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, 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.
Course objectives
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.
Syllabus
  • 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.
Literature
  • HODGES, Wilfrid. Model theory. 1st pub. New York [N.Y.]: Cambridge University Press, 1993, xiii, 772. ISBN 9780521066365. info
Teaching methods
Lectures with a few optional exercises meanwhile.
Assessment methods
Confirmation of an active participation.
Language of instruction
English
Further Comments
Study Materials
The course is taught only once.
  • Enrolment Statistics (recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2022/MMOTH