**Thomas JechThe Pennsylvania State University **

A * left-distributive algebra* is a set with one binary operation
satisfying the left distributive law

*a(bc)=(ab)(ac)*

(the operation of
conjugation in a group is an example).

The subject of my lectures is the free left-distributive algebra on one generator. This algebra arises in the study of elementary embeddings in set theory, and the word problem for it was first solved by R. Laver using the theory of large cardinals. Subsequently, P. Dehornoy found an elementary solution of the problem by giving a representation in the braid group.

The free algebra can also be expressed as a limit of finite algebras. These
*cyclic* algebras have been studied by R. Dougherty and myself. The
statement that the limit algebra is free is equivalent to a simple statement of
elementary number theory, and follows from Laver's work (using large cardinals).
It is however unprovable by elementary methods (viz. in primitive recursive
arithmetic), and it is an open problem whether the large cardinal assumptions
are necessary.

The study of cyclic left-distributive algebras leads to some interesting questions on computability and complexity.