Recently Toshiyasu Arai submitted "An ordinal analysis of $\Pi_{N}$-Collection", which claims ordinal analysis of full second-order arithmetic. There has been discussion before on MO about this, see:
- Consistency of Analysis (second order arithmetic)
- Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?
The consensus seemed to be that we were a long way away from proving anything like this. For such a big result, I am struggling to find much information on it. I think the two most obvious questions for a lay-person is:
- What does Arai claim to be proof-theoretic ordinal of $Z_2$?
- What does this mean for the ordinal analysis of $ZFC$?
Additionally what are the wider implications Proof Theory and Reverse Mathematics?