[FOM] Eliminability of AC and GCH

[Joe Shipman]

Aatu Koskensilta wrote:

"...by Shoenfield's absoluteness theorem, we know that any sentence of complexity below Delta-1-3 is necessarily decided the same way in any inner model. In the case of AC and GCH in particular we can do even better and show that conservativity holds in even higher reaches a bit farther up in the analytical hierarchy."

How much farther up? I am asking the following three distinct and well-defined questions: 

(1) what is the lowest quantifier complexity an analytical sentence that is theorem of ZFC can have and not be a theorem of ZF?

(2) what is the lowest quantifier 
complexity an analytical sentence that is a theorem of ZF+GCH can have and not be a theorem of ZF?

(3) what is the lowest quantifier complexity an analytical sentence that is a theorem of ZF+GCH can have and not be a theorem of ZFC?

(These are the only cases because GCH implies AC.)

-- JS

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