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Seminar on Mathematical Logic

2012/2013; 2nd Semester

Institute for Logic, Language & Computation
Universiteit van Amsterdam

Instructor: Prof. Dr. Benedikt Löwe
Vakcode: 5314SEML3Y
Time & Place: Wednesday 15-19, SP107 F1.16
Course language: English
Intended audience: M.Sc. students of Logic specializing in mathematical logic.

Topic of this course: Extensions of first-order logic.

Evaluation: The course is graded on a pass/fail basis. In order to pass, a student will have to give a presentation in class.


6 February 2013 15-17 Benedikt Löwe. Introduction to the theme of the seminar and the various topics. Distribution of tasks to the students.
17-19 Alexander Block. Lindström's Theorem.
20 February 2013 15-17 Yuning Feng, Iva Gornishka, Nikhil Maddirala, Deniz Sarikaya. Second-order logic vs. many-sorted logic. Syntax and Semantics of many-sorted logic. Syntax and Semantics of second-order logic. Expressivity results and impossibility results (status of compactness, completeness and Löwenheim-Skolem). Representation of second-order logic in many-sorted logic.


  • Jean H. Gallier, Many-Sorted First-Order Logic, Chapter 10 of "Introduction to the Theory of Computation", 2004: PostScript file.
  • Jouko Väänänen, Second-Order Logic and Foundations of Mathematics, Bulletin of Symbolic Logic 7:4 (2001), pp.504-520.
  • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas, Mathematical Logic, Springer-Verlag 1994 [Undergraduate Texts in Mathematics]
27 February 2013 15-17
6 March 2013 15-17 Simon Docherty, Evante Garza-Licudine. Infinitary logic. Syntax and Semantics of Lω1ω. Expressive power of infinitary logics in comparison to first-order logic. Status of completeness, compactness and Löwenheim-Skolem.


  • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas, Mathematical Logic, Springer-Verlag 1994 [Undergraduate Texts in Mathematics]
  • Dave Marker, A Primer on Infinitary Logic. PDF file.
17-19 Adam Crager, Justin Kruger, Andrea Nespola. Generalized quantifiers. Definitions of generalized quantifiers. Examples. Definability.


  • Per Lindstöm, First order predicate logic with generalized quantifiers, Theoria 32:3 (1966), pp. 186-195. PDF file
  • Lauri Hella, Definability hierarchies of generalized quantifiers, Annals of Pure and Applied Logic 43 (1989), pp. 235-271. publisher's webpage
  • Lauri Hella, Kerkko Luosto, Jouko Väänänen, The hierarchy theorem for generalized quantifiers, Journal of Symbolic Logic 61 (1996), pp. 802-817. PDF file.
13 March 2013 15-17
20 March 2013 15-17 Hans Grathwohl, Zhiguang Zhao. ω-logic. Syntax and Semantics of ω-logic. Proof of the completeness of arithmetic in ω-logic. Connections between ω-logic, ω-consistency and Gödel's second incompleteness theorem.

Literature. C.C. Chang, H. Jerome Keisler, Model Theory, North-Holland 1990 [Studies in Logic and the Foundations of Mathematics 73]

3 April 2013 17-19 Alexander Block, Julia Ilin, Tanmay Inamdar, Hugo Nobrega, Apostolos Tzimoulis. Subsystems of second-order arithmetic. Lévy hierarchy of formulas. The definitions of the five main systems of reverse mathematics. Separation of the systems by construction of models. β-models. ω-models.

Literature. Steve Simpson, Subsystems of Second Order Arithmetic, Second Edition, Cambridge University Press 2010 [Perspectives in Logic]

10 April 2013 17-19
17 April 2013 17-19
24 April 2013 17-19
Last update: 20 March 2013.