UC Irvine

It is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10].We consider several antichain-catching properties that are weaker than saturation, and prove: (1) If I is a normal ideal on w2 which satisfies stationary antichain catching, then there is an inner model with aWoodin cardinal; (2) For any n ∈ w, it is consistent relative to large cardinals that there is a normal ideal I on wn which satisfies projective antichain catching, yet I is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).