Characterize when a symmetric system adds no new sets of ordinals

Determine necessary and/or sufficient conditions on a symmetric system S = ⟨P,G,F⟩, used to form a symmetric extension over an intermediate model M in a two-step iteration of symmetric extensions ⟨P0,G0,F0⟩ ∗ ⟨P1,G1,F1⟩, that guarantee forcing by S over M adds no new sets of ordinals; equivalently, characterize the intermediate-model formulation of the upwards homogeneity property that ensures the second stage ⟨P1,G1,F1⟩ does not add new sets of ordinals.

Background

The main theorem of the paper shows that an iteration of symmetric extensions is upwards homogeneous if and only if the second step adds no new subsets of the ground model, which in the presence of the Axiom of Choice is equivalent to adding no new sets of ordinals. This is established from the viewpoint of the full two-step iteration.

The authors ask for an intrinsic characterization of this phenomenon within the intermediate model itself: rather than invoking the iteration-wide upwards homogeneity criterion, can one describe conditions directly on a single symmetric system S = ⟨P,G,F⟩ (used over the intermediate model) that ensure it does not add any new sets of ordinals? This seeks a local criterion mirroring the global property proved in the paper.