Inner Model Theory

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。内模理论的基本开放问题是将理论扩展到满足更强的大基数假设的模型，其中“存在一个超紧基数”是最著名的目标。尽管在60年代末、1974-1980年和1985-1994年取得了重大进展，但这个目标却阻碍了集合论者40年的努力。早期的工作让我们很好地理解了满足“有一个伍丁基数”的规范内部模型，甚至稍微强一点的假设，但它远远没有达到超强基数。自1994年以来，在这段时间内的进展一直缓慢而渐进，但斯蒂尔认为，这已经开辟了一条有希望的进攻路线。这条线建立在概念和方法的基础上，这些概念和方法来源于内模型理论和纯描述性集合理论的不断发展的合并。它以核心模型归纳技术和鼠标集猜想为中心。斯蒂尔建议朝这个方向努力。关于数学的集合论基础的研究揭示了一个显著的现象。似乎所有的自然数学理论都落入一个单一的层次，由理论中隐含的对大无限集存在性的承诺的程度来校准。此外，这种程度的承诺正好对应于理论解决具体对象问题的能力，如自然数、实数或实数集。集合理论家已经发展了两种互补的方法来定位逻辑强度层次结构中的理论，即强迫理论和内模型理论。虽然作用力的基本原理得到了全面的理解，但内模理论的基本原理却没有得到充分的理解，因此，关于特定基础理论的逻辑优势的许多问题仍然存在。Steel将致力于进一步发展内在模型理论的基础。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The fundamental open problem of inner model theory is to extend the theory to models satisfying stronger large cardinal hypotheses, with ``There is a supercompact cardinal" being the most well-known target. This target has resisted 40 years of effort by set theorists, despite major advances in the late 60's, in 1974-1980, and in 1985-1994. That earlier work gave us a good understanding of canonical inner models satisfying ``There is a Woodin cardinal", and even slightly stronger hypotheses, but it fell well short of superstrong cardinals. Since 1994, progress in this interval has been slow and incremental, but Steel believes it has opened up a promising line of attack. This line builds on concepts and methods derived from the evolving merger of inner model theory and pure descriptive set theory. It centers on the Core Model Induction technique, and the Mouse Set Conjecture. Steel proposes to work in this direction. Work on the set-theoretic foundations of mathematics has revealed a remarkable phenomenon. It appears that all natural mathematical theories fall into a single hierarchy, calibrated by the degree to which a commitment to the existence of large infinite sets is implicit in the theory. Moreover, this degree of commitment corresponds exactly to the power of the theory to decide questions about concrete objects, like natural numbers, real numbers, or sets of real numbers. Set theorists have developed two complementary methods for locating theories in this hierarchy of logical strength, namely Forcing and Inner Model Theory. While the basics of Forcing are understood in full generality, the basics of Inner Model Theory are not, and consequently many problems concerning the logical strengths of particular foundational theories remain open. Steel will work to further develop the basics of Inner Model Theory.