Re: naive question from a non-mathematician


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.

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?

.

Relevant Pages

  • Re: naive question from a non-mathematician
    ... There is a first order theory of algebraically closed fields of ... cardinality. ... cardinality C, without constructing it, from model theory; ... complete real closed field with cofinality the integers is R up to ...
    (sci.math)
  • Re: naive question from a non-mathematician
    ... cardinality. ... cardinality C, without constructing it, from model theory; ... first order theory has models for every infinite cardinal. ... complete real closed field with cofinality the integers is R up to ...
    (sci.math)
  • Re: What are numbers?
    ... Take cardinality as an undefined primitive. ... > Quantity refers to a property of a set. ... Worry: ... First order: nice and simple but not categorical. ...
    (sci.math)
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