Re: Is continuum completely filled up?

On 10 Jan 2007 13:42:58 -0800, Math1723 wrote:
Here's a paradox: The reals are a complete ordered field. If you extend
the reals by including infinitesimals, the resulting extension field is no
longer complete. By adding things to a complete ordered field, we get an
incomplete ordered field.

How does adding infinitessimals make *R incomplete? I haven't figured
out where that should fail.

*R does not satisfy the least upper bound property, which is what we mean
by saying that the field is incomplete. For example, the set of positive
infinitesimals is bounded above, but has no least upper bound. It may
seem paradoxical that you can turn a complete ordered field into an
incomplete one by adding more elements, but that's what follows from the
definition.


--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.

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