Logic

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M2
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Abstract syntax. Bound and free variables. Substitution. Natural deduction. Cut elimination. Consistency of natural deduction.
First-order languages and theories. Consistent and complete theories. Conservative extensions. Interpretation of a language and a model of a theory.
Soundness and Gödel completeness theorem. Compactness theorem and its consequences.
Peano arithmetic, Gödel incompleteness theorems.
Examples of first-order theories and applications of model theory.

Readings

N. Prijatelj: Osnove matematične logike, 2. del: Formalizacija, DMFA Slovenije, Ljubljana, 1992.
N. Prijatelj: Osnove matematične logike, 3. del: Aplikacija, DMFA Slovenije, Ljubljana, 1994.
W. Rautenberg: A Concise Introduction to Mathematical Logic, 3. izdaja, Springer, 2010.
E. Mendelson: Introduction to Mathematical Logic, 4. izdaja, Chapman and Hall, 1997.
A.S. Troelstra, H. Schwichtenberg: Basic Proof Theory, 2. izdaja, Cambridge University Press, 2000.

Objectives and competences

Basic knowledge of foundations of mathematics and mathematical logic.

Intended learning outcomes

Knowledge and understanding:
Understanding of logical foundations of mathematics and the fundamental limitations of the axiomatic method.
Application:
Logic, being the foundation of mathematics, provides the means for communication and methodology in mathematics.
Reflection:
The fact that there are mathematical problems without solutions invites a thorough reconsideration of the nature of mathematics.
Transferable skills:
Ability to formally express mathematical content. Ability to perform meta-mathematical analysis.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Type (examination, oral, coursework, project):
2 midterm exams instead of written exam, written exam
oral exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Andrej Bauer:
AWODEY, Steve, BAUER, Andrej. Propositions as [Types]. Journal of logic and computation, ISSN 0955-792X, 2004, vol. 14, no. 4, str. 447-471. [COBISS-SI-ID 13374809]
BAUER, Andrej, SIMPSON, Alex. Two constructive embedding-extension theorems with applications to continuity principles and to Banach-Mazur computability. Mathematical logic quarterly, ISSN 0942-5616, 2004, vol. 50, no. 4/5, str. 351-369. [COBISS-SI-ID 13378649]
BAUER, Andrej. A ralationship between equilogical spaces and Type Two Effectivity. Mathematical logic quarterly, ISSN 0942-5616, 2002, vol. 48, suppl. 1, str. 1-15. [COBISS-SI-ID 12033369]