  In this paper we present a new proof of Solovay's theorem on arithmetical
completeness of G\"odel-L\"ob provability logic GL. Originally, completeness of
GL with respect to interpretation of $\Box$ as provability in PA was proved by
R. Solovay in 1976. The key part of Solovay's proof was his construction of an
arithmetical evaluation for a given modal formula that made the formula
unprovable PA if it were unprovable in GL. The arithmetical sentences for the
evaluations were constructed using certain arithmetical fixed points. The
method developed by Solovay have been used for establishing similar semantics
for many other logics. In our proof we develop new more explicit construction
of required evaluations that doesn't use any fixed points in their definitions.
To our knowledge, it is the first alternative proof of the theorem that is
essentially different from Solovay's proof in this key part.
