中科院数学与系统科学研究院

数学研究所

数理逻辑讨论班

报告人：Toshimichi Usuba 副教授（Waseda University)

题  目：The Downward Directed Ground Hypothesis

时  间：2017.02.28（星期二）, 15:00--16:00

地  点：数学院南楼N818室

Abstract: Laver and Woodin independently proved that the ground model is first-order definable in its forcing extension. Afterward, Fuchs, Hamkins, and Reitz improved their result, and they studied the structure of all ground models of the universe, now it is called the set-theoretic geology. There are many open questions about this geology, an important one is the downward directedness of ground models.In this talk, we prove that ground models are always downward directed. Consequently, we establish some fundamental theorems on the set-theoretic geology. For instance, the mantle, the intersection of all ground models, must be a model of ZFC. We also show that if the universe has some very large cardinal, then the mantle must be a ground models.