Interface:Real number axioms

The axioms in this file describe the real numbers on top of set theory. This is an axiom set which is designed for ease of understanding; the axioms are not independent and it is possible to come up with a set of real number axioms which are considerably shorter. . They are based on a system of Tarski.

We build on propositional logic, first-order logic, and set theory.

 param (CLASSICAL Interface:Classical_propositional_calculus ) param (FIRSTORDER Interface:First-order_logic_with_quantifiability (CLASSICAL) ) param (SETS Interface:Set_theory (CLASSICAL FIRSTORDER) ) 

Real numbers
There is a set of real numbers. As a convention, we tend to use  and   to refer to real numbers, although using these names does not by itself ensure that a term is a real number rather than a set of numbers or a relation or something else.  term (object (ℝ)) var (object x y z) var (variable v) 

Ordered
We define a total order on the reals.  term (formula (< object object)) stmt (LessThanTotalityImplication  (((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) → ((x ≠ y) → ((x < y) ∨ (y < x))))) stmt (LessThanAsymmetry  (((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) → ((x < y) → (¬ (y < x))))) stmt (LessThanTransitivity  ((((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) ∧ (z ∈ (ℝ))) → (((x < y) ∧ (y < z)) → (x < z)))) 

Continuity
Continuity, or completeness, refers to the existence of numbers which bound given sets. It distinguishes the reals from the rationals and involves sets of numbers, not just individual numbers. The formulation in Tarski is the following. Given a set  and a set , such that every element in   is less than every element in  , there exists a number   such that every element of   is less than or equal to  , and every element of   is greater than or equal to.  var (object K L) var (variable separator k l) stmt (Continuity ((separator K L))  ((((K ⊆ (ℝ)) ∧ (L ⊆ (ℝ))) ∧  (∀ k (∀ l ((((value k) ∈ K) ∧ ((value l) ∈ L)) → ((value k) < (value l)))))) → (∃ separator (∀ k (∀ l ((((value k) ≠ (value separator)) ∧ ((value l) ≠ (value separator))) → (((value k) < (value separator)) ∧ ((value separator) < (value l))))))))) 

Addition
The real numbers are closed under an addition operation, which is commutative and associative.  term (object (+ object object)) stmt (RealAdditionClosure  (((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) → ((x + y) ∈ (ℝ)))) stmt (AdditionCommutativity  (((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) → ((x + y) = (y + x)))) stmt (AdditionAssociativity  ((((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) ∧ (z ∈ (ℝ))) → (((x + y) + z) = (x + (y + z))))) 

Addition is invertible. Invertibility states that given two numbers, there is a third number that when added to the second yields the first.  stmt (AdditionInvertibility ((v x) (v y))  (((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) → (∃ v (((value v) ∈ (ℝ)) ∧ (x = (y + (value v))))))) 

Addition is monotonic with respect to.  stmt (AdditionMonotonic  ((((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) ∧ (z ∈ (ℝ))) → ((y < z) → ((x + y) < (x + z))))) 

There is a number zero which is an additive identity.  term (object (0)) stmt (ZeroReal  ((0) ∈ (ℝ))) stmt (AdditiveIdentity  ((x ∈ (ℝ)) → ((x + (0)) = x))) 

Multiplication
The real numbers are closed under a multiplication operation, which is commutative and associative.  term (object (· object object)) stmt (MultiplicationClosure  (((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) → ((x · y) ∈ (ℝ)))) stmt (MultiplicationCommutativity  (((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) → ((x · y) = (y · x)))) stmt (MultiplicationAssociativity  ((((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) ∧ (z ∈ (ℝ))) → (((x · y) · z) = (x · (y · z))))) </jh>

The law of invertibility for multiplication is slightly different from the one for addition in that it excludes zero.  stmt (MultiplicationInvertibility ((v x) (v y))  (((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) → ((y ≠ (0)) → (∃ v (((value v) ∈ (ℝ)) ∧ (x = (y · (value v)))))))) </jh>

The law of monotony for multiplication only applies to positive multipliers.  stmt (MultiplicationMonotonic  ((((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) ∧ (z ∈ (ℝ))) → ((((0) < x) ∧ (y < z)) → ((x · y) < (x · z))))) </jh>

Multiplication distributes over addition.  stmt (Distributivity  ((((x ∈ (ℝ)) ∧ (y ∈ (ℝ))) ∧ (z ∈ (ℝ))) → ((x · (y + z)) = ((x · y) + (x · z))))) </jh>

There is a complex number, which serves as a multiplicative identity.  term (object (1)) stmt (OneReal  ((1) ∈ (ℝ))) stmt (MultiplicativeIdentity  ((x ∈ (ℝ)) → ((x · (1)) = x))) </jh>

Zero is not equal to one.

 stmt (ZeroOne  ((0) ≠ (1))) </jh>

Builders
This work by Tarski, as an introductory text, does not go into fine points of logic, but he does assume the ability to substitute equals for equals. Our equivalent is builders.

Although the builders are not interesting unless A0 and so on are real numbers, we state them without restricting them to real numbers. However we define the sum of two sets which are not real numbers, it is no hardship to make that definition obey the builders.  var (object A0 A1 B0 B1) stmt (AdditionBuilder  (((A0 = A1) ∧ (B0 = B1)) → ((A0 + B0) = (A1 + B1)))) stmt (MultiplicationBuilder  (((A0 = A1) ∧ (B0 = B1)) → ((A0 · B0) = (A1 · B1))))

stmt (LessThanBuilder  (((A0 = A1) ∧ (B0 = B1)) → ((A0 < B0) ↔ (A1 < B1)))) </jh>

Cited works

 * Tarski, Alfred (1946), Introduction to Logic and to the Methodology of Deductive Sciences, Dover edition of 1995, ISBN 978-0-486-28462-0