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

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2022-265).

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