Timeline for Belief in consistency of extremely large cardinals
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 17, 2021 at 19:57 | comment | added | Thomas Benjamin | Why the downvote? | |
| Dec 3, 2016 at 8:53 | comment | added | Thomas Benjamin | (cont.) of $L^{2}_{\omega}$). One might use Henkin models of $L^{2}_{\kappa}$ as the means to study such consistency results. Regarding the heuristic that "tells you where to draw the line"--finding such a heuristic would be an integral part of developing the "research program". | |
| Dec 3, 2016 at 8:40 | comment | added | Thomas Benjamin | @CameronZwarich: Regarding extendible cardinals, note the forward direction ($\Rightarrow$) of Magidor's Thm. 4: If $\kappa$ is extendible then $L^{2}_{\kappa}$ is $\kappa$-compact. This suggests (to me at least) the following: if one can show the relative consistency of some theory $T$+ "There exists an extendible cardinal" in which one can interpret $L^{2}_{\kappa}$ , one can prove $T$+"There exists an extendible cardinal"$\vDash$$L^{2}_{\kappa}$ is $\kappa$-compact (Question: can $\omega$ be deemed an extendible cardinal? If not, then one can use Thm. 4 to prove the non-compactness | |
| Dec 3, 2016 at 4:50 | comment | added | Cameron Zwarich | That's an interesting approach, and it certainly seems more sensible to describe large cardinals via model-theoretic properties than the usual definitions. In the case of extendible cardinals, it's interesting that the property asserted for $\kappa$ is even stronger than that of $\omega$, because compactness fails quite badly for $L^2_\omega$. Can it provide an answer by which to judge the consistency of these new model-theoretic axioms? I assume you can come up with some that are obviously inconsistent, but is there a heuristic that tells you where to draw the line? | |
| Dec 2, 2016 at 13:30 | history | edited | Thomas Benjamin | CC BY-SA 3.0 |
corrected spelling
|
| Dec 2, 2016 at 12:15 | history | edited | Thomas Benjamin | CC BY-SA 3.0 |
completed answer
|
| Dec 2, 2016 at 11:14 | history | answered | Thomas Benjamin | CC BY-SA 3.0 |