Academic year 2020/21:
you will get a question chosen randomly
from the following list.
(1)
The completeness theorem (its precise statement),
the compactness theorem and a proof
of the latter from the former.
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.
(2)
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 the field of rationals.
From completeness to decidability for recursive theories.
(3)
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.
(4)
Quantifier elimination and its proofs for DLO and ACF.
The strong minimality of ACF and the
o-minimality of RCF (assuming
QE for RCE).
(5)
Types, saturated structures and their properties and
existence (an example construction via ultraproduct).
Omitting types theorem and
MacDowell-Specker theorem (without proofs).