[FOM] CH and forcing

[Ali Enayat]

This is a response to Robert Lubarsky's recent posting (which was
written in response to a posting of Andreas Blass).

Lubarsky writes:

>If a Boolean-valued model is reduced by a non-generic ultrafilter, how does this get you anything coherent? ...[snip]...
>I don't doubt that V-check looks reasonable, but I'd think the new universe wouldn't even provably model ZF.

The following synopsis of the Boolean valued approach to forcing
addresses Lubarsky's skeptical remarks.

Given any model M of ZFC (where M is allowed to be a class model), and
any complete Boolean algebra B in M,  there are two canonical B-valued
universes of set theory associated with B:

I. The Boolean ultrapower of M,  let's call it M(B). This model is an
*elementary extension* of M. It can be described without any mention
of forcing, using an algebraic definition reminiscent of ordinary
ultrapowers. But it also can be described via "checks" , as mentioned
by Blass. Boolean ultrapowers have been investigated by a number of
people.
A key paper investigating Boolean ultrapowers is: Theory of Boolean
Ultrapowers*, Ann. Math. Logic (1971), by Richard Mansfield.

II. The (forcing) extension M^B. This is the usual B-valued
Scott-Solovay model as computed within M; it is discussed in most
expositions of forcing.

Now here is the connection between (I) and (II).

Theorem. For ANY ultrafilter U over B (note: U is allowed to be in M), we have:
(1) M is an elementary submodel of M(B)/U;
(2) M(B)/U is generically extended by M^B/U (in particular they have
the same ordinals).
(3) U is B-generic over M iff M(B)/U coincides with M.

The above Theorem has been known ever since the discovery of
Scott-Solovay Boolean valued models, but it is unfortunately absent
from most expositions of forcing. Its proof, as well as many other
interesting aspects of Boolean ultrapowers of models of set theory,
can be found in the lecture notes of Joel Hamkins at the 2011 Young
Set Theory Meeting, at the following URL:

http://www.math.uni-bonn.de/people/logic/events/young-set-theory-2011/Slides/Joel_David_Hamkins_slides.pdf