The origins of forcing in mathematical logic and other branches of mathematics 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) Fernando Tohmè, Gianluca Caterina, Rocco Gangle, "Forcing Iterated Admissibility in Strategic
Belief Models": Game theory (in particular epistemic game theory).
4) Solovay, Tennenbaum, "Iterated Cohen extensions and Souslin's problem" : Analysis-Topology.
5) Shelah, "Infinite abelian groups, Whitehead problem and some constructions" : Algebra.
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Edits:

1) I think, the work of Silver on the independence of gap two cardinal transfer principle, and Chang's conjecture is essentially the first application of forcing in model theory.
2) Are there any applications of forcing in dynamical systems?
3) What about "Recursion theory" and "Complexity theory"?
4) What about other branches of mathematics not mentioned above or in the answers?
 A: 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 $V$.
A: 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
A: 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.

A: 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.
A: 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.
A: 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.
