Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense the complexity or power of a theory. 


Does anyone know what is the proof theoretic ordinal of $ZFC$ or any non-trivial $ZFC$ extensions? Wikipedia says this is unknown for $ZFC$ as of 2008, but maybe there has been some recent progress? Thank you.



Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?
As Andres and I have stated in the comments, we are still nowhere near a proof-theoretic analysis of $ZFC$ or similar theories; even full second-order arithmetic remains well out of reach.


The paper "The Art of Ordinal Analysis" by Michael Rathjen does a good job of both describing ordinal analyses which have succeeded (e.g., $PA$), and showing how new difficulties arise as we climb towards higher and higher comprehension axioms (still around the level of $\Pi^1\_2$-$CA\_0$ by the end of the paper). I think this might be worth reading, if you want an explanation of why finding proof-theoretic ordinals of strong theories is "hard."


