# Number Theory/Axioms

## Axioms of the Integers[edit]

**Axioms** are the foundation of the integers. They provide the fundamental basis for proving the theorems that you will see through the rest of the book.

Here is a mostly complete list:

For *, **, and ** integers:
*

*Closure of and *: and are integers

*Commutativity of *:

*Associativity of *:

*Commutativity of *:

*Associativity of *:

*Distributivity*:

*Trichotomy*: Either , , or .

*Well-Ordered Principle*: Every non-empty set of positive integers has a least element. (This is equivalent to induction.)

*Non-Triviality*: . *This is actually unnecessary to have as an axiom, since it can be easily be proven that .
*Proof*: Assume .
There exists a positive integer such that is a member of the positive integers.
Then,
Therefore,
However, since tricotomy states that every integer is either equal to 0, positive, or negative, there is a contradiction such that is both 0 and a positive integer. Therefore, .
This simple proof provides a more powerful system since less has to be assumed.

*Existence*: is an integer.