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An ideal on $\omega_1$ is strong if it is precipitous and the associated generic elementary embedding always maps $\omega_1$ to $\omega_2$. This definition is from Baumgartner and Taylor, Saturation Properties of Ideals in Generic Extensions II (available online at http://www.jstor.org/stable/1998900.) Every pre-saturated ideal on $\omega_1$ is strong (this was well-known even before the terminology was introduced, I think) and in this paper the authors ask whether the converse is true.

Does anyone know the status of this question: "is every strong ideal on $\omega_1$ pre-saturated"?

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3 Answers

Unfortunately, I don't think Trevor's question is addressed by either of the papers that Matteo mentions.

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Welcome to MathOverflow, Sean! –  Trevor Wilson Sep 8 '12 at 21:30
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up vote 2 down vote accepted

It is consistent that the nonstationary ideal on $\omega_1$ is strong but not pre-saturated. Baumgartner and Taylor proved in the aforementioned paper that strong ideals are preserved by c.c.c. forcing and asked whether the same is true for pre-saturated ideals. The answer to this question is negative, implying a negative answer to the question I posted above. Apparently this was first proved by Veličković in the paper Forcing axioms and stationary sets (which I cannot seem to access online) from ZFC + SPFA. Another example of a c.c.c. forcing that destroys pre-saturation may be found in a more recent paper by Larson and Yorioka, Another c.c.c. forcing that destroys presaturation, assuming the consistency of ZF + AD.

I don't know if a negative answer can be forced from only a Woodin cardinal (which is equiconsistent with the existence of a presaturated ideal and also with the existence of a strong ideal.)

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A somewhat related problem - in their paper Baumgartner and Taylor asked if the existence of a strong ideal implies the existence of a normal one. Moti Gitik (in On normal precipitous ideals. Israel J. Math 175, 191-219 (2010)) proposition 3.1 shows that the projection of such a strong ideal to a normal one gives an isomorphic ideal, and hence is precipitous.

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