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 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. *This is actually unnecessary to have as an axiom, since it can be easily be proven that 0 \ne 1. Proof: Assume 0 = 1. There exists a positive integer a such that a is a member of the positive integers. Then, a x 0 = a x 1 Therefore, 0 = a 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, 0 \ne 1. This simple proof provides a more powerful system since less has to be assumed.

Existence: 1 is an integer.