[FOM] CH and forcing
Andreas Blass
ablass at umich.edu
Wed Jul 6 19:41:18 EDT 2011
Let me try to answer several of the questions and comments that have been
raised here about Boolean-valued models and non-well-founded forcing
extensions.
Roger Bishop Jones wrote:
> ------------------------------
> 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).
--------------------------------
The last sentence quoted here is incorrect. The truth value of CH in the
model under consideration depends also on the non-well-founded sets.
Indeed, CH is about the collection of all subsets of the natural numbers,
and, when interpreted in a model, this will refer to the natural numbers
of that model, including non-standard (non-well-founded) ones.
He also wrote:
---------------------------------
> 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?
----------------------------------
The sets being added in a Boolean-valued model are (in general)
Boolean-valued sets. The forcing extensions that produce violations of CH
add new Boolean-valued sets, i.e., sets x such that, for all sets y in the
ground model, the Boolean truth value of x=y is zero.
Drake O'Brien wrote:
----------------------------------
>> In this context, I would interpret "forcing extension" to mean a
> Boolean-valued
>> model. With this interpretation, the assertion is correct.
>>
>
> Could you please show this, to the uninitiated?
----------------------------------
In the usual Boolean-valued models for proving consistency of an assertion
A (like, for example, the negation of CH) with ZFC, all the following
statements have Boolean truth-value 1:
The axioms of ZFC;
A;
V-check is a transitive class containing all the ordinals and satisfying
ZFC;
[infinitary sentences expressing, for each set x in the original
universe:] The members of x-check
are exactly the y-check as y ranges over members of x.
In other words, with truth-value 1, the Boolean-valued model contains an
exact copy V-check of the original universe (the ground model) and it
satisfies ZFC plus A.
All this should be available in standard expositions of Boolean-valued
models, for example Bell's book, or the older book by Rosser, or Jech's
set-theory bible.
It may be helpful to add that the "exact copy" phenomenon remains true in
the two-valued models (if any) obtained by reducing the Boolean-valued
model by a *generic* ultrafilter (if such ultrafilters exist), but it need
not remain true if one uses a non-generic ultrafilter. Nevertheless, even
with a non-generic ultrafilter, all the finitary, first-order consequences
of "exact copy" remain correct; the reduction of V-check modulo any
ultrafilter is an elementary extension of the ground model.
Roger Bishop Jones also presented an argument against Woodin's
"weaker claim", namely:
>
> "This weak version simply asserts that if
> (M,E) |= ZFC,
> then there exists a structure
> (M**, E**) |= ZFC + "CH is false",
> such that M is contained in M** and such that
> E = E** intersection (M x M)"
>
The anti-Woodin argument contained several lemmas, including:
>
> Lemma2: ZFC |- (V(w+4) subset_of M) =>
> ((M |= CH) <=> (V(w+4) |= CH))
This lemma is false. In fact neither half of the bi-implication follows
unless one has some additional information about M, like well-foundedness.
In general, M could have new, non-standard natural numbers, in which case
its "subsets of w" would be quite different from those of V. (Like Jones,
I abbreviate omeega as w.) And its functions from the set of countable
ordinals to the set of subsets of w would also be quite different from
those of V.
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