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As everyone knows, forcing was created by Cohen to answer questions in set theory.

Question 1. What are the first applications of set theoretic forcing in other branches of mathematical logic, like number theory, computability theory, complexity theory and model theory.

Question 2. What are the first applications of set theoretic forcing in other branches of mathematics like topology, algebra, analysis, ....

Update. Here I will collect the answers and will add a few that I am aware:

1) Scott, "A proof of the independence of the continuum hypothesis": models of higher order theories of the Real numbers.

2) Feferman, "Some applications of the notions of forcing and generic sets": Number theory.

3) Abraham Robinson, "Forcing in model theory" : Model theory.

4) Fernando Tohmè, Gianluca Caterina, Rocco Gangle, "Forcing Iterated Admissibility in Strategic Belief Models": Game theory (in particular epistemic game theory).

5) Solovay, Tennenbaum, "Iterated Cohen extensions and Souslin's problem" : Analysis-Topology.

6) Shelah, "Infinite abelian groups, Whitehead problem and some constructions" : Algebra.

What about "Recursion theory" and "Complexity theory"?

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Since this question seems to ask for a list, and therefore does not have a unique correct answer, it should be community-wiki. –  HJRW Sep 16 '13 at 12:17
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In computability theory, arguably the first forcing argument there was Kleene-Post 1954, where an incomparable pair of $\Delta^0_2$ degrees is constructed by meeting countably many dense sets in a poset of approximations. Does this count as "set-theoretic forcing," or are you specifically asking about forcings over models of $ZF$ or a stronger theory (in which case I don't think Feferman's paper "Some applications of the notions of forcing and generic sets" counts)? –  Noah S Sep 16 '13 at 20:45
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The Baker-Gill-Solovay results on relativizations of P vs NP are a similar use of forcing as that of Kleene-Post. –  François G. Dorais Oct 13 '13 at 14:11
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You may have more answers if you asked for uses of "forcing" rather than "set-theoretic forcing" since the latter excludes the examples @NoahS and I gave. –  François G. Dorais Oct 13 '13 at 14:13
    
Also Friedberg-Muchnik Theorem –  Kaveh Oct 16 '13 at 1:18
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6 Answers 6

One application I know is Scott's construction of forcing extensions of models of higher order theories of the Real numbers. Scott quickly after the invention of forcing, used a forcing argument to show that the Continuum Hypothesis is independent of an axiomatization of Second Order Analysis.

He interpreted first order variables (usually real numbers) as Random Variables over a complete Boolean Algebra with the ccc, second order variables (usually functions from the reals to the reals) as functions from RV to RV, satisfying a certain technical condition and then defined the Boolean value of each statement, which is an element of the Boolean algebra. He then showed that the axioms have Boolean Value $\mathbb{1}$ and that the inference rules respect the Boolean value (i.e. do not decrease the Boolean value) and finally exhibits a Boolean algebra in which the statement $CH$ has Boolean value $\ne \mathbb{1}$. See here for more details: http://link.springer.com/content/pdf/10.1007%2FBF01705520.pdf

All in all this line of argumentation preshadows the way (set theoretic) forcing is presented in (for example) Jech's book, which explains forcing as forcing with Boolean valued models of the universe $V$.

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Dear Stefan, thanks. How are you? –  Mohammad Golshani Sep 16 '13 at 11:03
    
Thanks Mohammed, I am fine. The ESI conference has started here, so there are tons of interesting people around. Why didn't you come? –  Stefan Hoffelner Sep 16 '13 at 12:34
    
Oh, I just saw that you will show up anyway. So see you soon! –  Stefan Hoffelner Sep 16 '13 at 12:44
    
Yes, if everything goes well (I don't miss the plane, ...) I'll be there on Thursday, –  Mohammad Golshani Sep 17 '13 at 4:55
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First application of forcing to model theory: Abraham Robinson, Forcing in model theory (1970).

The forcing concept of Paul J. Cohen has had an immense effect on the development of Axiomatic Set Theory but it also possesses an obvious general significance. It therefore was to be expected that it would have an impact also on general Model Theory. I shall show that this expectation is indeed justified and that the forcing notion provides us with a new tool in Model Theory, which leads to a better understanding of concepts that have by now become classical in this area and to their further development.

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Boris Tsirelson constructed the first Banach space that did not have a subspace isomorphic to some $\ell_p$, $1\le p < \infty$ or $c_0$. This eventually led to the Gowers-Maurey construction of hereditarily indecomposable Banach spaces. Tsirelson writes that his forcing construction was motivated by Cohen's forcing arguments:

http://www.tau.ac.il/~tsirel/Research/myspace/remins.html

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Soon after discovering the forcing Solomon Feferman in his paper : "Some applications of the notions of forcing and generic sets, Fund. Math., vol. 56, pp. 325-345, 1965." applied forcing in number theoretic problems.

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There is a recent paper which is probably the first application of forcing in game theory (in particular epistemic game theory). The authors apply their ideas to the analysis of the very well known concepts of an iterated elimination of weakly dominated strategies and a common assumption of rationality.

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The first application of forcing to topology was probably Tenenbaum and Solovay's 1971 proof that, consistently with ZFC, there is no Suslin line$^*$. The existence of Suslin lines follows from $V=L$, and so was already known to be consistent with $ZFC$; showing the consistency of the opposite result required "killing off" all Suslin lines with an iterated forcing argument. I believe this was also the first iterated forcing argument.

(The article "Topology and Forcing" by Malykhin (http://iopscience.iop.org/0036-0279/38/1/R02/pdf/0036-0279_38_1_R02.pdf) is somewhat relevant.)


$^*$ A Suslin line is a complete dense linear order with no endpoints, satisfying the countable chain condition, which is not order-isomorphic to $\mathbb{R}$; see http://en.wikipedia.org/wiki/Suslin%27s_problem.

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