From: Virgil <Vmhjr@frii.com>
Subject: Re: What kind of infinity?
Date: Sat, 17 Feb 2001 16:29:36 -0700
Newsgroups: sci.math
Summary: Ideal-theoretic construction of field of non-standard real numbers

Once one has a countably infinite set, such as the set of all natural 
numbers or the set of integers, to get an uncountable set one need only 
consider the set of all subsets of the countably infinite set. Then 
repeat ad infinitum to get larger uncountabilities.


To get a non-standard model of the reals is a bit more complicated.

You must start with a countable set, often taken to be the set of 
positive intyegers, N, and the complete ordered filed of real numbers, 
(R,+,*,<). Construct the set, R^N, of all functions f:N -> R as an an 
ring by the point-wise operations (f+g)(x) = f(x) = g(x) and 
(f*g)(x) = f(x)*g(x).

Now in the ring R^N, find a maximal non-principal ideal, I (each of 
which will correspond to a non-principal ultrafilter on N), and form the 
field (R^N)/I. 

Note: The more common formulation is based on ultrafilters rather than 
maximal ideals, but the results are equivalent.
There is a straightforward mapping between ideals in R^N and filters on 
N.



Those that swallow the hyperreals and choke on the uncountables seem to 
me to be swalloing the camel and straining at the gnat.