The 13 Postulates
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
``of positive numbers'' so that the following 13
- Addition is associative:
(``'' means ``for every'').
- The number 0 is an additive identity:
- Additive inverses exist:
(``'' means ``there is'' or ``there exists'').
- Addition is commutative:
- Multiplication is associative:
- The number 1 is a multiplicative identity:
- Multiplicative inverses exist:
- Multiplication is commutative:
- The distributive law:
- The trichotomy for
: for every , exactly one of the
following holds: ,
- Closure under addition: if and are in , then so is
- Closure under multiplication: if and are in , then so is
- 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
Boolean operations , , and can be defined and they
have all the expected properties).
- The ``absolute value'' function
can be defined and for
all numbers and we have
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