[FOM] Cohen and Kunen as sources on forcing
Timothy Y. Chow
tchow at alum.mit.edu
Sun May 20 14:42:05 EDT 2007
Colin McLarty wrote:
> So I'd like advice on Cohen versus Kunen. I understand that to become
> a research set theorist the student would need Kunen. Is Cohen a
> better beginner's book?
I can give my perspective as a fellow novice.
The answer to your question depends somewhat on the learning style of your
student. Is he a top-down person who needs to have an intuitive big
picture first before slogging through the details, or is he a bottom-up
person who needs the details spelled out at each step? I think most
people need a little bit of the big picture at the beginning, but after
that need to work from the bottom up. For such a person, Kunen is a
better primary reference. For most students, Cohen's book is too sketchy
to serve as the primary treatment.
That's not to say that Cohen's book can't be used as a supplement. There
are illuminating ideas in Cohen's book that Kunen chooses to leave out.
It might not be too bad an idea to skim Cohen to start with to get an
overview, with the intention of moving to Kunen as soon as possible.
> So: would it be terribly unfair to use Cohen? Would I cheat the
> student of the benefit of later improvements? Or does Cohen give a
> still-usable introduction to the basics?
"Later improvements" in my opinion have mostly to do with slickening up
the machinery to make the proofs shorter and the method more powerful.
They aren't necessarily improvements from the pedagogical standpoint.
Cohen, for example, makes it seem like forcing is intimately related to L
and minimal models. We know now that forcing is a much more general
technique. So which is better for the student, an "advanced" treatment
that strips forcing down to its true essentials, or an early treatment
that gives you some idea of how someone might have invented this stuff?
Different people will answer this question differently, but in any case
this is the dichotomy that you're facing.
By the way, you haven't mentioned a third alternative, which is to
introduce forcing by way of Boolean-valued models. This is something that
I haven't investigated thoroughly myself, so perhaps others can comment
further on it. My understanding is that, from a pedagogical point of
view, Boolean-valued models may be a good compromise between the two
alternatives you're considering. It has the benefit of being fully
general, and therefore escapes the problem I mentioned above about Cohen's
treatment being somewhat limited in scope (from a modern standpoint). On
the other hand, the analogy between logical AND/OR on the one hand, and
meet/join in a Boolean algebra on the other hand, is very helpful
intuitively---certainly when compared to the mysterious concept of a
generic filter on an arbitrary poset. The price paid is that you do have
to build some machinery about Boolean algebras which some folks might feel
is an unnecessary detour since forcing can be developed without it.
By the way, I'm hoping to revise my "forcing for dummies" article later
this year, and clean it up into a publishable exposition:
http://alum.mit.edu/www/tchow/mathstuff/forcingdum
You might find the current version helpful nonetheless.
> Amazon.com believes that both Cohen and Kunen are out of print.
Bookfinder.com will give you more reliable information about this sort of
thing than Amazon.com will. For example, it will tell you that Kunen is
still in print in the U.K.
Tim
More information about the FOM
mailing list