  Varieties with log terminal and log canonical singularities are considered in
the Minimal Model Program, see \cite{...} for introduction. In
\cite{shokurov:hyp} it was conjectured that many of the interesting sets,
associated with these varieties have something in common: they satisfy the
ascending chain condition, which means that every increasing chain of elements
terminates. Philosophically, this is the reason why two main hypotheses in the
Minimal Model Program: existence and termination of flips should be true and
are possible to prove.
  In this paper we prove that the following two sets satisfy the ascending
chain condition:
  1. The set of minimal log discrepancies for $K_X+B$ where $X$ is a surface
with log canonical singularities.
  2. The set of groups $(b_1,...b_s)$ such that there is a surface $X$ with log
canonical and numerically trivial $K_X+\sum b_jB_j$. The order on such groups
is defined in a natural way.
