Large cardinals and the continuum

本课题主要研究大基数公理、大基数强迫、大基数公理与连续统之间的联系，以及集合论在逆向数学和一元可决性中的应用。具体的研究课题包括：强迫公理与连续统的大值相一致；与大基数相关的无穷组合问题，特别是树性质与小基数下的奇异基数假设之间的关系内部模型的可迭代性，以及与长扩展模型相关的适当强制公理的强度指标；强归纳假设在逆向数学中的应用以及受集合论公理影响的一元理论，包括序数的一元理论和限定于可定义集合的实数的一元理论。许多数学的基础问题可以用集合论的强公理来解决。也许最著名的例子是关于实数连续统的可定义子集的问题。即使是这些集合相当简单的性质，例如它们是否承认长度的鲁棒概念，现在已知也依赖于集合论的强公理。但是关于集合论的强公理和连续体之间的联系仍然有很多未知的地方。这个项目的激励目标是加深我们对这种联系的理解。这就要求我们对公理本身，以及这些公理和连续统的性质之间的中介原则有更深的了解：这些原则可以（可证明地或一致地）从公理中得到，并直接影响连续统。该项目寻求扩展这两个方面的工作，研究公理模型，无限集的组合原理，以及影响连续体性质的集合宇宙的饱和原理。

This project is concerned with the study of large cardinal axioms, forcing with large cardinals, connections between large cardinal axioms and the continuum through forcing axioms and through inner model theory, and applications of set theory to reverse mathematics and monadic decidability. Specific research topics to be addressed include: forcing axioms consistent with large values of the continuum; questions in infinitary combinatorics that are related to large cardinals, particularly on the relationship between the tree property and the singular cardinals hypothesis at small cardinals; iterability for inner models, and indicators of strength for the proper forcing axiom in connection with long extender models; uses of strong induction hypotheses in reverse mathematics; and monadic theories that are affected by set theoretic axioms, including monadic theories of ordinals and the monadic theory of the reals restricted to definable sets.Many of the foundational questions of mathematics can be addressed using strong axioms of set theory. Perhaps the most celebrated instances involve questions about definable subsets of the continuum of real numbers. Even fairly simple properties of these sets, for example whether they admit a robust notion of length, are now known to be dependent on strong axioms of set theory. But much still remains unknown about the connection between strong axioms of set theory and the continuum. The motivating goal for the project is to deepen our understanding of this connection. This requires a deeper understanding of the axioms themselves, and of intermediary principles between these axioms and properties of the continuum: principles that can be obtained (provably or consistently) granted the axioms, and directly affect the continuum. The project seeks to extend work on both fronts, with research into models for the axioms, combinatorial principles on infinite sets, and saturation principles of the universe of sets that affect properties of the continuum.