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: . *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 thricotomy states that every positive integer is either equal to 0, positive, or negative, there is a contradiction such that a 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.