Dror Bar-Natan: Classes: 2003-04: Math 157 - Analysis I: | (7) |
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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**- Addition is commutative: .
**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:

- Sums such as are well defined.
- The additive identity is unique. (Also multiplicative).
- Additive inverses are unique. (Also multiplicative).
- Subtraction can be defined.
- iff (if and only if) or .
- iff or .
- iff or .
- iff .
- A ``well behaved'' order relation can be defined (i.e., the booloean operations , , and can be defined and they have some expected properties).
- The ``absolute value'' function
can be defined and for
all numbers and we have

The generation of this document was assisted by
L^{A}TEX2`HTML`.

Dror Bar-Natan 2003-08-14