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In the last post, we mentioned that sometimes we identify 2={0,1}. So 3={0,1,2}, 4={0,1,2,3}?

Yes. It is the construction of natural numbers by polymath John von Neumann:

- 0={ }, the empty set.
*n*+1=*n*∪{*n*}.

I see, so

*n*is a set of

*n*elements, right?

This definition has many advantages:

*n*is a set of*n*elements.- The definition is recursive, it does not depend on anything outside the number system.
*n*>*m*if and only if n is an element of*m*.*n*≥*m*if and only if*n*is a subset of*m*.- This definition can be extended to infinite ordinals.

Are there any other definitions for natural numbers?

Yes. For instance, 0={ }, 1={0}, 2={1}, etc. As soon as the definition fits Peano axioms, that's okay.

What are Peano axioms?

Proposed by Giuseppe Peano, there are five axioms:

- There is an initial number, called 0. (Peano indeed use 1 as the initial number)
- There is a successor operation. If
*n*is a natural number, its successor is denoted by*n*+1. - For all nonzero natural number
*m*, there exists a unique*n*such that*n*+1=*m*. - There exists no natural number
*n*such that*n*+1=0. - If 0 has Property P and that "if
*n*has Property P then*n*+1 also have Property P", then all natural numbers have Property P.

The last property is the mathematical induction?

Well... the reason that mathematical induction works is that it is a property of the natural number system!