Field Axioms: there exist notions of addition and multiplication, and additive and multiplicative identities and inverses, so that: • (P1) (Associative law for addition): a + (b + c) = (a + b) + c • (P2) (Existence of additive identity): 9 0 : a + 0 = 0 + a = a • (P3) (Existence of additive inverse): a + (−a) = (−a) + a = 0 • (P4) (Commutative law for addition): a + b = b + a
• (P5) (Associative law for multiplication): a · (b · c) = (a · b) · c • (P6) (Existence of multiplicative identity): 9 1 6= 0 : a · 1 = 1 · a = a • (P7) (Existence of multiplicative inverse): a · a−1 = a−1 · a = 1 for a 6= 0 • (P8) (Commutative law for multiplication): a · b = b · a • (P9) (Distributive law): a · (b + c) = a · b + a · c
Order Axioms: there exists a subset of positive numbers P such that • (P10) (Trichotomy): exclusively either a 2 P or −a 2 P or a = 0. • (P11) (Closure under addition): a, b 2 P ) a + b 2 P
• (P12) (Closure under multiplication): a, b 2 P ) a · b 2 P Completeness Axiom: a least upper bound of a set A is a number x such that x _ y for all y 2 A, and such that if z is also an upper bound for A, then necessarily z _ x. • (P13) (Existence of least upper bounds): Every nonempty set A of real numbers which is bounded above has a least upper bound.
We will call properties (P1)–(P12), and anything that follows from them, elementary arithmetic. These properties imply, for example, that the real numbers contain the rational numbers as a subfield, and basic properties about the behavior of ‘>’ and ‘ on R. (That is, given any pair a, b then a > b is either true or false). It satisfies:
a) Trichotomy: For any a R exactly one of a > 0, a = 0, 0 > a is true. b) If a, b > 0 then a + b > 0 and a.b > 0
c) If a > b then a + c > b + c for any c
Something satisfying axioms I and II is called an ordered field. Examples
1. The field Q of rationals is an ordered field.
Define a/b > c/d provided that b, d > 0 and ad > bc in Z. One may...