In Mathematical Logic one studies mathematical reasoning using the methods of mathematics itself. During the last 50 years this effort has shed light on the nature and limitations of the axiomatic method and it has also yielded new tools for use within mathematics. The starting point of this study is the specification of an appropriate class of formal languages. For example, a first order language which is adequate to express the statements of elementary number theory might have special symbols 0, The main areas of mathematical logic are model theory, set theory, proof theory, and recursion theory. Below are some brief indications of the scope of these areas.
More information is given in the course descriptions.
A model of a set of sentences is given by specifying a universe of discourse and appropriate meanings for all the special symbols involved in such a way that all sentences in the set become true. For instance, the models of the axioms of group theory are precisely the groups. The results of model theory apply to many structures studied in other areas of mathematics and thus model theory has been applied to many problems in algebra and analysis. Also, certain new structures which are called non-standard models have been constructed by the methods of model theory. This subject provides a foundation for mathematics by specifying a language and an intuitive and compelling system of axioms in which all currently accepted mathematical reasoning can be formalized.