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).