Monday, February 18, 2013

3={0,1,2}

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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}.
Therefore we have: 0={ }, 1={0}, 2={0,1}, 3={0,1,2}, .... In general n={0,1,2,...,n-1} for n≥1.

I see, so n is a set of n elements, right?
This definition has many advantages:
  1. n is a set of n elements.
  2. The definition is recursive, it does not depend on anything outside the number system.
  3. n> m if and only if n is an element of m.
  4. nm if and only if n is a subset of m.
  5. 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:
  1. There is an initial number, called 0. (Peano indeed use 1 as the initial number)
  2. There is a successor operation. If n is a natural number, its successor is denoted by n+1.
  3. For all nonzero natural number m, there exists a unique n such that n+1=m.
  4. There exists no natural number n such that n+1=0.
  5. 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!