The exam is oral.
You will get a question from the following list.
If you want to take the exam earlier (or later) than in the exam period
write me.
(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 RCF).
(5)
Types, saturated structures and their properties and
existence (an example construction via ultraproduct).
Omitting types theorem and
MacDowell-Specker theorem (without proofs).