Peano axiomatized Arithmetic by defining the naturals, Addition and Multiplication. We can see the naturals as simply a "ruler" (as in a 12 inch piece of wood) that is infinite in one direction, with arbitrary symbols along it where the location of each symbol is a function.
Addition and Multiplication are ways to fill up one quadrant of the lattice squares on a plane. For Addition we have:
Imagine the ruler across the top and along the left side, and each square inside is the value of the sum of the two coordinates.
Starting with a blank quadrant, how can we take the infinite ruler and paste the correct numbers down on it? There are at least 2 ways to do it in 2 steps. For example, we can lay a copy of the ruler at the top of the quadrant as one step.
Multiplication is much trickier! We need to construct: