Define numbers sets (e.g. $\mathbb{N,Z}$ etc.) as being subsets of next one... Usually, numbers sets are constructed from $\mathbb N$ (with Peano's axioms) : then $\mathbb Z$ is viewed as a quotient set of $\mathbb N\times\mathbb N$, $\mathbb Q$ as a quotient set of $\mathbb Z \times \mathbb Z$, and $\mathbb R$ is defined with Dedekind cuts or Cauchy sequences of rationals. Well, we can define $\mathbb C$ as $\mathbb R\times\mathbb R$ or $\mathbb R[X]/(1+X^2)$.
These sets are linked through injective morphism, i.e. $\mathbb N \hookrightarrow \mathbb Z \hookrightarrow \mathbb Q \hookrightarrow \mathbb R \hookrightarrow \mathbb C$.
However, there is no inclusion like $\mathbb N \subset \mathbb Z \subset \mathbb Q \subset \mathbb R \subset \mathbb C$ because, there are different sets (by construction)...
How to define numbers sets as being subsets of next one ?
Does it possible, to construct $\mathbf N$, $\mathbf Z$, $\mathbf Q$, $\mathbf R$ and $\mathbf C$ instead of $\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$ and $\mathbb C$ then create some copies of these sets through isomorphism, for example, let define $\mathbb C := \mathbf C$ then
$$ \mathbb R := \{ x\in\mathbb C \:|\: \exists x'\in\mathbf R : x = \phi_{\mathbf R}(x')\} $$
with $\phi_{\mathbf R} : \mathbf R \to \mathbf C =:\mathbb C$ an isomorphism. This set $\mathbb R$ is a subset of $\mathbb C$.
Moreover, it's a field and we can call it the "set of real numbers", and so on... ? 
Then, if I want to construct quaternion $\mathbb H$, $\mathbb C$ become some copy of $\mathbf C$ and a subset of $\mathbb H$, et cætera...
 A: Most mathematician will be happy with just an embedding (i.e. injective homomorphism), not the exact inclusion relationship. However, we can make the set of reals really contains the set of natural numbers, by cutting copies of the image of embedding map $i:\mathbb{N}\to \mathbb{R}$ and putting the set of natural numbers. 
Here is a detail: Let $\mathbb{R}_0$ be a set of real numbers we constructed (via Cauchy sequence or Dedekind cut, it doesn't matter.) with an embedding $i:\mathbb{N}\to\mathbb{R}_0$. Define $\mathbb{R} = (\mathbb{R}_0\setminus i[\mathbb{N}])\cup\mathbb{N}$ with appropriate operations and relations; for example we define the addition of $\mathbb{R}$ as
$$a+b := \begin{cases} 
i(a+b) &\text{if } a, b\in \mathbb{R}_0\text { and }a+b\notin i[\mathbb{N}]
\\
i(a)+b& \text{if } a\in \mathbb{N} \text{ and } b\in \mathbb{R}_0
\\
a+i(b)& \text{if } a\in \mathbb{R}_0 \text{ and } b\in  \mathbb{N}
\\
a+b &\text{otherwise}
\\\end{cases}$$
(Note that we abuse notations: the addition operations that appeared are not same, though they use same symbol.)
and so on. Of course, we should care that our $\mathbb{R}_0$ and $\mathbb{N}$ have empty intersection. If not our addition could be ill-defined.
