Woodin gave a consistency proof of a normal $\omega_1$-dense ideal on $\omega_1$ from an almost-huge cardinal. He never published this argument, but it is written up by Foreman in the Handbook of Set Theory. In this article, and in a paper from the 90s, Foreman claims that this argument is adaptable to other cardinals to yield the consistency of an $\kappa$-dense ideal on $\kappa$ where $\kappa$ is the successor of a regular cardinal. I have had great trouble trying to prove this claim, as one crucial part of the argument seems specific to $\omega_1$ (which I will explain if you ask). So does anyone know how to prove Foreman's claim?
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asked Feb 3, 2013 at 9:04
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$\begingroup$ What is the crucial dependence on $\omega_1$? I'm looking at Foreman's article right now, and I'm not sure exactly what you mean. $\endgroup$– Noah SchweberCommented Feb 3, 2013 at 21:57
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$\begingroup$ Look at the paragraph that starts with the following:" We let W be the model V1[G0] and construct the partial ordering C ∈ W as in Lemma 7.61. In V1[G ∗ C], the cardinality of P (C)W is countable, so we can build a W -generic object for C." What are we supposed to do at higher cardinals? Let's say we try for ω2. The analogous forcing C for amalgamating small Levy generics is not countably closed, so how do we build a generic for it in the appropriate model? $\endgroup$– Monroe EskewCommented Feb 4, 2013 at 0:00
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$\begingroup$ What about $|C|$-strategically closure? $\endgroup$– EranCommented Feb 4, 2013 at 22:53
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$\begingroup$ Eran, that would be good, but how would you show it? $\endgroup$– Monroe EskewCommented Feb 4, 2013 at 22:53
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This is answered in chapter 2 of my thesis. As far as I know, this is essentially the only method for obtaining such ideals. I am very interested in finding alternative constructions. Please contact me if you have some ideas.
answered Feb 19, 2015 at 10:33