Section 4: Problem 1 Solution »

Section 4: The Integers and the Real Numbers

Real numbers

The set of real numbers is an ordered set with two binary operations and such that:
  1. Associative laws for addition and multiplication.
  2. Commutative laws for addition and multiplication.
  3. Existence of and .
  4. Existence of the negative of , and the reciprocal of .
  5. Distributive law of multiplication over addition.
  6. If and , then and .
  7. has the least upper bound property.
  8. If then there is such that .
[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 of is called inductive if and implies . Then, the set of positive integers is .
A section of the positive integers .
Well-ordering property: every non-empty subset of has a smallest element.
Archimedean ordering property: has no upper bound in .

Induction principle

Induction Principle for Positive Integers: if and is inductive, then .
Strong Induction Principle: if and for every : implies , then .