# Number Theory/Axioms

## Axioms of the Integers

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 $a$, $b$, and $c$ integers:

Closure of $\times$ and $+$: $a\times b$ and $a+b$ are integers

Commutativity of $+$: $a+b=b+a$

Associativity of $+$: $(a+b)+c=a+(b+c)$

Commutativity of $\times$: $a\times b=b\times a$

Associativity of $\times$: $(a\times b)\times c=a\times(b\times c)$

Distributivity: $a\times(b+c)=a\times b+a\times c$

Trichotomy: Either $a<0$, $a=0$, or $a>0$.

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

Non-Triviality: $0 \ne 1$.

Existence: $1$ is an integer.