Compactness principles, GCH and anti-guessing principles

שלחו לחבר
Seminar
Speaker
Jing Zhang (BIU)
Date
06/01/2020 - 15:00 - 13:00
Place
Area 502, Room 37
Abstract
Compactness principles often times imply guessing principles, for example, if kappa is measurable, then the diamond principle holds at kappa. Though small uncountable cardinals (like omega_2) is not a large cardinal, there are many compactness principles that can consistently hold at omega_2. A theorem of Shelah states in the model where GCH holds and omega_2 is ``sufficiently’’ compact, then the diamond principle holds at the points in omega_2 having cofinality omega_1. We will demonstrate a scenario where omega_2 is still pretty ``compact’’, but the diamond principle as above fails in a rather severe way. This shows Shelah’s theorem is optimal in some sense. Some relationships between our model and the model of GCH + aleph_2-Souslin hypothesis (if it exists) will also be discussed.