Woodin gave a consistency proof of a normal $\omega_1$dense ideal on $\omega_1$ from an almosthuge 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?

$\begingroup$ 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 Schweber Feb 3 '13 at 21:57

$\begingroup$ 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 Eskew Feb 4 '13 at 0:00

$\begingroup$ What about $C$strategically closure? $\endgroup$ – Eran Feb 4 '13 at 22:53

$\begingroup$ Eran, that would be good, but how would you show it? $\endgroup$ – Monroe Eskew Feb 4 '13 at 22:53
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.