## 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:

- 1 is in S,
- For each n S, n' S,

then S contains all of the numbers.
All models of Peano's Axioms are isomorphic.

**Addition:** Let n be a natural number. Then

- n + 1 = n'
- For all natural numbers m, n + m' = (n + m)'

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