Syllabus and literature for lectures on
Course code: NMAG407
Fall 2020 special page
Master Thesis topics: If you contemplate the idea to
write an MSc Thesis in logic, or specifically
in model theory, talk to me: there is a variety of possible suitable topics
and I usually put into the SIS only one or two.
Student logic seminar.
The course covers main topics of model theory with an emphasis
on examples and methods important for applications of model
theory in algebra, geometry and number theory.
Syllabus
Structures and an interpretation of a language.
Tarski's truth definition.
Embeddings and isomorphisms of structures, substructures.
Elementary equivalence and the Ehrenfeucht-Fraisse game,
example: the theory DLO
of dense linear orderings without end-points.
Preservation theorems, the diagram of a structure.
Algebraic
examples: ordered real closed fields (RCF) and algebraically closed
fields (ACF_p and ACF_0), vector spaces over a fixed field, groups.
The compactness theorem and its applications: elementary extensions,
the upward Lowenheim-Skolem theorem, non-standard models of RCF and
of the ring of integers. A transfer theorem from ACF_p to ACF_0.
The Ax-Grothendieck theorem.
Complete theories and kappa-categorical theories.
Vaught's test. Skolemization and the
downward Lowenheim-Skolem
theorem.
Quantifier elimination and its proofs for DLO
and ACF. The strong minimality of ACF and the o-minimality
of RCF (assuming QE for RCF).
Strongly minimal theories and their associated
(pre)geometries.
Types, their realization and omitting.
The Stone space of complete types, algebraic ex.:
Zariski spectrum.
Isolated types and the Omitting types theorem.
MacDowell-Specker's theorem.
kappa-saturated structures and their existence.
Countable saturated structures
and the size of the Stone space. Saturation of ultraproducts.
The number of non-isomorphic models of a given cardinality.
Vaught's conjecture. Morley's categoricity theorem.
Literature:
Main sources
J.Kirby, An Invitation to Model Theory (Cambridge U. Press, 2019)
D. Marker, Model Theory: An Introduction (Springer, 2002).
(More advanced. The MFF library has an access to the e-version of the book.
Almost all material can be also found in Marker's lecture notes
(online) listed below.)
Other classics
C.C.Chang, J.H.Keisler: Model theory, NHPC 1973.
W. Hodges: Model Theory, Cambridge Univ. Press, 1993.
W. Hodges, Shorter Model Theory (CUP, 1997).
G.Sacks, Saturated Model Theory (World Sci., 2nd ed., 2010)
Lecture notes on the web
D.Marker's lectures
at the MSRI.
D.Marker's course at Orsay in 2010.
A.Pillay's lecture notes, U. Leeds.
A.Wilkie's lecture notes,
U. Manchester.