Timeline for Consistency of the size of the least real-valued measurable cardinal, vis-a-vis the continuum
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 3, 2018 at 3:24 | comment | added | Andrés E. Caicedo | @Zoorado If you start with $L[\mu]$, the smallest model with a measurable cardinal $\kappa$ (with $\kappa$ as small as possible) and add $\kappa$ random reals, Solovay's argument shows that in the extension $\kappa$ is real-valued measurable and the continuum. If $\lambda<\kappa$ is also real-valued measurable, Solovay's argument using the null ideal for a witnessing measure on $\lambda$ would give you an inner model with $\lambda$ measurable. This contradicts the minimality of $\kappa$. (The one technical further detail is that minimality of $L[\mu]$ is absolute under forcing extensions.) | |
| May 3, 2018 at 2:42 | comment | added | Zoorado | Can you briefly illustrate how adding $\kappa$ random reals in Case 2 would not make any cardinal below $\kappa$ real-valued measurable? | |
| May 3, 2018 at 2:30 | vote | accept | Zoorado | ||
| May 2, 2018 at 23:31 | comment | added | Andrés E. Caicedo | Regarding 1: If by "real-valued measurable" we mean atomlessly measurable (rather than measurable in the usual sense of large cardinals), then any real-valued measurable cardinal is of size at most the continuum. | |
| May 2, 2018 at 23:12 | history | answered | Robert Furber | CC BY-SA 4.0 |