Teachings

Select Academic Year:     2016/2017 2017/2018 2018/2019 2019/2020 2020/2021 2021/2022
Professor
HECTOR CARLOS FREYTES (Tit.)
Period
First Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/65]  MATHEMATICS [65/40 - Ord. 2020]  MATEMATICA PURA 9 72
[60/65]  MATHEMATICS [65/50 - Ord. 2020]  MATEMATICA PER LA DIDATTICA E LA DIVULGAZIONE 9 72
[60/65]  MATHEMATICS [65/60 - Ord. 2020]  MATEMATICA APPLICATA 9 72

Objectives

Knowledge and understanding
The main objective of the course is devoted to provide students with the key tools for dealing with different kind logical and mathematical problems of various types (algebraic, digital techniques etc ...).

Making judgements
The student have to acquire an autonomous and critical reflection on the course’s issues.
Communication skills
At the end of the course, the student will be able to applied the arguments regarding to the basic mathematic of the course.
Learning skills
Finally, the student must be able to find sources to update and deepen autonomously and constantly knowledges and professional competencies

Prerequisites

Only basic notions on algebra

Contents

The initial purpose of the course is to give the theoretical, conceptual and methodological fundamental issues in mathematical logic. Application of the theoretical concept are considered.

1) Propositional logic
Propositional logic, Syntax
Propositional logic, Semantics
Boolean valuations
Hilbert style calculus (deductive systems)
Soundness and completeness studies.

2)First order logic
First order logic, Syntax
First order logic, Semantics
Hilbert style calculus (First order deductive systems)
Soundness and completeness studies.

3. Godel Theorems
Formal number theory
Primitive recursive and recursive functions
Arithmetization, Godel numbers
Diagonal Lemma
Incompleteness Godel Theorem
Church thesis

4) Order, Lattices and Boolean algebras
Order
Lattice structures
Boolean algebras
Boolean rings
Filters and ideals
Direct products
Quotient algebras
Sundirect product
Equational completeness
Lindenbaum algebra

Teaching Methods

Classroom lectures and online didactic

Verification of learning

Written and oral test in presence and/or steaming

The exams are expressed in thirtieths. The tests aim to attest the aims set up in the section "knowledge and comprehension".

The written exam covers chapters 1, 2 and 3 of the course, while the oral exam covers chapter 4. In order to pass the course, the student needs to pass both exams, and the final mark will be a scaled average (67% written, 33% oral).

In order to pass the exam that is with a minimum score of 18/30 the student must show a sufficient knowledge of all the addressed topics, with a proper use of the language. In order to achieve the maximum score of 30/30 cum laude, the student must show an excellent knowledge of all the dealt topics.

Texts

1) First-Order Logic and Automated Theorem Proving (Melvin Fitting), Springer-Verlag New York, 2nd edition 2012.

2) Introduction to Mathematical Logic, (Elliott Mendelson), Chapman and Hall ed., Sixth Edition (2015)
3) Sistemas electronicos digitales (Enrique mandado)
Marcombo, 10ma Ed, 2015.
4) Lectures on Boolean Algebras (Paul Halmos), Martino Fine Books, (2013)
5) A Shorter Model Theory, (Wilfrid Hodges), Cambridge University Press, 1997
6) Basic Category theory (Tom Leinster, Cambridge University Press, 2014 - freely available at arXiv:1612.09375)
7) Topoi: The Categorial Analysis of Logic (Robert Goldblatt) Dover Pubns; (2006)
8) Basic Category Theory for Computer Scientists (Benjamin C. Pierce) Mit Pr (1991)