Re: naive question from a non-mathematician

On 28 May 2006 12:34:28 -0700, "Gene Ward Smith"
<genewardsmith@xxxxxxxxx> wrote:


David C. Ullrich wrote:

That's certainly the right way to think of the reals. I object
to calling it a "definition" - before we can call it a definition
we need to either assume that a complete ordered field exists
(as is often done in "advanced calculus" courses, explicitly
or not) or construct one from something else we're assuming,
like maybe set theory.

There is a first order theory of algebraically closed fields of
characteristic zero; just add an axiom scheme for characteristic zero,
and another for algebraic closure, to the axioms for a field. It has a
model in the algebraic numbers, so it has models in every infinite
cardinality. It can be shown to be categorical in uncountable
cardinality. So, we do have an algebraically closed field of
cardinality C, without constructing it, from model theory; and hence we
could use the top-down method(s) I've outlined.

Um. Leaving aside questions of whether invoking model theory means
that you've proved the existence of something "without constructing
it", leaving aside questions of why anyone would think that this
was simpler that what we might venture to call the "standard"
construction, etc:

_How_ does it follow that the reals are least-upper-bound complete?
That's not a first-order property.

Elsewhere you say that it _does_ follow. I'm not sure what you
mean by "categorical in uncountable cardinality":

(i) If aleph is an uncountable cardinal then any two models
of cardinality aleph are elementarily equivalent

(ii) Any two uncountable models are elementarily equivalent.

Not that I have an actual proof, but if anything it seems
to me that this would tend to indicate that the
construction does _not_ imply the reals are lub-complete.

Real closed fields are not categorical, but you can of course show the
first order theory has models for every infinite cardinal. An order
complete real closed field with cofinality the integers is R up to
isomorphism, so I guess you could ask about a first order theories with
given cofinality. Given two infinite cardinals A < B, when does there
exist a real closed field with cardinality B and cofinality A? Does the
question of the size of the continuum screw things up here?


************************

David C. Ullrich
.

Relevant Pages

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