# Strongly proper forcing and some problems of Foreman

@article{Cox2018StronglyPF,
title={Strongly proper forcing and some problems of Foreman},
author={Sean D. Cox and Monroe Eskew},
journal={Transactions of the American Mathematical Society},
year={2018}
}
• Published 5 December 2016
• Mathematics
• Transactions of the American Mathematical Society
We answer several questions of Foreman, most of which are closely related to Mitchell’s notion of strongly proper forcing. We prove that presaturation of a normal ideal implies projective antichain catching, providing a solution to a problem of Foreman about ideal projections that is more comprehensive and simpler than the earlier solution obtained by Cox and Zeman. We answer an older question of Foreman about the relationship between generic hugeness and generic almost hugeness. Finally, we…
Destroying Saturation while Preserving Presaturation at an Inacceessible; an Iterated Forcing Argument
We prove that a large class of presaturated ideals at inaccessible cardinals can be de-saturated while preserving their presaturation, answering both a question of Foreman and of Cox and Eskew. We do
Local saturation and square everywhere
We show that it is consistent relative to a huge cardinal that for all infinite cardinals [Formula: see text], [Formula: see text] holds and there is a stationary [Formula: see text] such that
2019 EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC LOGIC COLLOQUIUM 2019 Prague, Czech Republic August 11–16, 2019
• Mathematics
The Bulletin of Symbolic Logic
• 2019
It is argued that also the successful use of proof-theoretic methods in core mathematics in recent decades was made possible by developing logical metatheorems tailored for applications to particular classes of theorems and proofs in specific areas of mathematics.

## References

SHOWING 1-10 OF 24 REFERENCES
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 of
IDEAL PROJECTIONS AND FORCING PROJECTIONS
• Mathematics
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.
Calculating quotient algebras of generic embeddings
Many consistency results in set theory involve forcing over a universe V0 that contains a large cardinal to get a model V1. The original large cardinal embedding is then extended generically using a
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) elementary
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 in
Saturated ideals
• K. Kunen
• Mathematics
Journal of Symbolic Logic
• 1978
In this paper, we give consistency proofs for the existence of a κ-saturated ideal on an inaccessible κ, and for the existence of an ω2-saturated ideal on ω1. We also include an historical survey
Saturation properties of ideals in generic extensions. I
• Mathematics
• 1982
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 a
SATURATION PROPERTIES OF IDEALS IN GENERIC EXTENSIONS
• Mathematics
• 2010
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 a
On the Hamkins approximation property