Mathematical Logic 1
A.Y. 2019/2020
Learning objectives
The courses have three aims.
First, teaching students how to reason using linear, intuitionistic, co-intuitionistic, and classical logic, and possibly the axiom of choice.
Second, familiarizing students with the way of thinking internal to category theory.
Third, showing some contemporary mathematical theories in action by means of the principles above.
First, teaching students how to reason using linear, intuitionistic, co-intuitionistic, and classical logic, and possibly the axiom of choice.
Second, familiarizing students with the way of thinking internal to category theory.
Third, showing some contemporary mathematical theories in action by means of the principles above.
Expected learning outcomes
Formalization abilities, use of a logical calculus, knowledge of the limits of logical mechanization.
Lesson period: First semester
Assessment methods: Giudizio di approvazione
Assessment result: superato/non superato
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
First order languages.
Tarski semantics for first order logic.
Theories and examples of theories.
Sequent calculus for classical logic (with some basic information on intuitionistic logic and
on cut elimination problems)
Completeness theorem for first order theories.
Hilbert calculi; formal systems for arithmetic.
Recursive functions and representability.
Goedelization techniques.
First Goedel incompleteness theorem.
Church theorem.
Tarski theorem.
Basic information on second Goedel incompleteness theorem.
Tarski semantics for first order logic.
Theories and examples of theories.
Sequent calculus for classical logic (with some basic information on intuitionistic logic and
on cut elimination problems)
Completeness theorem for first order theories.
Hilbert calculi; formal systems for arithmetic.
Recursive functions and representability.
Goedelization techniques.
First Goedel incompleteness theorem.
Church theorem.
Tarski theorem.
Basic information on second Goedel incompleteness theorem.
Prerequisites for admission
No specific preliminary knowledge is required
Teaching methods
Standard lecture
Teaching Resources
Electronic notes available from Ariel website
Assessment methods and Criteria
The final examination consists of two parts: an oral exam and a lab exam.
- In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding Galerkin methods in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
-The lab exam consists in developing a project, which will be assigned in advance by the professor. The project will be presented by the student during the oral exam. The lab portion of the final examination serves to assess the capability of the student to put a problem of numerical approximation of PDEs into context, find a solution and to give a report on the results obtained.
The complete final examination is passed if both the parts (oral, lab) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
- In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding Galerkin methods in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
-The lab exam consists in developing a project, which will be assigned in advance by the professor. The project will be presented by the student during the oral exam. The lab portion of the final examination serves to assess the capability of the student to put a problem of numerical approximation of PDEs into context, find a solution and to give a report on the results obtained.
The complete final examination is passed if both the parts (oral, lab) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/01 - MATHEMATICAL LOGIC - University credits: 6
Lessons: 42 hours
Professor:
Ghilardi Silvio
Shifts:
-
Professor:
Ghilardi SilvioProfessor(s)