[FOM] CH and forcing

Roger Bishop Jones rbj at rbjones.com
Tue Jul 5 15:54:39 EDT 2011

On Tuesday 05 Jul 2011 20:02, Aatu Koskensilta wrote:
> Quoting Roger Bishop Jones <rbj at rbjones.com>:
> > Consider specifically the interpretation of ZFC
> > consisting of all pure well-founded collections of
> > accessible rank.
>    Obviously we can't add stuff, e.g. new reals, to
> <V_kappa, epsilon>, with kappa the first inaccessible,
> and hope this gets us a model violating CH. However,
> there's nothing to stop us starting with V_kappa and
> using the usual machinery of forcing to construct a
> model in which CH fails. It's just that, once all the
> details are sorted out, we find that V_kappa is not a
> substructure of what we end up with, as one might expect
> based on the usual story. This is a matter of putting
> the oft repeated standard guiding intentions to one
> side, and simply concentrating on the mathematics of it.

Woodin is explicit in his claim, that the extension refuting 
CH contains the original model.
(see my most recent post for a more detailed statement of 
Woodin's claim).

Would you describe a model obtained by forcing which does 
not include the original as a "forcing extension"?

Roger Jones

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