Instructor: Prof. Dr. Benedikt
Löwe
Vakcode: 5314SEML3Y
ECTS: 3
Time & Place:
Wednesday 1519, SP107 F1.16
Course language: English
Intended audience:
M.Sc. students of Logic specializing in mathematical logic.
Topic of this course:
Extensions of firstorder 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.
Schedule.
6 February 2013 
1517
 Benedikt Löwe.
 Introduction to the theme of the seminar and the various topics.
Distribution of tasks to the students.


1719
 Alexander Block.  Lindström's Theorem.


20 February 2013 
1517

Yuning Feng,
Iva Gornishka,
Nikhil Maddirala,
Deniz Sarikaya.
 Secondorder logic vs. manysorted logic. Syntax
and Semantics of manysorted logic. Syntax and Semantics of secondorder
logic. Expressivity results and impossibility results (status of
compactness, completeness and LöwenheimSkolem). Representation of
secondorder logic in manysorted logic.
Literature.
 Jean H. Gallier, ManySorted FirstOrder Logic, Chapter 10 of
"Introduction to the Theory of Computation", 2004: PostScript file.
 Jouko Väänänen, SecondOrder Logic and Foundations
of Mathematics, Bulletin of Symbolic Logic 7:4 (2001),
pp.504520.
 HeinzDieter Ebbinghaus, Jörg Flum, Wolfgang Thomas,
Mathematical Logic, SpringerVerlag 1994 [Undergraduate Texts in
Mathematics]


1719


27 February 2013 
1517


1719


6 March 2013 
1517

Simon Docherty,
Evante GarzaLicudine.
 Infinitary logic. Syntax and Semantics of
L_{ω1ω}. Expressive power of infinitary logics in comparison to firstorder logic. Status of completeness, compactness and LöwenheimSkolem.
Literature.

HeinzDieter Ebbinghaus, Jörg Flum, Wolfgang Thomas,
Mathematical Logic, SpringerVerlag 1994 [Undergraduate Texts in
Mathematics]
 Dave Marker, A Primer on Infinitary Logic. PDF file.


1719

Adam Crager,
Justin Kruger,
Andrea Nespola.
 Generalized quantifiers. Definitions of generalized
quantifiers. Examples. Definability.
Literature.
 Per Lindstöm, First order predicate logic with generalized
quantifiers, Theoria 32:3 (1966), pp. 186195. PDF
file
 Lauri Hella, Definability hierarchies of generalized quantifiers, Annals of Pure and Applied Logic 43 (1989), pp. 235271. publisher's webpage
 Lauri Hella, Kerkko Luosto, Jouko Väänänen, The hierarchy theorem for generalized quantifiers, Journal of Symbolic Logic 61 (1996), pp. 802817. PDF file.


13 March 2013 
1517


1719


20 March 2013 
1517

Hans Grathwohl,
Zhiguang Zhao.

ωlogic. Syntax and Semantics of ωlogic. Proof of
the completeness of arithmetic in ωlogic. Connections between
ωlogic, ωconsistency and Gödel's second
incompleteness theorem.
Literature. C.C. Chang, H. Jerome Keisler, Model Theory,
NorthHolland 1990 [Studies in Logic and the Foundations of Mathematics
73]


1719


3 April 2013 
1719

Alexander Block,
Julia Ilin,
Tanmay Inamdar,
Hugo Nobrega,
Apostolos Tzimoulis.
 Subsystems of secondorder arithmetic.
Lévy hierarchy of formulas. The definitions of the five main systems
of reverse mathematics. Separation of the systems by construction of models.
βmodels. ωmodels.
Literature. Steve Simpson, Subsystems of Second Order Arithmetic,
Second Edition, Cambridge University Press 2010 [Perspectives in Logic]


10 April 2013 
1719


17 April 2013 
1719


24 April 2013 
1719


