Timeline for If $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, what properties does $κ$ have?
Current License: CC BY-SA 4.0
7 events
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| Oct 6, 2018 at 6:33 | comment | added | Keith Millar | Hmm... that's interesting. If we require $\mu<j(\kappa)$, does this become more 'large-cardinal-like?' More specifically, would it also hold for $\lambda<\mu$ if it held for $\mu$? | |
| Oct 5, 2018 at 15:15 | comment | added | Keith Millar | If you close $M$ under $\kappa$-sequences, and you make $\mu<j(\kappa)$ (like strong compactness) then it turns out to be equiconsistent to strongness; this is because every $\mu+2$-strong cardinal would have this property at $\mu$, and every such cardinal would be $\mu$-tall. Since tallness is equiconsistent to strongness, it's equiconsistent to both. | |
| Oct 5, 2018 at 10:23 | history | edited | Yair Hayut | CC BY-SA 4.0 |
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| Oct 5, 2018 at 10:22 | comment | added | Yair Hayut | You're right, this is true if we further assume that $M$ is closed under $\kappa$-sequences, but without this assumption I'm not certain what is the exact consistency strength. | |
| Oct 4, 2018 at 22:24 | comment | added | Keith Millar | Thanks! Why exactly does it imply that $o^K(κ)≥κ^{++}$? | |
| Oct 4, 2018 at 14:00 | vote | accept | Keith Millar | ||
| Oct 4, 2018 at 11:08 | history | answered | Yair Hayut | CC BY-SA 4.0 |