# Rigid ideals

@article{Cody2016RigidI,
title={Rigid ideals},
author={Brent Cody and Monroe Eskew},
journal={Israel Journal of Mathematics},
year={2016},
volume={224},
pages={343-366}
}
• Published 31 May 2016
• Mathematics
• Israel Journal of Mathematics
An ideal I on a cardinal κ is called rigid if all automorphisms of P(κ)/I are trivial. An ideal is called μ-minimal if whenever G ⊆ P(κ)/I is generic and X ∈ P(μ)V[G]V, it follows that V [X] = V [G]. We prove that the existence of a rigid saturated μ-minimal ideal on μ+, where μ is a regular cardinal, is consistent relative to the existence of large cardinals. The existence of such an ideal implies that GCH fails. However, we show that the existence of a rigid saturated ideal on μ+, where μ is… Expand
2 Citations
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#### References

SHOWING 1-10 OF 10 REFERENCES
SATURATION PROPERTIES OF IDEALS IN GENERIC EXTENSIONS
We consider saturation properties of ideals in models obtained by forcing with countable chain condition partial orderings. As sample results, we mention the following. If M[G] is obtained from aExpand
The number of normal measures
• Computer Science, Mathematics
• The Journal of Symbolic Logic
• 2009
This article treats all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinality-preserving forcing, and explores the possibilities for the number of normal measures on a cardinal at which the GCH fails. Expand
On closed sets of ordinals
We prove that every stationary set of countable ordinals contains arbitrarily long countable closed subsets. Call a set A of ordinals closed if and only if every nonempty subset of A which has anExpand
IDEAL PROJECTIONS AND FORCING PROJECTIONS
• Mathematics, Computer Science
• The Journal of Symbolic Logic
• 2014
This paper proves that there is a normal ideal on $\omega _2$ which satisfies stationary antichain catching, and provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory. Expand
Ideals and Generic Elementary Embeddings
This chapter covers the technique of generic elementary embeddings. These embeddings are closely analogous to conventional large cardinal embeddings, the difference being that they are definable inExpand
Martin's Maximum, saturated ideals and non-regular ultrafilters. Part II
• Mathematics
• 1988
We prove, assuming the existence of a huge cardinal, the consistency of fully non-regular ultrafilters on the successor of any regular cardinal. We also construct ultrafilters with ultraproducts ofExpand
A uniqueness theorem for iterations
• P. Larson
• Computer Science, Mathematics
• Journal of Symbolic Logic
• 2002
Abstract If M is a countable transitive model of , then for every real x there is a unique shortest iteration j: M → N with x ∈ N, or none at all.
Iterated Forcing and Elementary Embeddings
I give a survey of some forcing techniques which are useful in the study of large cardinals and elementary embeddings. The main theme is the problem of extending a (possibly generic) elementaryExpand
The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal
The second edition of a well-established monograph on the identification of a canonical model in which the Continuum Hypothesis is false is updated to take into account some of the developments in the decade since the first edition appeared. Expand
Forcing Closed Unbounded Sets
• Computer Science, Mathematics
• J. Symb. Log.
• 1983