0,$$
where the right hand side of this equation means, for every rational $\epsilon>0$ there exists $m\in N$ such that
$$|q_n-p_n|<\epsilon, \forall n\geq m.$$
We can thus define the reals as $R=C_Q/\sim$. Similar to before, we can define + and $\cdot$ as well-defined operation and confidently associate every real number with an equivalence class to obtain our standard notation and write $\mathbb{R}=:\mathbb{C_Q}/\sim$.
So what we have is that from the Peano axioms we can axiomatically characterize counting numbers, introduce equivalence relations to extend these numbers to integers (negatives and 0) and rationals (ratios of integers). If we go further and endow the rationals with topology (sequences and concepts of convergence), we can use equivalence classes once again to arrive at a conceptual definition for the reals. We can do so with any set of objects that adheres to the original Peano axioms, the set $\{1,2,\ldots\}$ is but one example of this.