Large cardinals and the continuum

大基数和连续体

• 批准号: 1101204
• 负责人: Itay Neeman
• 金额: $ 28.59万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101204&HistoricalAwards=false
• 关键词: Large; cardinals; continuum

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.

该项目涉及大基数公理的研究、大基数的强制、大基数公理与通过强制公理和内模型理论的连续统之间的联系，以及应用集合论来逆向数学和单子可判定性。要解决的具体研究主题包括：强制公理与连续统的大值一致；与大基数相关的无限组合问题，特别是关于小基数的树性质和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；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仅限于可定义集合的实数一元理论。数学的许多基本问题都可以使用集合论的强公理来解决。也许最著名的例子涉及有关实数连续体的可定义子集的问题。即使是这些集合的相当简单的属性，例如它们是否承认稳健的长度概念，现在也被认为依赖于集合论的强公理。但关于集合论的强公理与连续统之间的联系，仍有许多未知之处。该项目的激励目标是加深我们对这种联系的理解。这需要更深入地理解公理本身，以及这些公理和连续统属性之间的中间原理：授予公理即可（可证明或一致地）获得并直接影响连续统的原理。该项目旨在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统性质的集合论的饱和原理。