Rank-bounded preservation under the second iterand of an iteration of symmetric extensions

Determine conditions on a two-step iteration of symmetric extensions ⟨P0,G0,F0⟩ ∗ ⟨P1,G1,F1⟩ ensuring that, for a given ordinal α, the second iterand ⟨P1,G1,F1⟩ does not add any new sets of von Neumann rank less than α; and determine conditions ensuring it adds no new subsets of the ground model V of rank less than α. More generally, identify classes of sets that can be guaranteed not to be added by the second iterand under suitable upwards homogeneity-like hypotheses.

Background

While the main emphasis of the paper is on preventing the second iterand from adding any new subsets of the ground model, practical applications may only require preservation up to a certain rank or for specific classes of sets. The authors therefore ask for refined criteria that ensure partial non-addition properties.

This problem aims to extend the theory of upwards homogeneity to rank-bounded and class-specific preservation, seeking structural conditions on the symmetric systems or their iteration that guarantee no new sets (or no new ground-model subsets) below a given rank are introduced.