A Strongly Non-Saturated Aronszajn Tree Without Weak Kurepa Trees

TL;DR

This paper constructs models demonstrating the consistent existence of strongly non-saturated Aronszajn trees alongside the negation of Kurepa trees, using advanced set-theoretic forcing and large cardinal assumptions.

Contribution

It introduces new forcing techniques and models showing the coexistence of strongly non-saturated Aronszajn trees with the negation of Kurepa hypotheses under large cardinal assumptions.

Findings

Existence of strongly non-saturated Aronszajn trees is consistent with the negation of Kurepa trees.

Constructs models using Mahlo and supercompact cardinals.

Forcing methods preserve indestructibility of certain properties.

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.