On the least upper bound property

 

On the least upper bound property[edit]

Axiom 4, which requires the order to be Dedekind-complete, implies the Archimedean property.

The axiom is crucial in the characterization of the reals. For example, the totally ordered field of the rational numbers Q satisfy the first three axioms, but not the fourth. In other words, models of the rational numbers are also models of the first three axioms.

Note that the axiom is not firstorderizable, as it expresses a statement about collections of reals and not just individual such numbers. As such, the reals are not given by a first-order logic theory.

On models[edit]

A model of real numbers is a mathematical structure that satisfies the above axioms. Several models are given below. Any two models are isomorphic; so, the real numbers are unique up to isomorphisms.

Saying that any two models are isomorphic means that for any two models ${\displaystyle (\mathbb{�},0_{\mathbb{�}},1_{\mathbb{�}},{+}_{\mathbb{�}},{\times }_{\mathbb{�}},{\le }_{\mathbb{�}})}$ and ${\displaystyle (�,0_{�},1_{�},{+}_{�},{\times }_{�},{\le }_{�}),}$ there is a bijection ${\displaystyle �:\mathbb{�}\to �}$ that preserves both the field operations and the order. Explicitly,

• f is both injective and surjective.
• f(0_ℝ) = 0_S and f(1_ℝ) = 1_S.
• f(x +_ℝ y) = f(x) +_S f(y) and f(x ×_ℝ y) = f(x) ×_S f(y), for all x and y in ${\displaystyle \mathbb{�}.}$
• x ≤_ℝ y if and only if f(x) ≤_S f(y), for all x and y in ${\displaystyle \mathbb{�}.}$

Tarski's axiomatization of the reals[edit]

Main article: Tarski's axiomatization of the reals

An alternative synthetic axiomatization of the real numbers and their arithmetic was given by Alfred Tarski, consisting of only the 8 axioms shown below and a mere four primitive notions: a set called the real numbers, denoted ${\displaystyle \mathbb{�}}$, a binary relation over ${\displaystyle \mathbb{�}}$ called order, denoted by infix <, a binary operation over ${\displaystyle \mathbb{�}}$ called addition, denoted by infix +, and the constant 1.

Axioms of order (primitives: ${\displaystyle \mathbb{�}}$, <):

Axiom 1. If x < y, then not y < x. That is, "<" is an asymmetric relation.

Axiom 2. If x < z, there exists a y such that x < y and y < z. In other words, "<" is dense in ${\displaystyle \mathbb{�}}$.

Axiom 3. "<" is Dedekind-complete. More formally, for all X, Y ⊆ ${\displaystyle \mathbb{�}}$, if for all x ∈ X and y ∈ Y, x < y, then there exists a z such that for all x ∈ X and y ∈ Y, if z ≠ x and z ≠ y, then x < z and z < y.

To clarify the above statement somewhat, let X ⊆ ${\displaystyle \mathbb{�}}$ and Y ⊆ ${\displaystyle \mathbb{�}}$. We now define two common English verbs in a particular way that suits our purpose:

X precedes Y if and only if for every x ∈ X and every y ∈ Y, x < y.

The real number z separates X and Y if and only if for every x ∈ X with x ≠ z and every y ∈ Y with y ≠ z, x < z and z < y.

Axiom 3 can then be stated as:

"If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets."

Axioms of addition (primitives: ${\displaystyle \mathbb{�}}$, <, +):

Axiom 4. x + (y + z) = (x + z) + y.

Axiom 5. For all x, y, there exists a z such that x + z = y.

Axiom 6. If x + y < z + w, then x < z or y < w.

Axioms for one (primitives: ${\displaystyle \mathbb{�}}$, <, +, 1):

Axiom 7. 1 ∈ ${\displaystyle \mathbb{�}}$.

Axiom 8. 1 < 1 + 1.

These axioms imply that ${\displaystyle \mathbb{�}}$ is a linearly ordered abelian group under addition with distinguished element 1. ${\displaystyle \mathbb{�}}$ is also Dedekind-complete and divisible