You can imagine that there was quite some opposition from within the mathematics community when Georg Cantor (1845-1918) proposed the following theorem:
Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjective function, from A to B.
A sequence for $\mathbf{N}$ is $$s(n) = \sum_{k=1}^n 1 \ ,$$
$$ \begin{array}{cc}
n & s(n) \\
1 & 1 \\
2 & 2 \\
3 & 3 \\
4 & 4 \\
\cdots & \cdots \end{array} $$
Likewise a sequence for $\mathbf{Z}$ is: $$s(n) = \sum_{k=1}^n (-1)^{k+1} \cdot k \ ,$$
$$ \begin{array}{cc}
n & s(n) \\
1 & 1 \\
2 & -1 \\
3 & 2 \\
4 & -2 \\
\cdots & \cdots \end{array} \, $$ We can thus establish a bijective (one-to-one) map between $\mathbf{N}$ and $\mathbf{Z}$.
By the theorem above we can conclude that $\mathbf{N}$ and $\mathbf{Z}$ have the same cardinality ( 'number of elements' ).
