[FOM] CH and forcing
Roger Bishop Jones
rbj at rbjones.com
Mon Jul 4 16:53:59 EDT 2011
On Monday 04 Jul 2011 17:39, Andreas Blass wrote:
> Roger Bishop Jones asked about the assertion
> > "Every model of ZFC has a forcing extension in which CH
> > fails."
> In this context, I would interpret "forcing extension" to
> mean a Boolean-valued model. With this interpretation,
> the assertion is correct.
> If one insists on two-valued models, the assertion is
> still correct, provided one does not require models to
> be well-founded. One can get a two-valued model as the
> quotient of a Boolean-valued model by any ultrafilter
> (not necessarily generic).
I remain in doubt.
Consider specifically the interpretation of ZFC consisting of
all pure well-founded collections of accessible rank.
The only way to extend this interpretation is by adding sets
which have inaccessible rank or which are not well-founded.
In neither case could the extension change the truth value
of CH, which is fixed as soon as we have all the well-founded
sets of rank less than w+(a small natural number).
I am puzzled by the suggestion that a boolean valued model
could make any difference.
Surely if we want to use a boolean valued model to force the
negation of CH we must have an appropriate forcing condition
to determine the algebra of truth values, and this could
only be the case if there were some set which could be added
(to the model described above) which would change the truth
value of CH?
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