It is well known that ZFC + "There is a measurable cardinal" is equiconsistent with ZFC + "There is a precipitous ideal on $\omega_1$." Is ZFC + "There is a measurable $\kappa$ such that $2^\kappa > \kappa^+$" equiconsistent with ZFC + "There is a precipitous ideal on $\omega_1$" + CH + $2^{\omega_1}>\omega_2$?
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asked May 13, 2016 at 15:16
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$\begingroup$ The theory ZFC plus a measurable $\kappa$ with $2^\kappa>\kappa^+$ is equiconsistent with a $\kappa^{++}$-tall cardinal, which is equiconsistent with a $(\kappa+2)$-strong cardinal. $\endgroup$– Joel David HamkinsCommented May 13, 2016 at 16:43
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$\begingroup$ I'd say that the title is a bit misleading, and should probably be "Precipitous ideals and the negation of GCH". $\endgroup$– Asaf Karagila ♦Commented May 13, 2016 at 21:43
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Just a measurable cardinal is sufficient.
Assume $\kappa$ is a measurable cardinal and force with $Col(\omega, < \kappa)*\dot{Add}(\kappa, \tau^*)$, where $\tau^* \geq \kappa^{++}$ is suitably defined (see Gitik's paper ``On Generic Elementary Embeddings'').
As it is shown in the above cited paper, there is a nomral precipitous ideal on $\omega_1$ in the resulting generic extension.
answered May 16, 2016 at 3:59