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.
Axioms A1-A4 say that (Z, +) is a commutative (abelian) group. Axioms A1-A8 say that (Z, +, ·)
is a commutative ring.
∗
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+ .