Graduate Why Can't the Continuum Hypothesis Be Decided Using Standard Real Number Models?

[lavinia]

I know that there are several models of the real numbers, some where the Continuum Hypothesis holds, others where it does not. I would like to understand why the usual definition of the reals, limits of Cauchy sequences of rational numbers under the usual absolute value norm, isn't one of these models and why then one can not decide the Continuum Hypothesis for it in particular.

 

[AgentCachat]

Because the system of axioms and derived theorems leads to the undecidabity of the CH?

 

[fresh_42]

I've just yesterday looked into Hewitt, Stromberg, Real and Abstract Analysis, on the search for hints or ideas on one of @micromass' analysis challenges. Their entire first chapter deals with set theoretical basics, starting with the proof of the various equivalences for AC and ending with the construction of $\mathbb{C}$ as the algebraic closure of $\mathbb{R}$ as Cauchy-sequences modulo null-sequences. (Dedekind cuts are an exercise there.)

It also contains some considerations like, e.g. "For all cardinals $\mathfrak{a}$ with $2 \leq \mathfrak{a} \leq \mathfrak{c}$ is $\mathfrak{a}^{\aleph_0} = \mathfrak{c}$ and $\mathfrak{a}^{\mathfrak{c}} = 2^{\mathfrak{c}}$".

I haven't looked into greater detail, yet, (esp. where they use CH and where not), but if you have the chance, it might be a good reference for this.

 

[pwsnafu]

Because the system of axioms and derived theorems leads to the undecidabity of the CH?

To be more precise CH is undecidable in ZFC. We construct the rationals from the integers, which is constructed from naturals (Peano) which in turn can be constructed from ZFC.