Math 220, The Axioms of the Integers∗ A RITHMETIC AXIOMS. The set Z of integers has two binary operations: addition +, and multiplication · (usually the · is omitted). A1. Addition is associative: ∀x, y, z ∈ Z, (x + y) + z = x + (y + z). A2. Addition is commutative: ∀x, y ∈ Z, x + y = y + x. A3. Z has an additive identity: the integer 0 satisfies ∀x ∈ Z, x + 0 = 0 + x = x. A4. Each integer has an additive inverse: ∀x ∈ Z, ∃(−x) ∈ Z, x + (−x) = (−x) + x = 0. A5. Multiplication is associative: ∀x, y, z ∈ Z, (xy)z = x(yz). A6. Multiplication is commutative: ∀x, y ∈ Z, xy = yx. A7. Z has a multiplicative identity: the integer 1 satisfies ∀x ∈ Z, x1 = 1x = x. A8. Distributive law: ∀x, y, z ∈ Z, x(y + z) = xy + xz. Note that in general, elements in Z do not have multiplicative inverses. ∗ c 2013 by Michael Anshelevich. 1 O RDER AXIOMS. Denote by Z+ the positive integers. A9. Closure property: Z+ is close with respect to addition and multiplication. That is, x, y ∈ Z+ x + y ∈ Z+ , ⇒ xy ∈ Z+ . A10. Trichotomy law: ∀x ∈ Z, exactly one of the following holds x ∈ Z+ , or − x ∈ Z+ or x = 0. Define the order on Z by x < y ⇔ y − x ∈ Z+ . In terms of <, the two order axioms can be re-formulated as A9’. Closure property: ⇒ x > 0, y > 0 x + y > 0, xy > 0. A10’. Trichotomy law: ∀x ∈ Z, exactly one of the following holds x > 0, or − x > 0 or x = 0. T HE W ELL - ORDERING PRINCIPLE. For a subset S ∈ Z, we say that a ∈ S is a smallest element of S if ∀x ∈ S, a ≤ x. A11. The Well-Ordering Principle: Every non-empty subset of Z+ has a smallest element. Note that equivalently, the principle states that if S ⊂ Z+ does not have a smallest element, then S = ∅. The Well-Ordering Principle is also equivalent (assuming the rest of the axioms) to the Principle of Mathematical Induction. A11’. The Principle of Mathematical Induction: If S is a subset of Z+ such that (1) 1 ∈ S. (2) If n ∈ S, then n + 1 ∈ S. then S = Z+ .