集合论模型中的大基数与组合性质

The consistency and independency argument indicates that various types of set theoritical models possess distinct large cardinal and combinatorial properties. The study of such combinatorial properties in the presence of large cardinals plays an important role in justifying the choice of idealized set theoritical models. This may also provide further mathematical evidences and philosophical‎ reasoning for extending Hilbert's doctrine about Foundation of Mathematics. Three types of combinatorial properties to be studied are list as follows: First, investigate the large cardinal properties on small cardinals and their consequences, in particular their impact on the Continuum Hypothesis. Second, study the relationship between supercompact cardinals and the structure of the normal measures, including the number of the normal measures, the Mitchell order of normal measure and some other order structure. Third, explore the structure of uncountable linear order under forcing axioms, especially the relationship and claasification of the Countryman order and coherent order.

经由相容性和独立性论证，不同集合论模型中的大基数性质和组合性质具有不同的特点。对这些大基数影响下的组合性质的研究有助于研究者判断何为理想的集合论模型，从而为进一步拓展Hilbert的数学基础论提供佐证。我们将重点研究下列三大类组合性质。首先，研究小基数上的大基数性质的推论，特别是其对连续统假设的影响。其次，研究超紧基数对正规测度结构的影响，包括正规测度的计数，正规测度上的Mitchell序及其他一些序结构。最后，研究在力迫公理影响下的不可数线性序的分类问题，重点研究Countryman序和Coherent序的关系和分类问题。

本项目研究不同集合论模型中的大基数和组合性质。与合作者一起，在下面三个方面取得了进展：1）对小基数上的大基数性质，证明了否定连续统假设和半稳定集反射原理导出强树性质。2）对大基数上的组合性质，证明了I2、奇异基数假设和树性质的相容性。3）对小基数上的组合性质，证明了强染色性质Pr_0(omega_1,omega_1,n)，并构造了L平方空间和L群。项目计发表学术论文4篇，接收1篇。项目成员多次应邀在国际会议上做学术报告。