Repeating myself on RVM

[Monroe Eskew]

Here are some interesting statements independent of the existence of RVM:

(1) Suslin’s Hypothesis.  (Laver 1987)
(2) The continuum is a successor / regular cardinal.  (Why? If kappa is measurable, then adding *at least* kappa random reals forces that kappa is RVM.)
(3) What is the the size of {2^kappa : kappa < continuum}?  (See Gitik-Shelah 1993.)

Best,
Monroe



> On Aug 3, 2020, at 5:59 AM, Joe Shipman <joeshipman at aol.com> wrote:
> 
> I read a lot of papers which talk about the unsatisfactoriness of “new axioms” for non-absolute statements like CH. It seems clear that most people working in the area don’t like V=L and related axioms because they are too restrictive about that sets may exist, and feel like prospects for settling CH are dim.
> 
> But I have a still never heard a satisfactory explanation of what is wrong with the axiom that Lebesgue measure can be extended to all sets of reals in a way that remains countably additive (though no longer translation-invariant).
> 
> What is an example of an independent-of-ZFC statement anyone cares about that this axiom does NOT decide (apart from propositions implying the consistency of cardinals larger than “measurable“)?
> 
> — JS
> 
> Sent from my iPhone