What are Natural Numbers

Fri, 09/14/2012 - 12:30



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!

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