Section 4: The Integers and the Real Numbers

Real numbers

The set of real numbers is an ordered set $(\mathbb{R},<)$ with two binary operations $+$ and $\cdot$ such that:

Associative laws for addition and multiplication.
Commutative laws for addition and multiplication.
Existence of $0$ and $1$ .
Existence of the negative of $x$ , and the reciprocal of $x\neq0$ .
Distributive law of multiplication over addition.
If $x>y$ and $p>0$ , then $x+z>y+z$ and $x\cdot p>y\cdot p$ .
$<$ has the least upper bound property.
If $x<y$ then there is $z$ such that $x<z<y$ .

[note]A field is a set (without order) satisfying 1-5. An ordered field satisfies 1-6. A linear continuum satisfies 7-8.[/note]

Integer numbers

A subset $A$ of $\mathbb{R}$ is called inductive if $1\in A$ and $x\in A$ implies $x+1\in A$ . Then, the set of positive integers is $\mathbb{Z}_{+}=\cap_{A\mbox{ is inductive}}A$ .

A section of the positive integers $S_{n}=\{m\in\mathbb{Z}_{+}|m<n\}=\{1,\ldots,n\}$ .

Well-ordering property: every non-empty subset of $\mathbb{Z}_{+}$ has a smallest element.

Archimedean ordering property: $\mathbb{Z}_{+}$ has no upper bound in $\mathbb{R}$ .

Induction principle

Induction Principle for Positive Integers: if $A\subset\mathbb{Z}_{+}$ and $A$ is inductive, then $A=\mathbb{Z}_{+}$ .

Strong Induction Principle: if $A\subset\mathbb{Z}_{+}$ and for every $n\in\mathbb{Z}_{+}$ : $S_{n}\subset A$ implies $n\in A$ , then $A=\mathbb{Z}_{+}$ .

Subpages

Section 4: The Integers and the Real Numbers

Section 4: The Integers and the Real Numbers

Real numbers

Integer numbers

Induction principle