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In this article we show how to use the result in Jäger and Probst  to adapt the technique of pseudo-hierarchies and its use in Avigad  to subsystems of set theory without foundation. We prove that the theory KPi0 of admissible sets without foundation, extended by the principle (Σ-FP),...
A type of subtlety for P
λ called “strongly subtle” is introduced to show almost ineffability is consistencywise stronger than Shelah property. The following are also shown: [InlineMediaObject not available: see fulltext.] is strongly subtle” has rather strong consequences. (ii) The ideal...
It is proved that the forcing apparatus can be built and set to work in ZFCA (=ZFC minus foundation plus the antifoundation axiom AFA). The key tools for this construction are greatest fixed points of continuous operators (a method sometimes referred to as “corecursion”). As an application it is...
We show, using a variation of Woodin’s partial order ℙ
, that it is possible to destroy the saturation of the nonstationary ideal on ω
1 by forcing with a Suslin tree. On the other hand, Suslin trees typcially preserve saturation in extensions by ℙ
variations where one does not try to...
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