> Therefore the reals would have to include all kinds of numbers that
> no
> finite description at all. I am not sure I believe such things exist,
> 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

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

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