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Department of Mathematics and Applications
"Renato Caccioppoli"


Logic and Foundations of Mathematics

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Logic and Foundations of Mathematics

Credits

6

Prerequisites

Algebra 1, Geometry 1.

Scientific-disciplinary sector (SSD)

MAT/04 Complementary Mathematics.

Examination method

Oral exam.

Learning
objectives

Acquisition of a historical and critical view of the theories and methods of mathematics, with particular regard to the “naive” and axiomatic versions of set theory. Understanding of the issues relating to the notion of infinity. Introduction to the fundamental concepts of classical logic, to the role of logic in mathematics and to its relationship with natural language.

Syllabus

From naive set theory, through the crisis of foundations, to axiomatic set theory. The axioms of ZF theory. Ordinal and cardinal numbers. Construction of the natural numbers as finite ordinals and as elements of a Peano triple. Induction and recursion on the natural numbers and on the ordinals. Finite and infinite sets and the historical and epistemological problem of infinity. The axiom of choice. The axiom of foundation and the universe U of sets. Outline of some more recent developments. Fundamental concepts and results of classical propositional and predicate logic: formal language, syntax/semantics, proofs, models, etc.

Expected learning
outcomes

On completion of the course, students must demonstrate that they

  • know and understand the problems concerning the foundations of mathematics, in particular those that led to the formulation and development of ZF set theory and of classical logic or possible alternatives;
  • can apply the knowledge acquired to solving problems, using a formal mathematical language to describe the axioms and main results of ZF theory, as well as of propositional logic and predicate calculus;
  • can communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
  • can identify the most appropriate methods to analyse and solve a problem relating to the course topics and interpret the results correctly.

Learning outcomes
to be assessed

Formal correctness and completeness in presenting the course topics. Ability to recognise flexibly the use of set theory and of the formal apparatus of classical logic in the various areas of mathematics.