# Number Theory/Axioms

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## 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*: .

*Existence*: is an integer.