This is the first semester of a one-year course in mathematical logic. This course was first presented in the 2008-2009 academic year and has since been taught at the St. Petersburg Academic University as part of the Chair of Mathematics and Computer Science.

The first semester serves as a reminder of and/or introduction to the basic notions of mathematical logic. It builds up to two main results: the Tarsky theorem on quantifier elimination in real closed fields and cut elimination in Gentzen's calculus and its corollaries