[FOM] Simple and difficult

Joe Shipman JoeShipman at aol.com
Wed Apr 3 08:58:41 EDT 2013

My wording was imprecise, I meant (2) not refuted, but provably not a theorem, otherwise Ex~(x=x) would do. 

I believe Harvey found a sentence whose consistency strength is a subtle cardinal:

There exists κ such that every transitive set S into which κ can be injected contains x and y such that x is a proper subset of y and x ≠ Ø and x ≠ {Ø}. 

Does anyone know a simpler sentence of set theory which has been neither proven nor disproven?

-- JS

Sent from my iPhone

On Apr 2, 2013, at 11:13 PM, Joe Shipman <JoeShipman at aol.com> wrote:

It's easy to write down a sentence in the language of Peano Arithmetic which is both short and unsettled:

AxEyAzAw (x<y & ~(SSzSSw=y V SSzSSw=SSy))

What's the shortest or simplest sentence you can come up with in the language of set theory that is either (1) not settled (2) provably not a theorem of ZFC if ZFC is consistent?

-- JS

Sent from my iPhone
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