TITLE: What are some important but still unsolved problems in mathematical logic?
QUESTION [60 upvotes]: In the past, first-order logic and its completeness and whether arithmetic is complete was a major unsolved issues in logic . All of these problems were solved by  Godel. Later on, independence of main controversial axioms were established by forcing method. 
I wonder if there still exist some "natural" questions in mathematical logic that are still unsolved? Or is it the case that most of the major questions have been already answered? 
I'd love to know about some important, but still unsolved problems that puzzle logicians and why would the young logician\mathematician care about those? (that is, Whey they are important?)
I'm not an expert in logic (nor in any other mathematical field, I'm undergraduate) but I'm interested in logic so I would like to know about the current problems that logicians face and what are the trends of research in the discipline nowdays and what type of problems people are trying to solve.
I know that logic is a vast term which includes many sub-disciplines: model theory, proof theory, set theory, recursion theory,  higher-order logics , non-classical logics, modal logics, algebraic logic and many others. So feel free to tell us about problems form whichever topic you would love to.

REPLY [9 votes]: One of the very interesting open problems in Logic (and CS) I think:
Is the field of constructible numbers known to be decidable?