[FOM] Motivating the concept of a generic filter
jays at panix.com
Tue Oct 9 15:51:13 EDT 2007
On Tue, 9 Oct 2007, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
> I'm currently trying to improve my informal "forcing for dummies" article,
> with a view to publishing it as an expository article. One major change
> that I'm likely to make is to discuss Boolean-valued models, since they
> seem quite intuitive to me.
> There remains one sticking point, which is that I can't seem to find a way
> to motivate the concept of a *generic* filter in a satisfactory manner.
> By "a satisfactory manner" I mean, roughly speaking, that someone not in
> possession of the concept could see why one would be led to define it.
> In some texts, the approach is to discuss generic filters and Martin's
> axiom in the context of infinitary combinatorics, long *before* any
> mention of forcing. Then by the time you reach forcing, you're supposed
> to be comfortable with generic filters already. I don't find this to work
> very well as "motivation." Even if you've seen a generic filter before,
> why would you define p ||- phi in terms of generic filters? This
> definition seems to be pulled out of a hat.
> In Cohen's book, he takes the approach of starting with a minimal model
> and seeing what it would take to add the "missing" subset you want, while
> otherwise keeping things as similar to the minimal model as possible.
> Thus one restricts to transitive epsilon-models, and one wants to keep the
> same ordinals. He shows that a naive attempt to adjoin a missing subset
> fails, so that one is led to consider more carefully "all conditions at
> once" in some sense, to make sure once chooses a set that mesh together
> properly. So far so good. At this point, however, he says that the chief
> point is to consider generic elements, with no "special" properties that
> are the source of trouble in the naive attempt, and see what is "forced"
> to hold. This still seems like a leap to me. What would make you think
> that the seemingly hopelessly vague concept of a "generic element" makes
> any sense and would solve your problems?
Ah, because in algebraic geometry, and in the theory of
equational and Horn classes of models, and, even, as Boole points
out, in the case of finite probability algebras, there is always
a generic point/model/probability algebra. And perhaps,
seemingly, naively, there are other examples. In all these cases
(the Boole case is not well known) the generic thing may be
described as the model which satisfies all "sentences" of a
"theory" and no more sentences. One does have to be careful in
the Boole case. Thus in these examples, the concept of generic
point, most free algebra, Boole's probability algebra given by
generators and relations, is seen to be, each, a special case of
the general concept "the generic thing". So one might
conjecture, the concept of a "generic set" might also be well
defined, in a way very like the earlier definitions. As usual
the attainment of such a new concept comes with new conditions of
applicability; the algebraic set must be one piece, the sentences
must be equational, there must exist some probability algebra
satisfying the relations, etc.. The surprising thing to me is
that the set theory construction succeeds despite these
limitations, that is, that parts of set theory are near to being
an equational theory.
> The Boolean-valued model approach works nicely up to a point. We don't
> know which statements will hold and which ones won't hold in our new
> model, so we take them all at once and track their interdependencies using
> a complete Boolean algebra B. If M is a model of ZFC, it's maybe not
> immediately apparent why we need to choose B to be complete in M, rather
> than complete in V, but as soon as one tries to prove that M^(B) is a
> Boolean-valued model of ZFC one quickly sees the need for the sups and
> infs of subsets of B to be in M. Modding out by an ultrafilter to get an
> actual model rather than a Boolean-valued model is also pretty natural.
> But again, why generic filters? Generic filters aren't needed to get new
> models of ZFC; if M is a transitive epsilon-model and B is any Boolean
> algebra in M that M thinks is complete, and U is any ultrafilter of B,
> then M^(B)/U is a model of ZFC. Generic filters are needed if you want
> M^(B)/U to have some "nice" properties, but why would you think that you
> need those nice properties? Indeed, in Bell's book, the independence of
> the continuum hypothesis is proved before generic filters are discussed.
> He takes the relevant poset, embeds it in a complete Boolean algebra in a
> natural way, and shows that this works. It seems a bit like magic; and
> even if you don't blink at this proof, it's not at all clear why you would
> think that you could then dispense with the Boolean algebra and work with
> *generic* filters in an arbitrary poset.
> So, any suggestions for getting past this sticking point?
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