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