2025/06/07 by Krueger, John, Stejskalová, Šárka · 1 citation
#03E05 #03E35 #03E40 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2506.06878
Assuming the negation of Chang's conjecture, there is a c.c.c. forcing which adds a strongly non-saturated Aronszajn tree. Using a Mahlo cardinal, we construct a model in which there exists a strongly non-saturated Aronszajn tree and the negation of the Kurepa hypothesis is c.c.c. indestructible. For any inaccessible cardinal κ, there exists a forcing poset which is Y-proper and κ-c.c., collapses κ to become ω2, and adds a strongly non-saturated Aronszajn tree. The quotients of this forcing in intermediate extensions are indestructibly Y-proper on a stationary set with respect to any Y-proper forcing extension. As a consequence, we prove from an inaccessible cardinal that the existence of a strongly non-saturated Aronszajn tree is consistent with the non-existence of a weak Kurepa tree. Finally, we prove from a supercompact cardinal that the existence of a strongly non-saturated Aronszajn tree is consistent with two-cardinal tree properties such as the indestructible guessing model principle.