> Therefore the reals would have to include all kinds of numbers that
have
> no
> finite description at all. I am not sure I believe such things exist,
and
> for a similar reason I am not sure I believe that every member of the
> hypothetical "power set of the integers" exists either.
Hi Jesse,
I think you are asking an important question. I think it relates to
self referencing definitions - which is implicit in the axiomatic
approach.
Here's a go at defining the describable reals...
The formal axiom that distinguishes the reals from the rationals is the
completeness axiom. This can take a number of different forms. One
version is : Every set bounded above has a supremum.
Let Q be the set of rationals. This is a solid starting point because
every rational has a finite description.
Consider that we restrict ourselves to sets of the form
X = { x in Q | p(x) }
where p(x) is a predicate on a rational number x. Given that every
rational has a finite description, and p(x) is associated with a finite
description, we deduce that X and all its members are describable.
The completeness axiom says that X has upper bound => sup(X) exists.
It seems to me that we can take this as a way of defining the
*describable* reals like sqrt(2) or pi - because it is precisely the
"holes" in the set of rationals that make us want to "plug them up".
Eg sqrt(2) = sup { x in Q | x^2 < 2 }
Naturally, we will only be able to describe countably many numbers in
this way. Let's call this set R.
Note that R was defined in terms of completing the rationals, rather
than "completing itself" which is the normal axiomatic way of defining
the reals.
A self reference approach would have been to write
X = { x in R | p(x) }
So that we are using the reals to define the reals!
- David
