INNER MODEL THEORY AND LARGE CARDINALS

在这个项目中，我们开发了内部模型理论、大基数理论、强迫理论、连续体和设置理论的其他子领域的基数变化，作为在它们之间进行交互，并将其应用到纯数学的其他领域。我们简要地绘制了主要的主题和结果：1.内部模型和基本嵌入的最大性能。例如，我们在其中研究了Jonsson的财产持有，如果它持有在宇宙V.2中的无限时间转动机器上的附加基数。我们特别展示了这一点，如果λ是可写普通件的最高值，i.e.普通件在输入0上的计算输出，而γ是可时钟普通件的最高值，i.e.普通件在输入0上的半导体计算长度，然后γ =γ 3.我们展示了一套真实的科恩，超越了一个ZFC模型，要么要么空，要么不空。meager. 4. The effect of cardinal invariants of the continuum on combinatorial properties of uncountable cardinals。例如，我们证明了马丁的轴心MA对任何科恩代数的无理想附加性的影响。On the other hand， we showed it is consistent that the continuum c is large and covering of the null ideal as well as the combinatorial principle * hold. 5.说that PSP（κ， Γ） holds if every set in the pointclass Γ of size at least κ has a perfect subset。我看过PSP（D21 D2，D2 D2 D21 D2） holds if and only if σ [greater than or equal] D22 D2 is the class of sets which are intersections of （at most）D21 D2 many open sets and σ is the dominating number. 6. We proved that every max？confinitary subgroup of Sym（ω），the permutation group of the natural numbers， has size at least the cardinality of the smallest non-meager set reals.7.Completing a cycle of results initiated by？ah and Spinas， we obtained that if c=？イイD22イエD2 there is a Gross space over every countably infinite field。

Research in this project was devoted to inner model theory, large cardinals, forcing theory, cardinal invariants of the continuum and other subfields of set theory, as well as to the interplay between them and to applications to other areas of pure mathematics. We briefly sketch the main topics and results.1.Maximality properties of inner models and elementary embeddings. For example, we investigated under which circumstances the Jonsson property holds for a given cardinal in the core model K if it holds in the universe V.2.Research on infinite time Turning machines. We showed in particular that, if λ is the supremum of the writable ordinals, i.e. ordinals which arise at outputs of computations on input 0, and γ is the supremum of clockable ordinals, i.e. ordinals which are lengths of halting computations on input 0, then λ=γ.3.We proved the set of reals Cohen over a model of ZFC must either be empty or non-meager.4.The effect of cardinal invariants of the continuum on combinatorial properties of uncountable cardinals. For example, we proved that additivity of the null ideal implies that Martin's axiom MA holds for any Cohen algebra. On the other hand, we showed it is consistent that the continuum c is large and covering of the null ideal as well as the combinatorial principle * hold.5.Say that PSP(κ, Γ) holds if every set in the pointclass Γ of size at least κ has a perfect subset. We showed that PSP(ΝィイD21ィエD2, GィイD2ΝィエD2ィイD21ィエD2) holds if and only if σ 【greater than or equal】 ΝィイD22ィエD2 where GィイD2ΝィエD2ィイD21ィエD2 is the class of sets which are intersections of (at most) ΝィイD21ィエD2 many open sets and σ is the dominating number.6.We proved that every maximal confinitary subgroup of Sym(ω), the permutation group of the natural numbers, has size at least the cardinality of the smallest non-meager set reals.7.Completing a cycle of results initiated by Shelah and Spinas, we obtained that if c=ΝィイD22ィエD2 then there is a Gross space over every countably infinite field.