[FOM] Motivating the concept of a generic filter

[Jay Sulzberger]

On Wed, 10 Oct 2007, Timothy Y. Chow <tchow at alum.mit.edu> wrote:

> Jay Sulzberger wrote:
>> 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.
>
> Can you be more explicit?  For example, can you write down a detailed
> dictionary that gives exact parallels between a generic (ultra)filter and
> these other "generic" objects?  And on the basis of that analogy, what
> would you predict would be the result of imposing the "generic" condition?
> Remember, modding out by *any* ultrafilter yields a model of ZFC.  Why
> would you then be led to consider generic filters in your quest to prove,
> say, the consistency of ~CH?
>
> Tim Chow

Tim, I will attempt a very short inadequate answer to the first
of your two questions.  I rephrase your two questions:

1. What is the common concept 'generic' that these three cases
are good examples of?

2. What about Cohen's use of 'generic sets' in his independence proofs?

The version of 2 I would like to answer goes:

What is it about ZF that we can transfer the meta-statement
'ultrafilters give old fashioned two valued models' into the
statement 'here is a set in a new model which is, itself,
suitably generic within the model'?  That is, as I, perhaps
obscurely, stated, how come some parts of set theory are
surprisingly like an equational theory.

I have just looked at your "Forcing for Dummies" at

http://www-math.mit.edu/~tchow/mathstuff/forcingdum

Clearly you are acutely aware of the puzzlement of 2.  It is
indeed the central puzzle of forcing.  The general concept of
generic is a crude thing, but, in some of its paradigm cases we
can see a bit of the puzzle of 2 peeping through.  Your
"Fundamental Theorem" is nearly a version of question 2.

I will not attempt an answer to 2 due to incompetence.  I have
not looked at forcing since taking Jack Schwartz's wonderful
course in the late Sixties of the last century.  There is work
done since Cohen's first proofs which sheds light on issues near
to question 2.  Indeed, people on this list have done serious
work on these things.

Here is a very feeble partial answer to question 1:

The first most naive example of genericity is defined in terms of
the Galois connection given by a relation between two sets.  Let
|= be a subset of the cartesian product of sets S x M. As usual
we write "m |= s" to mean that the pair (s, m) lies in |=.
Usually, "m |= s" is read "Model m satisfies sentence s." and it
is also read "Sentence s is true in model m.".  We have our two
Galois closures, one for S, and one for M.  Then:

Definition: Let A be a closed subset of M.  Let a lie in A.  a is
a generic point for A when the closure of {a} is A.

This definition handles in a straightforward way the algebraic
geometry case and the universal algebra case.  The probability
algebra case is some different.

I just glanced at the Wikipedia entry:

http://en.wikipedia.org/wiki/Generic_point

and I think it is good, though too short.  Linked articles also
seem good, and this one offers a useful, again too short,
overview of one of the three cases:

http://en.wikipedia.org/wiki/Scheme_theory


Here is part of the Dictionary:

algebraic geometry case: S is the set of all polynomial equations
in some given set of "variables", with coefficients in some given
fixed field. A model m is an affine space over any field
extending the field of coefficients, with coordinates places
labeled by variables, with a distinguished point, call the point
p.  Put another way, a model is a field with a map from the set
of variables into (the carrier of) the field.  Then we define
(m, p) |= s when s is true at p, in the usual sense that when you
substitute the values of corresponding coordinates into s, you
get that s is true.  Often, sentences are normalized, so that the
right hand side is 0, in which case, (m, p) |= s when s evaluated
at p is zero.  Loose remark on limitations: Every prime ideal has
a generic point, that is, the set of all models of a prime ideal
has a generic point.  But the non-prime ideal generated by x*y
does not.


universal algebra case: Fix a signature for an algebra.  Fix a
set of variables, as above.  A model, just as above, is an
algebra of the signature, with a map from the set of variables
with target the carrier of the algebra.  A sentence is two
polynomials in the variables, with a "=" between them.
Satisfaction is as usual.  This case is important because there
are no limitations: You can always give an algebra by generators
and relations.  That is, any set of sentences has a generic
model.


Boole's finite probability algebra case: This is set forth in
Boole's "Laws of Thought" and Hailperin's "Boole's Logic and
Probability".  I learned this case from the second edition of
Hailperin's book.  There is an abstract of an abstract at

http://www.panix.com/~jays/Boole.four.page.summary.for.1.August.2006.Cork.conference.Sulzberger.v.4.pdf

As in the first two cases, we fix a finite set of variables.

A probability algebra is a Boolean algebra with a map p, from
elements, often called events, of the Boolean algebra to the
reals.  p is required to satisfy the usual probability axioms.
Note that, seemingly, this is not an equational class of
algebras.

A model is a probability algebra with a map from the set of
variables, with target the carrier of the Boolean algebra part of
the probability algebra, as in the first two cases.  The
signature of a probability algebra is the usual (finite) Boolean
operation-symbols, often called union, intersection, complement,
0, and 1, with an extra one place operation-symbol, call it p.
The set of sentences is the union of two disjoint sets: first the
set of all Boolean algebra sentences, as in the universal algebra
case, and second, the set of all expressions of the form

p(b) = r

where b is a Boolean polynomial in our variables, and r is a real
number.  Boole's method may be extended to handle more sentences.

We have again the usual relation of satisfaction between a
sentence, of either kind, and a model.  We cannot use the whole
satisfaction relation |= to handle Boole's genericity, but must be
more careful.  I will not lay out in this post what is done.

Boole's passage to the most free probability algebra has
limitations: Given a set of sentences we require that the
sentences be strictly consistent in order to get the generic
probability algebra.  The case of inconsistency never arises in
the universal algebra case.  And a set of sentences in the
probability algebra case can fail to be strictly consistent, but
still be consistent.  Boole's theory deals with this case also.


I beg the indulgence of FOMers!  I hope to write a tiny bit more
next week.  Of course, many FOMers could do a better job on this
than I, and so, come on down!

oo--JS.


PS. In all three cases to get uniqueness of the generic point, as
is often asked for today, you add the condition that, in every
model, the evaluated variables generate the structure.