The idea of Hilbert's program was to start with a simple finitary logic and proof the consistency of more complex systems in this system. Of course, this turned out to be problematic. Even when absolute consistency is replaced by relative consistency.
But I just wondered, if someone attempted to tried it the other way. So, what is the simplest logical system X, such that X can prove the consistency of ZFC relative to X?
In other words, are there things in ZFC that can be dropped from foundational point of view?
Is there any scientific article about this?