# MAT-60556 Mathematical Logic, 5 cr

Esko Turunen

#### Lessons

 Implementation Period Person responsible Requirements MAT-60556 2019-01 4 Esko Turunen Lecture diary and examination.

#### Learning Outcomes

The course is based on the first and second chapter of Gaisi Takeuti’s in 1987 published book Proof Theory (Studies in Logic and the Foundations of Mathematics). The first chapter introduces Gentzen's sequent calculi for Intuitionistic Logic (LJ) and Classical Logic (LK) and proves Gentzen's cut-elimination theorem as well as completeness theorems for both LJ and LK. The second chapter is Gentzen's second proof of the eliminability of essential cuts from a derivation of the empty sequent in Peano Axioms. All these results are needed to prove the main result of the course: Gödel’s two incompleteness theorems and Tarski’s teorem. After completing the course, the student knows what the fundamental results of mathematical logic are, in particular the proof theoretical approach, Completeness of propostional and first order logc, Gödel's incompleteness theorems, Tarskis theorem, Löwenheim-Skolem theorem, Herbrand theorem. A rudimentary understanding of proof theory is formed, and the student is able to apply some basic results and knows some details of basic techniques such as truth tables.

#### Content

 Content Core content Complementary knowledge Specialist knowledge 1. The basic ideas of Gentzen's proof theory, logical foundations of classical and intuitionistic mathematical theories. Gödel's incompleteness theorems 2. Equational, propositional and predicate calculus. Peano Axioms od arithmetic 3. Connections to meta-mathematics and computation. Lindenbaum algebra, connection to Boolean and Hayting algebras

#### Instructions for students on how to achieve the learning outcomes

Visit the lectures, do meticulous lecture notes, do all the home exercises in time and participate actively in exercises. Inquire about things that are unclear to you.

#### Assessment scale:

Numerical evaluation scale (0-5)

#### Partial passing:

Completion parts must belong to the same implementation

#### Study material

 Type Name Author ISBN URL Additional information Examination material Book Proof Theory Gaisi Takeuti No Lecture slides Esko Turunen Yes