Some consequences from Proper Forcing Axiom together with large continuum and the negation of Martin's Axiom

摘要

Recently, David Aspero and Miguel Angel Mota discovered a new method of iterated forcing using models as side conditions. The side condition method with models was introduced by Stevo Todorcevic in the 1980s. The Aspero-Mota iteration enables us to force some ∏_2-statements over H(N_2) with the continuum greater than N_2- In this article, by using the Aspero-Mota iteration, we prove that it is consistent that V fails, there are no weak club guessing ladder systems, p = add(N) = 2~(N_0)_ > N_2 and MA_(N1) fails.