The Ultimate L Project

现代数学对无穷大的研究始于1879年至1884年间，当时康托尔发表了一系列论文，确定了这一学科的基本框架。在40年里，集合论的关键原理被发现了，这些就是ZFC公理，为超越数学的详细发展奠定了基础，至少看起来是这样。然而，在一个完全出乎意料的发展中，科恩在1963年表明，即使是集合论中最基本的问题也不能仅仅根据这些原理来解决。这个问题就是被广泛讨论和著名的康托尔连续统假说问题。自从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，人们意识到，在集合理论中，无法解决的问题无处不在。这无疑挑战了集合论所基于的康托尔的概念。然而，在同一时期，对连续统假设的特殊情况的详细研究取得了显著的成功。这是对二阶数论的一个关键新原理的发现和验证。二阶数论是对所有可计数数集合结构的研究，这只是集合论最简单的化身。由此产生的理论基本上不受科恩方法的影响。这一理论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公理的一个附加公理），这种前景似乎总是完全没有希望。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。这些发现是美国国家科学基金会先前支持的研究的结果。该项目继续并扩大了基于这些发现的研究。Gödel对选择公理和连续统假设的一致性证明涉及到他对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，Cohen的强迫方法不能在公理“V = L”的背景下使用。然而，公理V = L是假的，因为它限制了无限的基本性质。特别地，这个公理驳斥了（大多数）强的无限公理。一个关键问题出现了。是否存在Gödel的可构造宇宙L的“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，但它不反驳强大的无限公理？这个模糊的问题已经被重新塑造，通过先前支持的研究，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.