Math 3000 Lectures 16 & 17

|A| |(A)| for any set A.

| | = c

Continuum Hypothesis

Binary operations on a set S (i.e., maps S S S )

Commutative, Associative Laws

Closure on a s/set.

Identity elements

Inverse elements

Field Axioms

Ex: Finite Fields

Peano's Axioms

a. 1 is a number.
b. For each number n, there is another number n' called the successor of n.
c. For each number n, n' does not equal 1.
d. For all numbers m and n, if m' = n' then m = n.
e. Inductive Property: If a set S of numbers has the properties:

then S contains all of the numbers.

All models of Peano's Axioms are isomorphic.

Addition: Let n be a natural number. Then

  1. n + 1 = n'
  2. For all natural numbers m, n + m' = (n + m)'
Multiplication: Let n be a natural number. Then
  1. n 1 = n,
  2. For all natural numbers m, mn' = mn + m.