How small is aleph_1?

[Monroe Eskew]

Weak Diamond is a good example. It’s actually equivalent to c<2^{aleph_1}, as shown by Devlin-Shelah.


> On 10.04.2021, at 03:56, JOSEPH SHIPMAN <joeshipman at aol.com> wrote:
> 
> 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
> 
> Sent from my iPhone