Exam questions from "Model theory":

Academic year 2018/19: You will get a question chosen randomly from the first five in the following list.

  • (1) The completeness theorem (its precise statement) and a proof of the compactness theorem via the ultraproduct method.

  • (2) Applications of the compactness theorem: constructions of non-standard models of the ring of integers and of the ordered real closed field, a proof of the Ax-Grothedieck theorem on injective polynomial maps on the field of complex numbers.

  • (3) Skolemization of a theory and the Lowenheim-Skolem theorem. Vaught's test and its applications to theories DLO, ACF_p and to the theory of vector spaces over a fixed field. From completeness to decidability for recursive theories.

  • (4) Countable categoricity of DLO, the Ehrenfeucht-Fraisse games and elementary equivalence of structures. The theory of random graphs and the 0-1 law for first-order logic on finite graphs.

  • (5) Quantifier elimination and it proofs for DLO and ACF. The strong minimality of ACF and the o-minimality of RCF (assuming QE for RCE).

  • (6) Types, saturated structures and their properties and existence (without proofs). Omitting types theorem and MacDowell-Specker theorem (with proofs).