Natural numbers

#matematik-discmath

Definition: Axioms of natural numbers

We define the natural numbers to be a set together with an operation addition and an operation multiplication such that is a commutative semigroup and is a commutative monoid with unit , with the following axioms:

  1. If for any , then .
  2. for every
  3. Law of trichotomy
  4. Principle of induction
Definition: Multiple of

is a multiple of , denoted , if there exists a such that .

Theorem

If and , then for every .

Proof:
We have and . This gives by the distributative property.

Definition: Less than, greater than

Let . We say that is less than , denoted , if there exists an such that . If is less than , we say that is greater than and write .

Axiom: Law of trichotomy

Exactly one of the three statements is true for any .

An important consequence of this definition is that for every .

Theorem: Reflexivity of

If and , then .

Proof:
We have and , so .

Theorem

Let . If , then .

Proof:
Assume . WLOG, let . Then , so which means which is clearly false since and only one can be true at once.