Large cardinals and the continuum

这个项目关注的是大基数公理的研究，强迫与大基数，大基数公理和连续体之间的联系，通过强迫公理和通过内部模型理论，以及集合论的应用，以扭转数学和一元决策。要解决的具体研究课题包括：强制公理与连续体的大值一致;在无穷组合学的问题，涉及到大基数，特别是在树的属性和小基数的奇异基数假设之间的关系;内部模型的迭代性，并与长扩展模型适当的强制公理的强度指标;在逆向数学中使用强归纳假设;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制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使用集合论的强公理来解决。也许最著名的例子涉及到关于连续真实的数的可定义子集的问题。即使是这些集合的相当简单的性质，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统之间的联系，仍然有许多未知之处。该项目的激励目标是加深我们对这种联系的理解。这需要对公理本身以及这些公理和连续统性质之间的中介原则有更深入的理解：这些原则可以（可证明地或一致地）被赋予公理，并直接影响连续统。该项目旨在扩展两个方面的工作，研究公理模型，无限集合的组合原理，以及影响连续体属性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.