Timeline for Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 5, 2013 at 20:49 | vote | accept | user40919 | ||
| Oct 5, 2013 at 20:35 | comment | added | user40919 | That's helpful, thank you both Noah and Andres. I will take a look at that Rathjen reference. | |
| Oct 5, 2013 at 20:24 | answer | added | Noah Schweber | timeline score: 16 | |
| Oct 5, 2013 at 20:05 | comment | added | Noah Schweber | For example, already an analysis of $\Pi^1_2$-$CA_0$ by Rathjen required very complicated ordinal notations, and recently (see the intro to miami.uni-muenster.de/servlets/DerivateServlet/Derivate-5965/…) a serious error was found (and patched, I think) in Rathjen's work around this level. | |
| Oct 5, 2013 at 20:04 | comment | added | Andrés E. Caicedo | No real progress towards this goal, in the sense that proof theory has a long way to go to reach $\mathsf{ZFC}$ or comparable theories. In some of his latest talks (on the consistency of $\mathsf{PA}$), Cohen suggested he had a way of understanding this ordinal $\alpha$, but I could never see a coherent presentation, and I doubt there was something sufficiently developed to allow us to unambiguously identify an ordinal as the proof-theoretic ordinal for $\mathsf{ZFC}$. | |
| Oct 5, 2013 at 20:03 | comment | added | Noah Schweber | I believe that proof-theoretic ordinals for much weaker theories (e.g., $\Pi^1_3$-$CA_0$, a subtheory of second-order arithmetic) are still unknown; the state of the art appears to be around $\Pi^1_2$-$CA_0$, if I understand the state of things correctly. | |
| Oct 5, 2013 at 19:18 | review | First posts | |||
| Oct 5, 2013 at 19:20 | |||||
| Oct 5, 2013 at 18:59 | history | asked | user40919 | CC BY-SA 3.0 |