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.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Tomek Bartoszynski
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Harvard University
United States
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