TITLE: Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,...)
QUESTION [25 upvotes]: Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many applications in algebraic geometry and Diophantine geometry.
(A) I wonder to know if there are any nontrivial applications of set theory in branches like algebraic geometry, Diophantine geometry, K-theory or number theory (algebraic or analytic)? In particular:
1) are there statements in these fields which are independent from $ZFC$? 
2) Are there $ZFC$ provable statements  in these fields whose proofs are known just using set theoretic methods?
(B) On the other hand are there any results in set theory whose proofs are based on some techniques from the above quoted fields?
Giving references is appreciated.
(C) Are there any connections between model theory and algebraic or analytic number theory?

REPLY [3 votes]: I would like to point out a work that may be an answer to your question (B).
Misha Gavrilovich constructs a certain model structure on a category of sets (rather sets of sets), and argues that the covering number (of Shelah's PCF theory) can be obtained as a value of certain derived functor (in the sense of Quillen) with respect to this model structure.