© | Dror Bar-Natan: Classes: 2004-05: Math 157 - Analysis I: (6) Next: Class Notes for Tuesday September 14, 2004 Previous: Homework Assignment 1

# The 13 Postulates

this document in PDF: Postulates.pdf

Everything you ever wanted to know about the real numbers is summarized as follows. There is a set of real numbers'' with two binary operations defined on it, and (addition'' and multiplication''), two different distinct elements 0 and 1 and a subset of positive numbers'' so that the following 13 postulates hold:

P1
Addition is associative: ('' means for every'').
P2
The number 0 is an additive identity: .
P3
Additive inverses exist: ('' means there is'' or there exists'').
P4
P5
Multiplication is associative: .
P6
The number 1 is a multiplicative identity: .
P7
Multiplicative inverses exist: .
P8
Multiplication is commutative: .
P9
The distributive law: .
P10
The trichotomy for : for every , exactly one of the following holds: , or .
P11
Closure under addition: if and are in , then so is .
P12
Closure under multiplication: if and are in , then so is .
P13
The thirteenth postulate is the most subtle and interesting of all. It will await a few weeks.

Here are a few corollaries and extra points:

1. Sums such as are well defined.
2. The additive identity is unique. (Also multiplicative).
3. Additive inverses are unique. (Also multiplicative).
4. Subtraction can be defined.
5. iff (if and only if) or .
6. iff or .
7. iff or .
8. iff .
9. A well behaved'' order relation can be defined (i.e., the Boolean operations , , and can be defined and they have all the expected properties).
10. The absolute value'' function can be defined and for all numbers and we have

The generation of this document was assisted by LATEX2HTML.

Dror Bar-Natan 2004-09-13