5.3 Axioms (optional)

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Definition

An axiom is a statement which says something which does not need to be proved. Or rather, something which we decide does not need to be proved. Something which we will assume to be true. We will then try to derive everything else from our axioms.

Example: Peano's axioms

Axioms are most useful in mathematics where we are dealing with abstractions. For example, the following five axioms (Peano's axioms) define the members of the set of natural numbers. Therefore, the type NaturalNumber can be defined by the following five axioms:

  1. 0 is a natural number.
  2. If a is a natural number then so is a+1.
  3. If you can prove something about a and that implies that you can prove it for a+1, and if you can prove the very same thing for 0, then will this hold for all natural numbers.
  4. If a+1=b+1 then a=b.
  5. You can not add 1 to a natural number to get 0.

Everything that does not satisfy these five axioms is not a natural number. Conversely, everything that satisfies these five axioms is a natural number. Thus these five axioms define the type NaturalNumber by axioms.


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