Timeline for How can any theory prove well-foundedness of ordinals above $\omega_1^{\text{CK}}$?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 27, 2016 at 16:14 | vote | accept | John Gowers | ||
| Jan 27, 2016 at 13:29 | answer | added | user57888 | timeline score: 14 | |
| Jan 27, 2016 at 12:20 | comment | added | Emil Jeřábek | ... as the objects never live in the same model. With recursive ordinals as a common representation that is reasonably absolute, we can ask about the same ordinal in two different theories. | |
| Jan 27, 2016 at 12:19 | comment | added | Emil Jeřábek | Basically, yes. But more precisely, you first take a recursive representation of the ordinal, and then ask if the theory proves it well-founded (which, for ZFC, might involve unwinding the recursive definition into the von Neumann representation that ZFC internally uses for ordinals). One reason for this is to have a comparison across theories: say, ZFC has an ordinal it calls $\omega_3$, and a fancy theory X (that does not look at all like set theory) has an ordinal it calls PSM9. Are they the same? Which one is longer? These would be meaningless questions, ... | |
| Jan 27, 2016 at 12:15 | answer | added | Peter LeFanu Lumsdaine | timeline score: 7 | |
| Jan 27, 2016 at 12:02 | comment | added | John Gowers | @EmilJeřábek Ah, I see. So really it is a double condition: the smallest ordinal that the theory can prove is recursive, but can't prove is well founded. | |
| Jan 27, 2016 at 12:00 | comment | added | Emil Jeřábek | The proof theoretic ordinal is the smallest recursive ordinal the theory cannot prove well-founded. ZFC thinks it can prove the well-foundedness of crazy huge ordinals, but it does not know how to represent them by recursive relations on the integers. | |
| Jan 27, 2016 at 11:47 | history | asked | John Gowers | CC BY-SA 3.0 |