bij means bijection. If a set bij the natural numbers set,then the set is countably infinite
For $\mathbb{N}^2$ bij $\mathbb N$,we could assume that is a $\mathbb N \times \mathbb N$ matrix such as: \begin{pmatrix} (1,1)&(1,2)&(1,3)&\cdots&(1,N)\\ (2,1)&(2,2)&(2,3)&\cdots&(2,N)\\ &&\vdots\\ (N,1)&(N,2)&(N,3)&\cdots&(N,N) \end{pmatrix} and we could easily prove that for a random $(m,n)$,it will correspond to a nature number which is $\frac{(m+n-1)(m+n-2)}{2}+m$. Thus we could prove $\mathbb N^2$ bij $\mathbb N$.But how about $\mathbb N^4$? This is four-dimensional and I don't think I could still use the above way to prove $\mathbb N^4$ bij $\mathbb N$. So is there any other way to prove this?
If you already have a bijection $\ b_2:\mathbb{N}^2\rightarrow \mathbb{N}\ $ then $$ b_4(i,j,k,l) \overset{\text{Def}}{=} b_2 (b_2(i,j),b_2(k,l)) $$ defines a bijection $\ b_4:\mathbb{N}^4\rightarrow \mathbb{N}\ $.
Proof: First suppose that $ b_4\left(i_1,j_1,k_1,l_1\right)=b_4 \left(i_2,j_2,k_2,l_2\right)\ $. Then, by definition, \begin{align} b_2\left(b_2\left(i_1,j_1\right),b_2 \left(k_1,l_1\right) \right)&=b_4\left(i_1,j_1,k_1,l_1\right)\\ &=b_4\left( i_2,j_2,k_2,l_2 \right)\\ &= b_2\left(b_2 \left(i_2,j_2 \right),b_2\left(k_2,l_2\right)\right)\ . \end{align} It follows that $\ b_2\left(i_1,j_1\right)= b_2 \left(i_2,j_2 \right)\ $ and $\ b_2 \left(k_1,l_1\right)=b_2\left(k_2,l_2\right)\ $, because $\ b_2\ $ is a bijection, and then, that $\ i_1=i_2, j_1=j_2, k_1=k_2\ $ and $\ l_1=l_2\ $, for the same reason. Thus $\ b_4\ $ is injective.
Now suppose that $\ n\ $ is any natural number. Then since $\ b_2\ $ is a bijection, there must exist natural numbers $\ n_1, n_2\ $ such that $\ n= b_2\left(n_1,n_2\right)\ $. For the same reason, there must exist natural numbers $\ i,j,k\ $ and $\ l\ $ such that $\ n_1= b_2\left(i,j\right)\ $ and $\ n_2= b_2\left(k,l\right)\ $. We then have \begin{align} n&= b_2\left(n_1,n_2\right)\\ &= b_2 (b_2(i,j),b_2(k,l))\\ &= b_4(i,j,k,l)\ , \end{align} so $\ b_4\ $ is surjective. Since $\ b_4\ $ is both injective and surjective, it is a bijection.