Timeline for How similar are large cardinals, over $L$?
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 5, 2016 at 21:35 | vote | accept | Noah Schweber | ||
| Nov 5, 2016 at 12:32 | comment | added | Joel David Hamkins | I made the update this morning. If this is correct, it seems to settle the issue. | |
| Nov 5, 2016 at 12:28 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
handled general case.
|
| Nov 5, 2016 at 3:16 | comment | added | Joel David Hamkins | I think this idea will generalize beyond $\Pi_2$, so that there can be no $\gamma$-decisive first-order large cardinal notion, if there can be a stationary proper class of such cardinals. I'm going to sleep on it; check back tomorrow! | |
| Nov 5, 2016 at 3:08 | comment | added | Joel David Hamkins | I think I can prove now that if the class of unfoldable cardinals (or uplifting cardinals) is a stationary class (or even just stationary with respect to $\Sigma_2$-definable class clubs), then unfoldable cardinals cannot be $\gamma$-decisive for any $\gamma$. It's late now, but I'll post the details tomorrow. This may generalize to any $\Pi_2$-definable large cardinal, which will cover essentially all the standard large cardinal concepts. | |
| Nov 5, 2016 at 2:06 | comment | added | Joel David Hamkins | By the way, uplifting cardinals are another $\Pi_2$ case, and very weak (below Mahlo in consistency strength). | |
| Nov 5, 2016 at 2:06 | comment | added | Joel David Hamkins | In general, you cannot expect that this is even a cardinal---this would make a strictly stronger notion. So you can't expect to get $\gamma$-decisiveness by making them all align like that to each other. | |
| Nov 5, 2016 at 2:03 | comment | added | Noah Schweber | Probably a naive question, but: if $\kappa$ is unfoldable, what can be said about the set of cardinals $\kappa$ "unfolds" to, that is, the set of $j(\kappa)$s for $j$ an unfoldability embedding? | |
| Nov 5, 2016 at 1:02 | comment | added | Joel David Hamkins | Oh, sure, it is fine to unaccept. | |
| Nov 5, 2016 at 1:02 | comment | added | Noah Schweber | That looks really interesting, thanks! | |
| Nov 5, 2016 at 1:02 | comment | added | Joel David Hamkins | Incidentally, I have a blog post on $\Sigma_2$ properties and the idea of local verification. See jdh.hamkins.org/local-properties-in-set-theory. | |
| Nov 5, 2016 at 1:01 | comment | added | Noah Schweber | Oh, I was unaware of those! Would it be rude of me to unaccept this answer, since I am interested in such properties now that I wiki them? (Of course I will not remove the upvote, this is a very nice answer!) | |
| Nov 5, 2016 at 1:01 | comment | added | Joel David Hamkins | There are many $\Pi_2$ large cardinal notions, consistent with $L$, such as: unfoldable, strongly unfoldable, and others. Those notions are defined by properties that involve a universal $\forall\theta$, and then having an embedding jumping beyond $\theta$. Violations of the property are locally verifiable, hence $\Sigma_2$, and so the notion is $\Pi_2$. My argument does not work for that case, so unfoldable might be a natural case. | |
| Nov 5, 2016 at 0:57 | vote | accept | Noah Schweber | ||
| Nov 5, 2016 at 1:02 | |||||
| Nov 5, 2016 at 0:57 | comment | added | Noah Schweber | Ouch, now I feel silly. Thanks! Since I can't think of any interesting $\Sigma_3$ large cardinal properties compatible with $V=L$, I've accepted. | |
| Nov 5, 2016 at 0:55 | comment | added | Joel David Hamkins | Every $L_\theta$ is $\Sigma_1$ elementary in $L$, for $\theta$ a cardinal, and so $\Sigma_2$-properties are upward absolute from $L_\kappa$. | |
| Nov 5, 2016 at 0:53 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
added 135 characters in body
|
| Nov 5, 2016 at 0:53 | comment | added | Noah Schweber | Alright, now I don't see why $L_{\kappa+\gamma}$ won't have any of the relevant kind of cardinals: what if $L_{\kappa+\gamma}$ thinks some $\lambda<\kappa$ satisfies the $\Sigma_2$ property, even though it doesn't in reality? (I'm sure I'm just missing something basic.) | |
| Nov 5, 2016 at 0:51 | comment | added | Joel David Hamkins | I see, I didn't explain it correctly. The point is that $L_{\kappa+\gamma}$ won't have any of those cardinals below $\kappa$, but larger $L_{\delta+\gamma}$ will have such cardinals below $\delta$. I'll edit to explain it correctly. | |
| Nov 5, 2016 at 0:49 | comment | added | Noah Schweber | Sorry, I'm being slow: why will $L_{\kappa+\gamma}$ realize that $\kappa$ is indeed a large cardinal of the relevant type? (That is, I don't see why $L_{\kappa+\gamma}$ is correct about $\Sigma_2$ facts - what if the witness lives in $L_\mu$ for some $\mu>\kappa+\gamma$?) | |
| Nov 5, 2016 at 0:46 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |