The side condition method and the strong properness
by
Teruyuki Yorioka
Kobe University

In this talk, I argue analogies between w₁-trees and (w₁, w₁)-gaps in Shelah's strongly proper forcing extensions. Abraham and Todorcevic said in their paper that an Aronszajn tree and an (w₁, w₁)-gap have a similar characterization and a Suslin tree and a destructible gap have also a similar characterization. In fact, the existences of a Suslin tree and a destructible gap have many analogical statements. For example, Jensen's diamond implies the existence of a Suslin tree and a destructible gap, and adding a Cohen real adds both a Suslin tree and a destructible gap.

We can show that a strongly proper forcing notion does not collapse an Aronszajn tree and an (w₁, w₁)-gap, and also a Suslin tree. However, a destructible gap can be destroyed by some strongly proper forcing extensions. This means that the analogy between a suslin tree and a destructible gap is broken down with respect to the strongly proper forcing extensions.

Date received: February 25, 2005