Is it consistent to have (at the same time) a real-valued measurable cardinal and precipitous ideals on small cardinals such as $\omega_1$? What about saturated ideals on small cardinals?
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asked Sep 16, 2011 at 23:53
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Suppose that there are two measurable cardinals $\kappa\lt\delta$ in $V$. Perform forcing in two steps: first, we form the forcing extension $V[G]$ by the Levy collapse making $\kappa$ become $\omega_1$. The standard arguments (I think due to Prikry) show that $\kappa=\omega_1^{V[G]}$ now carries a precipitous ideal in $V[G]$. Next, we form the extension $V[G][H]$ by pumping up $2^\omega=\delta$ by random real forcing in such a way that $\delta$ is now real-valued measurable, a result due to Solovay. (Both of these arguments are in Kanamori's text.)
Since $V[G][H]$ is a ccc extension of $V[G]$, it follows by a theorem of Kakuda that the precipitous ideal of $V[G]$ generates a precipitous ideal on $\omega_1$ in $V[G][H]$. Thus, in $V[G][H]$ we have both a precipitous ideal on $\omega_1$ and the continuum is a real-valued measurable cardinal.
More generally, if you don't insist that the real-valued measurable cardinal is actually the continuum or less, then all that you need is just a measurable above, and so the forcing $H$ is not required. That is, $V[G]$ already has a precipitous ideal and a measurable cardinal $\delta$, which is also real-valued measurable.
I find it likely, although it would take an inner-model-theory expert to confirm, that the situation you request requires at least two measurable cardinals, because if you have a precipitous ideal on $\omega_1$ and a real-valued measurable cardinal above this, then I expect that there is a fine-structural inner model of ZFC in which both of these cardinals are fully measurable. Perhaps this can be confirmed in another answer by the inner-model-theory experts.
answered Sep 17, 2011 at 0:19
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$\begingroup$ Why does Product Measure Forcing preserve the precipitousness of k? is it omega_2 closed? $\endgroup$– EranDec 20, 2011 at 13:46
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$\begingroup$ 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. $\endgroup$ Dec 20, 2011 at 14:26
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$\begingroup$ 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.) $\endgroup$– EranDec 21, 2011 at 20:12
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$\begingroup$ 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. $\endgroup$ Dec 22, 2011 at 2:13