Timeline for Large cardinals without choice?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Apr 22, 2016 at 16:22 | vote | accept | Tim Campion | ||
| Apr 22, 2016 at 12:06 | comment | added | Tim Campion | Thanks so much! It occurs to me that maybe I should ask this as a separate question. | |
| Apr 21, 2016 at 17:59 | comment | added | Joel David Hamkins | Yes, I agree with you about (1), and that is indeed the only reason I added the axioms $\text{AC}\to\sigma$. For method 2, the basic fact is that every model of ZFC is the HOD of another model---you can find this in my paper Set-theoretic geology (jdh.hamkins.org/set-theoreticgeology). So you can do that forcing, and then form a symmetric extension, which will preserve it. Lastly, I'll give some thought to your question about Reinhardt and $I_0$. | |
| Apr 21, 2016 at 17:45 | comment | added | Tim Campion | @JoelDavidHamkins Thanks, this is interesting! I noticed that in (1), the axioms $\{AC \to \sigma \mid \sigma \in T\}$ sort of "ride for free", and are only necessary to get $T' + AC = T$ -- without them everything else works and we still have $T \vdash T'$ which I think I'd still consider to be a "nice" way to get $Con(T) \to Con(T')$. Regarding the existence of $W$ in (2) -- is this something I can find in, say, Kunen's book? Lastly, I don't suppose you know anything about how $Con(ZF + Reinhardt) \implies Con(ZFC+I_0)$ is proved? If $M \vDash ZF + Reinhardt$, does $HOD^M \vDash ZFC + I_0$? | |
| Apr 21, 2016 at 17:26 | comment | added | Joel David Hamkins | I have edited to use HOD instead of ground models, which I think makes a nicer theory. Basically, T' asserts that T holds in HOD, if AC fails. | |
| Apr 21, 2016 at 17:07 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Apr 21, 2016 at 17:03 | comment | added | Joel David Hamkins | There is no issue if $T$ extends ZFC with a single axiom. But in the general case, I was thinking of the third bullet as a scheme. But we can assume that $W$ is definable, so they are all using the same $W$. In fact, thinking about it now, we can just take $W$ as the HOD of $V$. Perhaps that is better than talking about ground models. I'll edit to this. | |
| Apr 21, 2016 at 16:57 | comment | added | Andreas Blass | In the third bullet item of Method 2, do you allow different models $W$ for different sentences $\sigma$? If not, how do you avoid undefinability-of-truth problems there? | |
| Apr 21, 2016 at 16:44 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Apr 21, 2016 at 16:35 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Apr 21, 2016 at 16:24 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Apr 21, 2016 at 16:17 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |