[FOM] CH and forcing - counter claim

Roger Bishop Jones rbj at rbjones.com
Tue Jul 5 15:45:20 EDT 2011

My doubts about Woodin's claim under discussion are here
crystalled in a sketch for a proof in ZFC that Woodin's
claim is not provable in ZFC.

The form of the statement which I used in my first post was
the form in which it was stated by Hamkins.

"every model of set theory has a forcing extension in
which CH fails"

Woodin offers the following weaker claim (WWC):

"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)"

I see that both authors claim that their statement was
proven by Cohen.

It seems to me that even the weaker of these claims is
stronger than the result obtained by Cohen, and that the
independence of CH can be used to prove in ZFC that neither
claim is provable in ZFC.

I confine my attention to the weaker claim.

The proof goes like this:

Lemma1: ZFC |- (V(w+4) |= CH) <=> CH

[note that there is no appeal to second order logic here, or
anywhere in this sketch, the "V" here is as defined in ZFC]

Lemma2: ZFC |- (V(w+4) subset_of M) =>
		 ((M |= CH) <=> (V(w+4) |= CH))

Lemma3: ZFC |- (M |= ZFC) =>
		Exists N st V(w+4) subset_of N /\ (N |= ZFC)

using WWC we then obtain:
	ZFC |- (M |= ZFC) =>
		Exists N st V(w+4) subset_of N
			/\ (N |= ZFC + not CH)

and using lemmas 1 and 2 we obtain:
	ZFC |- (M |= ZFC) => not CH

contradicting the independence of CH from ZFC.

Hence:  ZFC |- (M |= ZFC) => (ZFC |-/- WWC)

However, I can't see how to prove lemma3.
I would guess that it might be done by starting with V(w+4) 
and then iterating the godel operations along the ordinals 
in the hypothesized model of ZFC.

Roger Jones

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