Repeating myself on RVM

[Joe Shipman]

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

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