The Ultimate L Project

现代数学研究无穷开始于1879年至1884年期间的一系列文件，由康托，确定了基本框架的主题。在40年内，集合论的关键原则被发现，这些是ZFC公理，并且为超限数学的详细发展奠定了基础，或者看起来是这样。然而，在一个完全出乎意料的发展，科恩在1963年表明，即使是最基本的问题集理论是不能解决的基础上，这些原则单独。这个问题是广受讨论和著名的问题康托的连续统假设。自科恩发表声明以来的50年里，科恩的方法得到了巨大的发展，人们认识到，在集合论中，不可解问题的出现是普遍存在的。这可以说是挑战康托的概念，而集合论的基础。然而，在同一时期，对连续统假说的特殊情况的详细研究取得了显着的成功。这是二阶数论的一个关键新原理的发现和验证。二阶数论是研究所有计数集合的结构，这只是集合论最简单的体现。由此产生的理论基本上不受科恩方法的影响。这种前景可以以某种方式扩展，为集合论本身产生一个类似的新原理（作为ZFC公理的一个单独的附加公理），这似乎总是完全没有希望的。但这种信念本身是基于一种误解，最近的发现表明有一个解决方案。这些发现是先前NSF支持的研究的结果。哥德尔对选择公理和连续统假设的一致性证明涉及到他的可构造集合宇宙的发现。公理“V = L”是断言每个集合都是可构造的公理。这个公理解决了连续统假设，更重要的是，科恩的强迫方法不能在公理“V = L”的上下文中使用。然而，公理V = L是错误的，因为它限制了无穷大的基本性质。特别是公理反驳（大多数）强公理的无穷大。一个关键问题出现了。哥德尔的可构造的宇宙L是否有一个“终极”版本，产生一个公理“V =终极L”，它保留了公理“V = L”的力量来解决像连续统假设这样的问题，它也不受科恩的强迫方法的影响，但它不反驳强无穷公理？这个模糊的问题已经被改写，通过先前的研究，成为一个具体而精确的猜想;终极L猜想。这个项目的目标就是解决这个猜想。

The modern mathematical study of infinity began in the period 1879-84 with a series of papers by Cantor that defined the fundamental framework of the subject. Within 40 years the key principles of Set Theory were discovered, these are the ZFC axioms, and the stage was set for the detailed development of transfinite mathematics, or so it seemed. However, in a completely unexpected development, Cohen showed in 1963 that even the most basic problem of Set Theory was not solvable on the basis of these principles alone. That problem was the widely discussed and celebrated problem of Cantor's Continuum Hypothesis. The 50 years since Cohen's announcement has seen a vast development of Cohen's method and to the realization that the occurrence of unsolvable problems is ubiquitous in Set Theory. This arguably challenges the very conception of Cantor on which Set Theory is based. However, during this same period, the detailed study of special cases of the Continuum Hypothesis led to a remarkable success. This was the discovery and validation of a key new principle for Second Order Number Theory. Second Order Number Theory is the study of the structure of all sets of counting numbers, and this is just Set Theory in its simplest incarnation. The resulting theory is largely immune to Cohen's method. The prospect that this could somehow be extended to produce an analogous new principle for Set Theory itself (as a single additional axiom to the ZFC axioms) has always seemed completely hopeless. But that belief was itself based on a misconception and recent discoveries suggest there is a resolution. These discoveries were the result of prior NSF supported research. This project continues and expands the research based on these discoveries.Gödel's consistency proof for the Axiom of Choice and the Continuum Hypothesis involves his discovery of the Constructible Universe of Sets. The axiom "V = L" is the axiom which asserts that every set is constructible. This axiom settles the Continuum Hypothesis and more importantly, Cohen's method of forcing cannot be used in the context of the axiom "V = L". However the axiom V = L is false since it limits the fundamental nature of infinity. In particular the axiom refutes (most) strong axioms of infinity. A key question emerges. Is there an "ultimate" version of Gödel's constructible universe L yielding an axiom "V = Ultimate L" which retains the power of the axiom "V = L" for resolving questions like that of the Continuum Hypothesis, which is also immune against Cohen's method of forcing, and yet which does not refute strong axioms of infinity? This vague question has been recast, through previously supported research, into a specific and precise conjecture; the Ultimate L Conjecture. The goal of this project is to resolve that conjecture.