Timeline for Can the incompleteness of set theory be isolated to questions about arithmetic?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| 11 hours ago | comment | added | zeb | How strange - you answered the question which was originally in my heart, but the technical question I actually ended up asking, about the existence of a CCMA theory, now seems like the more interesting question. I suppose that if no one can resolve that within a day or two I will accept this answer. | |
| 12 hours ago | history | edited | Noah Schweber | CC BY-SA 4.0 |
deleted 878 characters in body
|
| 12 hours ago | comment | added | Noah Schweber | @zeb Ah yes I messed things up with existence. Removing for now ... | |
| 12 hours ago | comment | added | zeb | The existence argument seems a bit dubious - there's no guarantee that $\theta_0$ isn't already causing an inconsistency with true arithmetic, regardless of how the later $\theta_i$ are defined. The correctness claim seems to hold up - perhaps a simpler reason is that truth would become definable in any $\omega$-model of $T^+$. | |
| 12 hours ago | history | edited | Noah Schweber | CC BY-SA 4.0 |
added 26 characters in body
|
| 12 hours ago | comment | added | Joel David Hamkins | Noah, in the existence part, your definition is not recursive, since $\theta_{i+1}$ does not depend on $\theta_i$ or the earlier part of the construction. So I think you've haven't said what you intended. It seems you want to add them successively, not independently. | |
| 12 hours ago | history | edited | Noah Schweber | CC BY-SA 4.0 |
added 708 characters in body
|
| 12 hours ago | comment | added | Joel David Hamkins | To attempt to describe things for the OP a bit more softly: The truths of higher levels of set theory, such as truth in $H_{\omega_1}$ or $V_{\omega^2}$ and so forth, are more complicated than arithmetic truth, and so we cannot expect that those truths would be settled correctly by the desired sort of theory. | |
| 12 hours ago | history | answered | Noah Schweber | CC BY-SA 4.0 |