Here is a plan for “constructing” the real numbers.
From here on I’ll use R
to denote the reals.
- Define
R as a “complete, ordered
field”. To do that we’ll have to:
- define
“field” axiomatically.
- define
“ordered field” axiomatically.
- define
“complete, ordered field” axiomatically.
- While
doing this we’ll want to become convinced that we’ve captured the
properties of R. I. e.
we’ll want to become convinced that a COF (guess what this means) has all
the properties we usually associate with R
and doesn’t have any properties which R
lacks.
- Because
a COF was defined axiomatically, we’d need to show that a COF exists. We could do this by constructing a set,
which we’d call R, using things with which we’re familiar,
such as the rationals, and then showing that this new set has all the COF
properties. WE WILL NOT DO THIS. I
mention it here to point out that it is an absolutely necessary part of a
formal construction of R.
- Finally
we’d need to define a “COF isomorphism” and then show that every COF is
isomorphic to the set R that we
constructed in C. WE WON’T DO
THIS. But if we did, we could then
say that R is the unique, complete, ordered field, where by “unique” we mean “up to
isomorphism”.