What are Natural Numbers

Natural numbers are one of the first things that we all have learnt first at school. When getting more involved we learn also about other type of numbers: Integer, Rational, Real and Complex ones... Natural numbers seems to be a so root concept that it seems non-sense to have a strict definition of the natural number concept. However from a logic point of view also Natural Numbers can be formally be defined. The definition that I propose here may not be the only possible one I think, but I have ever liked this definition for its simplicity and brightness at the same time.

So, let's go to the point!

Natural Numbers Set N is a set having these properties:

1. Has infinite elements;
2. Every contained element has exactly one successor inside the same set;
3. Contains exactly one element that is not the successor of any other element in the set;
4. Any element in the set is the successor of at most one element of the set;
5. Among all the sets that has the preceding 4 properties N is the smaller one.

That's all!
I always found very clever the fifth property.
I let to you to think about why it is needed as well as the other four ones: if you like Math I'm sure you're gong to find all this interesting!