TITLE: Connections between Complexity Theory & Set Theory
QUESTION [12 upvotes]: Inspired by Joshua Grochow and Iddo Tzameret's answers in a post on http://cstheory.stackexchange.com , I would like to get more references on possible connections between complexity theory and set theory here in this post. My question simply is: 

What are examples of those points where set theoretical axioms, concepts and tools like large cardinals and forcing appear in the complexity theory? 

As a matter of fact note that the statement $\mathsf P=\mathsf{NP}$ (and other similar assertions) remains unaffected by most set theoretic methods and assumptions. So this may help you restrict the searching domain for finding possible connections between set theory and complexity theory through excluding straightforward independence results.

REPLY [6 votes]: As far as I'm aware, large cardinals have not found any application in complexity theory. The main concepts from set theory that have found some application in complexity theory are forcing and generic filters. In addition to the references mentioned by Mohammad Golshani, there is also the earlier book by Krajíček, Bounded Arithmetic, Propositional Logic and Complexity Theory, as well as Gaisi Takeuti's papers, e.g., Forcing on bounded arithmetic (with Yasumoto).
It is not clear, however, how useful these connections are for addressing the fundamental questions in complexity theory.  In this MO question I raised this question and learned that there are some serious obstacles to using forcing to construct models of arithmetic that satisfy (or refute) arithmetic sentences of interest.
In a different direction, Joel Friedman has attempted to phrase certain problems about circuit lower bounds in the language of sheaves on a Grothendieck topology.  I don't know if this counts as "set theory" in your book, but if you're interested, I commented briefly on Friedman's work at cstheory.stackexchange.com.