Re: Interesting (IMO) question about the reals...

On 7 Feb 2007 03:26:19 -0800, sg552@xxxxxxxxxxxxx wrote:

I'm not certain that this question is well formed, but I will try
asking it anyway.

Given some set-theoretic construction of the real numbers, e.g.
Dedekind cuts, one can divide statements about the reals into two
types: those that are not meaningful in a purely axiomatic treatment,
and those that are. The first type would include, for example, any
statement that refers to the union of a real number, while the second
would include statements that only refer to operations which are
referred to in the real number axioms, such as addition of two reals
or the sup of a bounded set of reals.

I think it's pretty clear what you mean - since you say it's
not clear that you've stated it precisely, here's a possible
reformulation: The second type of statement is one that it
would make sense to ask about _any_ complete ordered field.

My question is this: is it possible that there is a statement of the
second type, which can be proved using Dedekind cuts (or whatever),
but which is not provable from the real number axioms alone?

No. The reason is that it's a theorem that any two complete
ordered fields are isomorphic. Hence anything true about
Dedekind cuts, _if_ it's something that makes sense for
any complete ordered field, must be true in every
complete ordered field, and hence must follow from
the axioms.

(Technicality: This assumes that we're talking about the
theory of complete ordered fields as a first-order
theory using set theory, so the Completeness Theorem
from logic applies...)

-Rotwang


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

David C. Ullrich
.

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