What are the alternatives?

[Joe Shipman]

A recent high point for Set Theory and Foundations was this popular article by Natalie Wolchover:

https://www.quantamagazine.org/to-settle-infinity-question-a-new-law-of-mathematics-20131126

She discusses four alternatives, but only the first two in depth:

1) Inner model axioms such as Woodin’s V=Ultimate-L (which implies CH)
2) Forcing axioms such as Martin’s Maximum (which implies the continuum equals aleph_2)
3) Multiverse axioms which don’t pick out a preferred “the” universe V but investigate models of ZFC on an equal footing
4) Skeptics who deny not only the determinateness of set-theoretical propositions but even their meaningfulness

It seems to me that BOTH the inner model axioms AND the forcing axioms are inconsistent with there being a real-valued measurable cardinal. Are any set theorists working on alternative axioms which are consistent with RVM?  (I’m not proposing RVM itself as an axiom, I just want to understand if anyone working on new axioms is even allowing it as possible!)

— JS


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