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What were the initial motivations of the use of the proper forcing.?

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Harrington once told me that Shelah called it proper forcing cause he thought that was the proper way to do forcing. –  Amit Kumar Gupta Nov 26 '10 at 0:32
@Amit: Haha. That sure sounds like Shelah. –  Andrés Caicedo Nov 26 '10 at 1:31
Gapo: I think your question, as stated, is a bit too ambitious for this site. You are asking us for something that could easily end up being a short paper if answered properly. Would you mind saying something about what you are looking for, where in the literature you have looked, that sort of thing? It may help us. –  Andrés Caicedo Nov 26 '10 at 1:33
I believe that the initial motivations for proper forcing are well explained throughout Shelah's book. If you want a brief look at the evolution of the subject, a good place to start reading at is the "Proper forcing" chapter in the handbook of set theory (written by Uri Abraham). Regarding the open problems, you can check this paper: –  Haim Nov 26 '10 at 1:59
@Andreas: I did not expect to have one good answer to my question but a sort of collective answer where every one could bring one original information as indeed Amit Haim and Stefan did. It was my first question I'm sorry if is not in the spirit of mathoverflow next time I will ask thing more precisely. If I should be more precise now I would say that initial motivation was the most important part of the question for me so that Stefan answer me. –  gapo Nov 26 '10 at 11:08

4 Answers 4

up vote 7 down vote accepted

I agree with Andres that this is a very ambitious question. Let me throw in a tiny little bit of information:

As far as I know, Jensen's construction of a model of CH without Souslin trees was one of the first uses of both countable support iteration and master conditions (conditions generic over a countable elementary submodel of a sufficiently large initial part of the universe).

One of the first publications using proper forcing seems to be Shelah 100, with the hilarious title "Independence results" (JSL 45 (1980), 563-573).
This paper not only introduces proper forcing (without proofs of the the iteration theorem and so on), but also oracle-ccc forcing and oracle-proper forcing.

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What I remember is that first Laver solved the Borel conjecture by an countable support iteration which added a real at every stage (1976). That such an iteration can be of any use, was very surprising at the time. Then Baumgartner introduced the very general notion of property A, which included most (all?) standard forcings known then which added reals and showed that countable support iteration of them behaves nicely (1978). Then came Shelah, who gave the proper definition (1980). This was again a surprising thing, as Baumgartner's definition was combinatorial (containing combinatorial properties of the poset, eh, almost) while Shelah simply required that P should preserve all stationary subsets of all sets of the form $[A]^{\aleph_0}$, that is, a semantic definition. Notice that Shelah's new theory gave new and elegant proofs to old theorems, as Baumgartner's consistency of that any two $\aleph_1$-dense sets of reals are isomorphic or Mitchell's consistency of the tree property of $\aleph_2$.

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Roslanowski once asked Shelah about this, and has kindly typed down the answer he got:

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This would have been my answer too, Assaf! –  Todd Eisworth Mar 19 '12 at 17:15
What a fun article! Thanks. –  Andrés Caicedo Mar 19 '12 at 17:19
Thanks to Assaf for pointing out Andrzej's paper, and to Andrés for linking this on twitter! –  Andrés Villaveces Mar 19 '12 at 17:48

My understanding is that part of the early motivation was the observation of the attractive features of two main classes of forcing:

  • ccc forcing. Preserves all cardinals. Preserves stationary subsets of $\omega_1$. Closed under finite support iterations.
  • countably closed forcing. Preserves $\omega_1$. Preserves stationary subsets of $\omega_1$. Closed under countable support iterations.

And what was wanted was a class of forcing notions in which these two kinds of forcing could be mixed together in iterations, while still preserving $\omega_1$. That is, what is wanted is a class of forcing that contains all ccc forcing, all countably closed forcing, which all preserve $\omega_1$ and which support some kind of iteration theorem. The full class of all $\omega_1$-preserving forcing does not fit the bill, since it is not closed under iterations. But meanwhile, the class of proper forcing does have the desired features...

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