A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees

Abstract

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 $\kappa$, there exists a forcing poset which is Y-proper and $\kappa$-c.c., collapses $\kappa$ to become $\omega_2$, 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.