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The construction of R is different (and more awkward to understand) because
we must ensure that the cardinality of R is greater than that of Q.
Set c is a Dedekind cut iff
-
(strict inclusions!)
- c is closed downward:
if
and
then
- c has no largest element:
there isn't an element
such that
for all
You can think of a cut as taking a pair of scissors and cutting Q in two
parts such that one part contains all the small numbers and the other part
contains all large numbers. If the part with the small numbers was cut in such
a way that it doesn't have a largest element, it is called a Dedekind cut.
. We will refer to the elements of
R by giving them a subscript . The elements of Q can be embedded as
follows:
such that
. Furthermore we can define:
-
iff
(strict inclusion!)
-
-
there exists an
such that
-
-
is defined as:
- if not
and not
then
- if
and
then
- otherwise
There exists an alternative definition of R using Cauchy sequences: a
Cauchy sequence is a
such that
can be made arbitrary near to
for all
sufficiently large
and
. We will define an equivalence relation
on the set of Cauchy sequences as:
iff
can be made arbitrary close to
for all
sufficiently large
.
.
Note that this definition is close to ``decimal'' expansions.
Alex Lopez-Ortiz
Mon Feb 23 16:26:48 EST 1998