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?
Best Answer
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 $V$.