Real numbers

Theorem 1.1.2:

NOTE

$\mathbb{R}$ is a commutative, ordered field that is order-complete. $\mathbb{R}$ is non-trivial

Axioms:

Axioms of Addition (A):

$(\mathbb{R},+)$ is an abelian group: (A1) Associativity: $\forall x,y,z\in \mathbb{R}:(x+y)+z=x+(y+z)$ (A2) Neutral Element: $\exists 0\in \mathbb{R},\forall x\in \mathbb{R}:x+0=x$ (A3) Inverse Element: $\forall x\in \mathbb{R},\exists y\in \mathbb{R}:x+y=0$ This $y$ is denoted as $-x$. (A4) Commutativity: $\forall x,y\in \mathbb{R}:x+y=y+x$

Axioms of Multiplication (M):

$(\mathbb{R}\setminus \{0\},\cdot )$ is an abelian group: (M1) Associativity: $\forall x,y,z\in \mathbb{R}:(x\cdot y)\cdot z=x\cdot (y\cdot z)$ (M2) Neutral Element: $\exists 1\in \mathbb{R},\forall x\in \mathbb{R}:x\cdot 1=x$ (M3) Inverse Element: $\forall x\in \mathbb{R}\setminus \{0\},\exists y\in \mathbb{R}:x\cdot y=1$ This $y$ is denoted as $x^{-1}$ or $\frac{1}{x}$. (M4) Commutativity: $\forall x,y\in \mathbb{R}:x\cdot y=y\cdot x$

Distributivity (D):

$\mathbb{R}$ is a field: $\forall x,y,z\in \mathbb{R}:x\cdot (y+z)=x\cdot y+x\cdot z$

Order Axioms (O):

$\le$ is a total order (O1) Reflexivity: $\forall x\in \mathbb{R}:x\le x$ (O2) Transitivity: $\forall x,y,z\in \mathbb{R}:(x\le y\wedge y\le z)\Rightarrow x\le z$ (O3) Antisymmetry: $\forall x,y\in \mathbb{R}:(x\le y\wedge y\le x)\Rightarrow x=y$ (O4) Totality: $\forall x,y\in \mathbb{R}:x\le y\vee y\le x$

Compatibility Axioms (K):

(K1) $\forall x,y,z\in \mathbb{R}:x\le y\Rightarrow x+z\le y+z$ (K2) $\forall x,y\in \mathbb{R}:(x\ge 0\wedge y\ge 0)\Rightarrow x\cdot y\ge 0$

Order Completeness

This is the crucial axiom that distinguishes $\mathbb{R}$ from $\mathbb{Q}$.

Let $A\subseteq \mathbb{R}$ and $B\subseteq \mathbb{R}$ be non-empty sets such that

$$A\ne \varnothing ,B\ne \varnothing$$

and

$$\forall a\in A,\forall b\in B:a\le b.$$

Then there exists a $c\in \mathbb{R}$ such that

$$\forall a\in A,\forall b\in B:a\le c\le b.$$

Interpretation of Order Completeness: This axiom states that there are no “gaps” in the real number line. Every set bounded above has a least upper bound (supremum), and every set bounded below has a greatest lower bound (infimum).

From the view of topology space

This is saying:

NOTE

In the order topology, $\mathbb{R}$ is connected (equivalently, every continuous map $\mathbb{R}\to \{0,1\}$ is constant), and it is a linear order without gaps.

This is a stronger statement than Connectedness

Definition: Supremum & Infimum

Suppose $(X,\le )$ is a poset with $A\subseteq X$

• Supremum: least upper bound
  • $\forall a\in A(a\le \text{sup }A)$ (the least such bound)
  • similar to closure in topology under $\subseteq$
  • $\text{sup }A=\text{inf }$upper bounds
• Infimum: greatest lower bound
  • $\forall a\in A(a\ge \text{inf }A)$ (the greatest such bound)
  • similar to interior in topology under $\subseteq$

Corollary: Archimedean Principle

• $\forall x\in \mathbb{R},y>0\exists n\in \mathbb{N}:ny>x$
  • Let $y=1$ , then $\forall x\in \mathbb{R}\exists n\in \mathbb{N}$ such that $n>x$.
• $\forall \epsilon >0\exists n\in \mathbb{N}:\frac{1}{n}<\epsilon$