[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:


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.


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