MMOTH Model theory

Faculty of Science
Autumn 2022
Extent and Intensity
0/2/0. 2 credit(s). Type of Completion: z (credit).
Teacher(s)
Kristóf Kanalas, Ph.D., MSc (lecturer)
prof. RNDr. Jiří Rosický, DrSc. (alternate examiner)
Guaranteed by
prof. RNDr. Jiří Rosický, DrSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Fri 16:00–17:50 M5,01013
Prerequisites
A few basic facts and notions from set theory and topology: basics of cardinal arithmetic, definition of a compact/ Haussdorff space, dense subset, etc.
Course Enrolment Limitations
The course is offered to students of any study field.
Abstract
The course introduces the main tools and theorems of first-order model theory, including the study of saturated and atomic structures.
Learning outcomes
Student will obtain knowledge of first-order model theory.
Key topics
  • Ultrafilters and ultraproducts. Spectrum of a Boolean algebra. Regular ultrafilters, cardinality of the regular ultrapower.
  • Structures, interpretation of formulas. Elementary equivalence, elementary substructures, Tarski-Vaught test. Downward Löwenheim-Skolem theorem.
  • Ultraproduct of structures, Łoś-lemma. Upward Löwenheim-Skolem theorem. Compactness. A class of L-structures is first-order axiomatizable iff it is closed under ultraproducts and elementary equivalence.
  • Universal structures, universality of regular ultrapowers, Frayne's theorem.
  • Types, types as ultrafilters, realizing types. Saturated models. Elementary equivalent saturated structures of the same cardinality are isomorphic. (CH) Two countable structures over a countable signature are elementary equivalent iff any of their non-trivial coutable ultrapowers are isomorphic.
  • Weak existence of saturated models. Theory of the random graph: if GCH fails at \kappa then this theory has no \kappa ^+-saturated model.
  • Every compact Haussdorff space is Baire. In a countable compact Haussdorff space the set of isolated points is dense. Omitting types theorem. Atomic and prime models. Equivalent conditions for the existence of atomic models.
  • Countable categoricity. The existence of a countable saturated model. Vaught's theorem.
  • If time permits: good ultrafilters, Keisler's isomorphism theorem.
Study resources and literature
  • HODGES, Wilfrid. Model theory. 1st pub. New York [N.Y.]: Cambridge University Press, 1993, xiii, 772. ISBN 9780521066365. info
Approaches, practices, and methods used in teaching
Lectures with a few optional exercises meanwhile.
Method of verifying learning outcomes and course completion requirements
Confirmation of an active participation.
Language of instruction
English
Further Comments
Study Materials
The course is taught only once.
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