Large Cardinals, Left-Distributivity, and Laver Tables

A large cardinal axiom is roughly a set-theoretic statement asserting that the universe of sets is large. A left-distributive algebra is a set $X$ together with a binary operation $*: X \to X$ set $A$ together with a binary operation $*: A \times A \to A$ satisfying $a * (b * c) = (a * b) * (a * c)$ for all $a, b, c \in A$. This talk is about a series of discoveries from the 1990s which connected the two concepts: one large cardinal axiom, called \textit{Axiom I3}, gives rise to a left-distributive algebra $\mathcal A_j$. While $\mathcal A_j$ arises most naturally in set theory, it can also be expressed in terms of \textit{Laver tables}, a sequence of finite algebras that are definable in arithmetic. However, some of the properties of the Laver tables seem to require large cardinals to be proven. The question of how strong a theory is needed to prove these properties is deeply unsolved, and the gap between the upper and lower bounds could hardly be larger. We give a gentle introduction to all of this open problem and all of the concepts that define it. This talk is based on thesis research advised by Peter Koellner and W. Hugh Woodin.

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