11
$\begingroup$

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?

$\endgroup$
4
  • $\begingroup$ 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. $\endgroup$ Feb 3, 2013 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? $\endgroup$ Feb 4, 2013 at 0:00
  • $\begingroup$ What about $|C|$-strategically closure? $\endgroup$
    – Eran
    Feb 4, 2013 at 22:53
  • $\begingroup$ Eran, that would be good, but how would you show it? $\endgroup$ Feb 4, 2013 at 22:53

1 Answer 1

4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.