[FOM] CH and forcing

Mitchell Spector spector at alum.mit.edu
Sat Jul 9 18:42:19 EDT 2011

```Bob Lubarsky <lubarsky.robert at comcast.net> wrote:
> ...
> If a Boolean-valued model is reduced by a non-generic ultrafilter, how does
this get you anything coherent? I thought the point of genericity is that if an
existential statement is forced then so is a witness. If you don't have that
then the truth lemma doesn't go through, and you end up having no control over
the resulting structure. I don't doubt that V-check looks reasonable, but I'd
think the new universe wouldn't even provably model ZF.

The point of genericity is that you don't add new ordinals, or, in fact, new
members of any set in the ground model.

For a slightly different perspective on this, look at the partial-ordering
version I posted.  Here's a link:

http://www.cs.nyu.edu/pipermail/fom/2011-July/015607.html

In this construction, the "generic filter" is still generic, but it's generic
over a different ground model from the one you started with.  V is elementarily
embedded in a new class model <M;E> (a limit ultrapower of V), and the generic
filter is generic over M, not over V.

The nice thing is that M and E are definable in V (with parameters), and an
M-generic filter G exists in V, so you don't need to "extend the universe".

M will usually have non-standard ordinals (even non-standard integers), but it's
still true that the ordinals of M[G] are the same as the ordinals of M.  And
both M and M[G] satisfy ZFC.

By the way, I have one small correction to the description of the construction
in my post.  At one point, I wrote:

> Given a partial ordering P, the collection of dense open sets is a filter.

I meant that the collection of dense open sets _generates_ a filter.  The filter
is the collection of all subsets of P which contain a dense open set.

Mitchell Spector

```