Repeating myself on RVM

[Gabriel Goldberg]

The independence of Suslin's Hypothesis from ZFC + RVM is proved in
Caicedo's paper "Real valued measurable cardinals and definable
well-orderings of the reals," Corollary 1.33. In "Real valued measurability
and Lebesgue mesurable sets," Caicedo shows that the measurability of all
projective sets is independent of ZFC + RVM; anyone who cares about RVM
should care about this question. Finally, obviously RVM tells you almost
nothing about sets larger than the powerset of the continuum, a topic that
I feel confident someone cares about, at least on good days.

Best,
Gabe







On Mon, Aug 3, 2020 at 12:30 PM 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
>
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