Are we still Researching the Foundations of Math? To my understanding, most of math can be founded in ZFC set theory.  And, because of Gödel, we can never really prove the consistency of ZFC and hence mathematics as a whole.  My question is, are we still researching the foundations or does it end at Gödel?  Is there any research about the soundness of ZFC axioms, or is that more of a philosophical question?  Is there a possibility to, say, found set theory in a larger system that is even more self evident and prove ZFC from there?  Or is that unlikely?  Is there debate about ZFC (I know there is about C) being taken as a priori?
I listened to a talk from W. Hugh Woodin who studies the continuum hypothesis.  Because CH is independent from ZFC, Woodin is doing research to found a new axiom of set theory that is "true" and will make CH decidable.  (I'm altogether unfamiliar with how studying math inside a system can give any insight to the soundness of that system.)  But, it got me thinking about if there is any research existing or ongoing, of the same breed as Woodin's addressing the truth of the other axioms of ZFC.  Are there mathematical results that tell us that set theory has to be true similar to Woodin's research?
 A: Of course we are doing research in the foundations of mathematics.


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*We want to understand the necessity of ZFC, and in particular the role of the Axiom of Choice, in our mathematics. This means that we need to understand how mathematics in ZF could look like.

*We want to understand the influence of set theoretic axioms on the mathematical universe, and vice versa. For example, if we wanted to add the assumption "Every set of reals of size $<2^{\aleph_0}$ is a null set" to our axiomatic system. Is this going to add a contradiction? Can we maybe prove this? What are the consequences of such assumption, or its failure?
For this we need to develop tools for studying the set theoretic universe, for studying independence in set theory, and then we need to study the "actual mathematics" one can do when assuming this statement or its negation.
For the independence results, we have forcing and inner models in the study of set theory, and we study forcing axioms and cardinal characteristics of the continuum, and other combinatorial axioms with these tools, as well as their influence on the rest of mathematics.

*We may want to study other foundations. Perhaps you want to formalise your proof into a computer program, and perhaps your approach makes ZFC an unsuitable foundation. Maybe you simply don't want to accept the Axioms of Power set and Choice. Maybe you reject the law of excluded middle.
I don't know you. The point is, some people do these things. Some people want to understand how mathematics looks like in other foundations, as well as the interactions between these foundations and ZFC. Examples would be Homotopy Type Theory, intuitionistic logic, Quine's New Foundations, category theory, and more.
All these things, of course, make the study of foundations of mathematics vibrant, deep, and interesting. And we have yet to begun discussing topics like philosophy of mathematics.
A: There are numerous research activities within axiomatic set theory that aim to further understand various foundational issues. Not so much to 'resolve' what is true and what is not, if that is at all possible (or even, in that context, desirable) but rather to get a better understanding of the foundations offered by set theory. I'm not an expert on that and I'm sure insightful answers touching on set-theoretic foundations will follow. 
Since your title mentioned foundations of mathematics I will mention that there are foundations which are not set theoretic, or at least do not take $\in $ as primitive. There are categorical foundations (topos theory, Lawvere's axiomatization of the category of sets) and, more recently, there is homotopy type theory and Voevodsky's univalent foundations. 
