Natural born builder

As it happened with the universe, everything is born from nothing; and so are the natural numbers. I might have succeeded in confusing you a bit, and now you might be wondering what it does mean for numbers to be built from nothing. If so, you are not alone, and neither the first person to be stumbled by this revelation. The very concept of zero has bugged many mathematicians, philosophers and scientists before you and it is still difficult to grasp in its true essence.

How can nothing be something?

Well, it is not. Zero indeed is just a symbol, a placeholder, a representation of the absence of quantity. You can think of it as the most basic building block of mathematics. It is our first brick to then construct arithmetics, and then algebra, and then geometry, and so on. As a matter of fact, the definition of zero is not a definition at all, but a convention. We choose to define zero as the number of elements in the empty set. Even better, we can identify the number zero with the empty set:

Ok, so what?

Well, now that we have our first brick, we can start building. First of all, we need to define what it means to add a brick to our tower. Start with the empty set, and put it in a box. Now we have a set containing the empty set; it is not empty, since it contains one element (the empty set). We call this set $1$:

GREAT! Can we do it again?

Of course we can! Start with the set $1$, and put it in a box together with the empty set. Now we have a set containing the set $1$ and the empty set; it is not empty, since it contains two elements (the set $1$ and the empty set). We call this set $2$:

Can you see where we are going?

We could keep going, all day long, and construct every natural number you want. It is not a problem at all, but it is not the most efficient way to do it. We can do better, and define something that does the job for us.

We are going to call it the successor function. It is a function that takes a natural number $n$ and returns the natural number $n+1$. Mathematically, we can write it as:

We could mention some of its properties, such as the fact that it is injective, surjective, and bijective. Or, that there is no natural number $n$ such that $S(n) = 0$. Or that $S(n) = S(m)$ if and only if $n = m$. But let's not get carried away, all we care about is that now we know we have every natural number we need, and now we can start building our tower.

Try yourself: