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We investigate the consistency strength of the forcing axiom for Σ3 formulas, for various classes of forcings. We review that the consistency strength of Σ3-absoluteness for all set forcing or even just for ω1-preserving forcing is that of a reflecting cardinal. To get the same strength from the forcing axiom restricted to proper forcing, one can add the… (More)

- Vera Fischer, David Schrittesser, Asger Törnquist
- J. Symb. Log.
- 2017

Assuming that every set is constructible, we find a Π1 maximal cofinitary group of permutations of N which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a Π1 maximal… (More)

- Σ2-INDESCRIBABLE CARDINALS, David Schrittesser
- 2006

Σ3-absoluteness for ccc forcing means that for any ccc forcing P , Hω1 V ≺Σ2 Hω1 P . “ω1 inaccessible to reals” means that for any real r, ω1 < ω1. To measure the exact consistency strength of “Σ3-absoluteness for ccc forcing and ω1 is inaccessible to reals”, we introduce a weak version of a weakly compact cardinal, namely, a (lightface) Σ2-indescribable… (More)

In this article we aim to survey some of the work that has been done to extend Jensen’s original Coding Theorem from L to core models witnessing large cardinal properties. The original result of Jensen can be stated as follows. Theorem 1 (Jensen, see [1]) Suppose that (V,A) is a transitive model of ZFC + GCH (i.e., V is a transitive model of ZFC + GCH and… (More)

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