Construction of R

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.