June 21 (Sun), 2020 Lecture Hall (Room No. 420) Research Institute for Mathematical Sciences Kyoto University, Kyoto, Japan |

Lecture 2: The Ultimate L Conjecture

W. Hugh Woodin

(Harvard University)

(Harvard University)

**Abstract**

The study of descriptive set theory in the context of determinacy axioms began over 50 years ago. The context for this study is now understood to be the Axiom AD$^+$, which is a refinement of the Axiom of Determinacy (AD). The objects of this study is a class of sets of reals, which extends the Borel sets.

This has led to what is arguably the main duality program of Set Theory, which is the connection between the sets of reals $A$ for which AD$^+$ holds, and generalizations of L, the inner model of the universe of sets constructed by Gödel.

This in turn has led to the identification of an ultimate version of Gödel's axiom $V = L$. The key conjecture now is the Ultimate L Conjecture which if true yields a single axiom compatible with all axioms of infinity, and which if added to the ZFC axioms resolves all the questions, such as that of Cantor's Continuum Hypothesis, that have been shown to be unsolvable by Cohen's method of forcing (modulo axioms of infinity).

The Ultimate L Conjecture is a number theoretic statement and so it must be either true or false, under any reasonable conception of mathematical truth.