How small is aleph_1?

[JOSEPH SHIPMAN]

Both Martin’s axiom (MA) and the axiom of a real-valued measurable cardinal (RVM) imply that if CH is false, 2^(aleph_1)=c (and RVM also implies that CH is false).

Are there any well-known extensions to ZFC which nontrivially imply that if CH is false, 2^(aleph_1)>c? 

Axioms like GCH, or the Powerset Size Axiom (a<b implies 2^a<2^b) don’t count since the implication is trivial.

— JS

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