Timeline for Why believe in the existence of large cardinals rather than just their consistency?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 26 at 7:17 | comment | added | Monroe Eskew | @მამუკაჯიბლაძე Not among the “standard” ones, meaning those that can be characterized by definable elementary embeddings. | |
| Sep 26 at 6:03 | comment | added | მამუკა ჯიბლაძე | How about alternatives within large cardinal system itself? Are not there kinds of large cardinals which cannot coexist? | |
| Sep 24 at 9:22 | comment | added | Monroe Eskew | @JesseElliott It seems you’re saying something more. Empirically, there doesn’t seem to be an alternative system to large cardinals with similar properties, and this calls out for explanation. I think that’s a good point. | |
| Sep 24 at 5:26 | comment | added | Monroe Eskew | @JesseElliott Thanks, but honestly I think Joel’s answer is better. As to your challenge, what I am claiming in my answer is that there is a naturalistic explanation for the consistency of large cardinals, but I’m not claiming to explain why set theorist tend to favor them over competing hypotheses when it comes to adopting axioms. There may be other frameworks whose consistency can be explained in similar ways, but I’d say large cardinals have the strongest case, both in terms of empirical evidence and theoretical coherence. (Although $V=L$ is a different case.) | |
| Sep 23 at 23:09 | comment | added | Jesse Elliott | ..I find your answer to be a novel and compelling perspective. If you can address the challenge above, I'd be happy to accept your answer. | |
| Sep 23 at 22:48 | comment | added | Jesse Elliott | I also think this is a nice perspective. However, a natural question arises. Are there any other theoretical frameworks extending ZFC that preclude the existence of nearly all large cardinals? One might be $V = L$ or even the axiom of restriction, but most set theorists dismiss them and instead seek core models that accommodate nearly all large cardinals. There are also forcing axioms. One is hard-pressed to find an alternative coherent system of hypotheses that "complete" the axioms of ZFC that preclude the large cardinals besides those that preclude them by fiat. | |
| Sep 21 at 14:11 | comment | added | Monroe Eskew | @JosephVanName Because we focused a lot of energy on finding the boundary between consistency and inconsistency. When we play around near the edge, we encounter “near inconsistency”. As for why the boundary lies exactly where it does (which we don’t know with certainty), I think that’s just a bare fact. | |
| Sep 21 at 12:41 | comment | added | Joseph Van Name | How would this explanation explain both the apparent consistency along with the overabundance of near inconsistencies in the large cardinal hierarchy? | |
| Sep 18 at 15:03 | comment | added | Joel David Hamkins | This is a nice perspective. | |
| Sep 18 at 9:17 | history | answered | Monroe Eskew | CC BY-SA 4.0 |