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?**