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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
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:
- (strict inclusions!)
- c is closed downward:
if and then
- c has no largest element:
there isn't an element such that for all
- iff (strict inclusion!)
- there exists an such that
- is defined as:
- if not and not
- if and then
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.
Mon Feb 23 16:26:48 EST 1998