Steel and Laver proved from a very large cardinal, a theorem about a tower of finite left-distributive ( a(bc) = (ab)(ac) ) algebras. Laver is studying a family of other finite problems about l.d. algebras and the braid groups, using large cardinal axioms. He proved a partial solution to the strong Halpern-Lauchli problem on binary trees, under the assumption of a measurable cardinal, and is working on solving the full problem. In all the above problems it is not known whether a large cardinal is needed. The project also involves studying a very large cardinal axiom of Woodin that behaves like the axiom of determinacy on higher cardinality versions of the real line. This axiom is being studied for its own interest and for the possibility of variants of it shedding some light on the continuum hypothesis. The CH is known to be immune to standard large cardinal axioms, but not necessarily to strong variants. Almost all theorems in mathematics can be proved from a simple set of basic principles, the Zermelo-Fraenkel (ZFC) axioms for sets. But, as first discovered by Goedel im the 1930's, there are some statements that can neither be proved nor disproved from ZFC. However, at least for some of these ``undecidable'' statements there is hope of their resolution. Namely, they might be proved when one augments ZFC by ``large cardinal'' axioms, axioms beyond the strength of ZFC, which assert the existence of very large infinite numbers. The undecidable statements are in some cases about finite mathematics. The project involves studying some large cardinal axioms and their potential to solve some well studied open questions in classical math. Since some of these problems are about the finite world, they cannot be ruled out as sources of concrete applications; if this happens, an area of high philosophical interest will also have a striking practical use. ***
