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.

# Logic

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.

Basic knowledge of foundations of mathematics and mathematical logic.

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.

Lectures, exercises, homeworks, consultations

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)

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]