This must be known inside out by model theorists by I have no cluse whether the following is true or not:
Denote by $F_n$ the free group on $n$ generators. Suppose that $n\neq m$. Are the groups $F_n$ and $F_m$ elementarily equivalent? What if we allow the set of generators to be countably infinite?
I'd appreciate any hints for the proof of the answer to this question. Looking at ultrapowers seem to me to be intractable, but who knows?
Yes they are. This is know as the Tarski Problem, and was recently solved by Sela and Kharlampovich-Myasnikov. The solution however is very difficult and spans more than 100 pages. Sela's approach at least uses sophisticated ideas from geometric group theory.