Interesting (IMO) question about the reals...

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.

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?

-Rotwang

.

Relevant Pages

  • Re: Are the Dedekind cuts uncountable?
    ... >> Of course the Dedekind cuts are uncountable as the reals are defined by ... >> of the subsets of the rationals and linearly moving along the rationals. ... sandwich theorem states that in between any two real numbers there is ...
    (sci.math)
  • Re: Dedekind Cuts, Fundamental Sequences: why?
    ... sequences nor Dedekind cuts add anything essential, ... view of what sort of reasoning about reals, functions of reals, and so on, ... and compare it with our expectations of completeness, ...
    (sci.math)
  • Re: The Dedekind Snap
    ... real line is _defined_ to be the set of all Dedekind cuts. ... It's made on the reals in the wiki article. ... This nonsense about creation by the mathematician is your own fevered ... "But then it looks as if the rationals can express the irrationals. ...
    (sci.logic)
  • Re: Are the Dedekind cuts uncountable?
    ... > Of course the Dedekind cuts are uncountable as the reals are defined by ... this sequence does not terminate, then the a_i's, being a nonempty ... while every countable subset has measure 0. ...
    (sci.math)
  • Are the Dedekind cuts uncountable?
    ... Of course the Dedekind cuts are uncountable as the reals are defined by ... of the subsets of the rationals and linearly moving along the rationals. ... A lower set B of rationals, is a subset of Q for which ...
    (sci.math)
Loading