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}
}
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… 
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2019 EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC LOGIC COLLOQUIUM 2019 Prague, Czech Republic August 11–16, 2019
TLDR
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.

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