28101 - Logic (1) (2nd cycle)

Learning outcomes

At the end of the course, students are supposed to become acquainted with fundamental knowledge on the metatheory of some selected formal systems concerning crucial topics in Logic, such as validity, completeness, decidability, and possibly referring to well known limiting results

Course contents

SETS AND MODELS

The course will introduce the foundations of Set Theory and some fundamental concepts of Model Theory, two central branches of formal Logic.

It is known how Georg Cantor introduced actual infinity into mathematics, which previously was treated primarily as a potential notion. Cantor went even further, finding a way to compare different degrees of infinity with each other and showing that increasingly larger infinite quantities exist. However, his intuitive conception of a set ran into paradoxes, which were solved with the introduction of the Zermelo-Fraenkel axiomatic set theory. In this respect, in the course we will analyze the concept of set, function and cardinality, and we will introduce then the axiomatic system ZFC.

Closely connected to Set Theory is Model Theory. We will study one of the crucial results of this area, the Compactness Theorem with some of its consequences, and the Herbrand Theorem and its application to the verification of validity and of logical consequence for first-order formulas.

 

Readings/Bibliography

About Set Theory, handouts provided by the teacher.

About Model Theory, G. Gherardi: Introduzione alla Teoria dei Modelli, Archetipo Libri, 2023.

Non attending students must study also the chapters on the Multiplicative Axiom and on the Infinity Axiom in B. Russell, Introduction to Mathematical Philosophy.

Teaching methods

Lessons in classroom with electronic blackboard. Lessons will be recorded and uploaded on line.

Assessment methods

The final exam will consist in an oral test, in which students are asked to prove their correct comprehension of the notions dealt with during the course, by oral explanation and also by written reconstruction of the fundamental definitions, results and proofs. 

 

To take the exam, it is possible not only to register for the exams set by the teacher (approximately every two months) but also to agree on individual exams with the teacher.

 

Assessment criteria and thresholds of evaluation:

30 cum laude: Excellent as to knowledge, terminology and critical expression.

30: Excellent, knowledge is complete, well articulated and mostly correctly expressed, although with some slight faults.

27-29: Good, knowledge comprehensive and satisfactory, essentially correct expression.

24-26: Fairly good, knowledge present in significant points, but not complete and not always expressed with correctness.

21-23: Sufficient, knowledge is sometimes superficial, but the guiding general thread is included. Expression and articulation incomplete and often not appropriate

18-21:.Almost sufficient, but knowledge present only on the surface. The guiding principle is not included with continuity. The expression and articulation of the speech show important gaps.

<18: Not sufficient, knowledge absent or very incomplete, lack of guidance in discipline, expression seriously deficient. Exam failed.

Teaching tools

- Electronic whiteboard. 

- Lessons recording.

Office hours

See the website of Guido Gherardi