Logic and Foundations of Mathematics

Credits

Prerequisites

Algebra 1, Geometry 1.

Examination method

Passing an oral exam.

Learning
objectives

Acquiring a historical-critical view of the theories and methods of mathematics. “naive” set theory and elements of axiomatic set theory. The notion of infinity. Ordinals and cardinals. Fundamental concepts of classical logic, the role of logic in mathematics and its relationship with natural language.

From naive set theory to the foundational crisis and to axiomatic 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 naturals and on the ordinals. Finite and infinite sets and the historical-epistemological issues surrounding infinity. Construction of the number systems N, Z, Q, R. 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.

Logic and Foundations of Mathematics