It is well known that using the compactness theorem for classical predicate logic, also known as the Godel–Maltsev local theorem, one can easily obtain non-standard models of many theories. Furthermore, it is practically impossible to get rid of non-standard models: when it seems that they have been eliminated (for example, by adding new axioms), they come back in a more sophisticated form. In particular, even the complete theory of the standard model of arithmetic has countable non-standard – more precisely, not isomorphic to the standard one – models. Moreover, by the completeness theorem for classical predicate logic, a sentence is derivable in a given theory iff it is true in all models of this theory. Hence the study of (non)derivability in various theories is closely connected with the study of their non-standard models.

At the same time, non-standard models may have interesting applications that go far beyond the scope of mathematical logic. A prominent example is the use of non-standard models for a rigorous treatment of infinitesimal ("non-standard") analysis, which was proposed by Abraham Robinson and made it possible to legitimize the method of actual infinitely small quantities.

The present course offers an accessible introduction to non-standard models of arithmetic and analysis. We shall only assume familiarity with the basic model-theoretic notions of classical predicate logic.

Lecturer
Speranski Stanislav Olegovich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center