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

SET AND PROOF THEORY

During the course the foundations of Set Theory and Proof Theory will be introduced.

It is well known how Cantorian Set Theory introduced the actual infinite in the mathematical ontology, after many centuries in which it was only accepted as a potential notion. Cantor went even further, by showing how always increasing infinite cardinalities exist.

In the course we will analyze the notions of set, funtion, and of transfinite cardinality. We will introduce the axiomatic system ZFC for Set Theory and we will analyze the mathematical and philosophical consequences of the Axiom of Choice and of the Axiom of the Infinity.

Moreover, we will deal with the foundations of Proof Theory by the sequent calculus approach, focusing on the proof of the Cut Elimination Theorem, the well-known "Hauptsatz" by which Gerhard Gentzen proved the consistency of Peano Arithmetic (we will focus on the proof of the Hauptsatz for the propositional and the predicative calculus, obtaining in this way a consistency proof for such calculi, and for recent extensions to the so-called geometrical theories).

The treatment of these topics will be homogeneously distributed during the course.

Non-attending students are referred to the instructions given in the section "Readings/Bibliography".

 

Readings/Bibliography

Handouts provided by the teacher.

P. Maffezioli: Il teorema di interpolazione in logica classica, intuizionista e nelle teorie del primo ordine. CLUEB, 2020.

M. Borga, D. Palladino: Oltre il mito della crisi. Editrice La Scuola, chap. 1-section 2.4, chap. 2-sections 2,3.

B. Russell: Introduzione alla Filosofia Matematica (qualsiasi edizione), chapters 12-13.

J-Y Girard: La logica lineare. In "Logica", Le Scienze Quaderni n. 60, a cura di C. Mangione, Ed. Le Scienze.

 

Non attending students must study also M. Borga, D. Palladino: Oltre il mito della crisi. Editrice La Scuola, chapter 2-sections 4,6 (under agreement with the teacher).

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.

The final exam will consist in the exposition of a topic selected by the student and in another question chosen by 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 blackboard.

- Handouts of the teacher.

- Lessons recording.

Office hours

See the website of Guido Gherardi