Finite Left-Distributive Algebras and Embedding Algebras

We consider algebras with one binary operation · and one generator (monogenic) and satisfying the left distributive lawa·(b·c)=(a·b)·(a·c). One can define a sequence of finite left-distributive algebrasA_n, and then take a limit to get an infinite monogenic left-distributive algebraA_∞. Results of Laver and Steel assuming a strong large cardinal axiom imply thatA_∞is free; it is open whether the freeness ofA_∞can be proved without the large cardinal assumption, or even in Peano arithmetic. The main result of this paper is the equivalence of this problem with the existence of a certain algebra of increasing functions on natural numbers, called anembedding algebra. Using this and results of the first author, we conclude that the freeness ofA_∞is unprovable in primitive recursive arithmetic.