Title A Philosophical Approach to Model Theory
Lecturer Péter Mekis
Codes BBN-FIL-401.104, BMA-LOTD-203.03, BMA-FILD-401.104, TANM-FIL-401.104, BMI-LOTD-203E.03
Time Monday 16:00-17:30
Venue i 224
Prerequisites The course assumes familiarity with the syntax and semantics of standard predicate logic (as discussed in the Logic Lecture).
Exam The course ends with an oral exam; students can choose one of the three main theorems covered in the course.
Description Model theory studies the relationship between formal theories and the structures that satisfy them. This course offers a technically mild, philosophically detailed introduction to the subject. After setting the stage, we will prove the following three basic theorems following the course:
  1. Compactness theorem: if every finite part of a first-order theory T has a model, then T has a model, too.
  2. Downward Löwenheim--Skolem theorem: If a first-order theory (expressed in a countable language) has a model, then it also has a countable model. (We'll actually prove a stronger version: every first-order structure with a countable signature has a countable elementary substructure.)
  3. Lindström's theorem: First-order logic is the strongest logical system of which both the compactness theoreem and the downward Löwenheim--Skolem theorem are true.
All three of these theorems will be discussed along with their prilosophical implications, with special regard to the differences between first-order and higher-order logics, and the so-called "Skolem's paradox".

Students who intend to have a deeper understanding of the subject will find a more advanced discussion in Zalán Gyenis' model theory courses.

  • C. C. Chang & H. J. Kreisler, Model Theory. (3rd edition.) Elsevier, 1993.
  • W. Hodges, A Shorter Model Theory. Cambridge UP, 1997.