A Models of the Real Numbers

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1. Aug 26, 2016

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.

Last edited: Aug 26, 2016
2. Aug 26, 2016

AgentCachat

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

3. Aug 26, 2016

Staff: Mentor

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.

Last edited: Aug 26, 2016
4. Aug 29, 2016

pwsnafu

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.