Timeline for How would one formulate large cardinals beyond rank into rank?
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5 events
| when toggle format | what | by | license | comment | |
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| Aug 2, 2022 at 18:41 | answer | added | Dmytro Taranovsky | timeline score: 3 | |
| Jul 28, 2022 at 1:17 | comment | added | Noah Schweber | (In my previous comment, "elementary" should be "$\mathcal{L}$-elementary.") | |
| Jul 27, 2022 at 13:15 | comment | added | Noah Schweber | The difficulty is not in formulating higher-consistency-strength principles, the difficulty is in isolating useful ones. For example, by replacing first-order logic with a different logic $\mathcal{L}$ we get a corresponding variant of the relativized constructible universe(s) $L^\mathcal{L}(-)$, and we can define an $I_0^\mathcal{L}$-cardinal as the critical point of a nontrivial elementary embedding $$L^\mathcal{L}(V_{\lambda+1})\rightarrow L^\mathcal{L}(V_{\lambda+1})$$ for some $\lambda<\kappa$. However, there's currently no reason to believe that any of them are useful or interesting. | |
| Jul 27, 2022 at 8:26 | comment | added | Wojowu | I'm assuming you mean in the ZFC context, as in ZF alone many strengthenings like Reinhardt and Berkeley cardinals are available. Dimonte has proposed some axioms strengthening I0. He dubbed those Icarus sets, after the idea of "flying so close to Kunen inconsistency", and are based on assuming that elementary self-embeddings exist for some inner models larger than $L(V_{\lambda+1})$. See here | |
| Jul 27, 2022 at 7:35 | history | asked | littleman | CC BY-SA 4.0 |