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: $0\neq 1$. *This is actually unnecessary to have as an axiom, since it can be easily be proven that $0\neq 1$. Proof: Assume $0=1$. There exists a positive integer $a$ such that $a$ is a member of the positive integers. Then, $a\times 0=a\times 1$ Therefore, $0=a$ However, since trichotomy states that every integer is either equal to 0, positive, or negative, there is a contradiction such that $a$ is both 0 and a positive integer. Therefore, $0\neq 1$. This simple proof provides a more powerful system since less has to be assumed.

Existence: $1$ is an integer.