Timeline for Real-valued measurable cardinals and strong ideals

7 events

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Dec 22, 2011 at 2:13	comment	added	Joel David Hamkins		Eran, the combined forcing is $\kappa$-c.c., since the first step is, and the second step is c.c.c. So one can use the original measure, which is $\kappa$-complete.
Dec 21, 2011 at 20:12	comment	added	Eran		Corollary 13.7.17 in the handbook requires that I is k-complete. But is this necessarily the case ? (Foreman's definition doesn't require it and corollary 2.6 implies only countable completion.)
Dec 20, 2011 at 14:26	comment	added	Joel David Hamkins		Eran, the precipitous ideal on $\kappa$ in $V[G]$ is preserved to $V[G][H]$, since the forcing to add $H$ is c.c.c., and as I mentioned, Kakuda proved that $\kappa$-c.c. forcing preserves the existence of a precipitous ideal on $\kappa$. (And this is how one can see that $V[G]$ has a precipitous ideal, generated from the dual ideal to the measure on $\kappa$ in $V$, since the $G$ forcing is $\kappa$-c.c.) The forcing to add $H$ is the forcing to add $\delta$-many Cohen reals, which is definitely not $\omega_2$-closed.
Dec 20, 2011 at 13:46	comment	added	Eran		Why does Product Measure Forcing preserve the precipitousness of k? is it omega_2 closed?
Sep 17, 2011 at 4:26	vote	accept	Monroe Eskew
Sep 17, 2011 at 0:24	history	edited	Joel David Hamkins	CC BY-SA 3.0	added 467 characters in body
Sep 17, 2011 at 0:19	history	answered	Joel David Hamkins	CC BY-SA 3.0