Diamond and Square

[JOSEPH SHIPMAN]

Thanks! That reminds me, it’s annoying that Diamond means something in modal logic as well. I think it was Solovay who suggested the set-theoretic interpretation of the modal quantifiers:
Possibly P = V_k |= P for some inaccessible k
Necessarily P = V_k |=P for all inaccessible k.
So one could ask if <> <> ? It seems we don’t know the answer to that, but because it is (in its original form) a statement about sets of low rank, if it’s true in any V_k with k inaccessible, it’s true in all of them, thus 
<> <> -> [] <> []
Here those 6 “symbols” each of which is represented by two ASCII characters concatenation are respectively 
Modal quantifier “possibly”
Set-theoretic principle “Diamond”
Implication symbol
Modal quantifier “necessarily”
Set-theoretical principle “Diamond”
End-of-proof box symbol!

There is also a combinatorial “square” principle, which ordinarily is used with a cardinal subscript, but Jensen and Solovay showed that the existence of a regular uncountable cardinal not satisfying it is equiconsistent with the existence of a Mahlo cardinal. So if there are provably no Mahlos, [] [] []

;)
— JS

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.

Sent from my iPhone
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