Covering with universally Baire operators∗†

We introduce a covering conjecture and show that it holds below ADR + “Θ is regular”. We then use it to show that in the presence of mild large cardinal axioms, PFA implies that there is a transitive model containing the reals and ordinals and satisfying ADR + “Θ is regular”. The method used to prove the Main Theorem of this paper is the core model induction. The paper contains the first application of the core model induction that goes significantly beyond the region of AD+ + θ0 < Θ. One of the central themes in set theory is to identify canonical inner models which compute successor cardinals correctly. A prototype of such results is Jensen’s famous covering theorem which in particular implies that provided 0 doesn’t exist, for every cardinal κ ≥ ω2, cf((κ)) ≥ κ where L is the constructible universe. Clearly “canonical inner model” is open for interpretations. For an inner model theorist, the canonical objects of a set theoretic universe are the sets coded by a mixture of fine extender sequences and the universally Baire sets. Recall that a set of reals is universally Baire if its continuous preimages in all compact Hausdorff spaces have the property of Baire. ∗2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. †