Complements of Logic
Code
9645
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Oleksiy Karlovych
Teaching language
Português
Objectives
At the end of this course students will have acquired advanced knowledge and skills in the area of logic, in particular set theory, in order to:
- Understand advanced contents in the area;
- Being able to execute research on a topic in the area.
Subject matter
Sets and Foundations: Russell''s paradox. The ZFC axiomatization. Classes vs. sets. Sketch of the development of mathematics within ZFC. Cantor''s and Schröder-Bernstein Theorem. Equipotence of P(N) and the real numbers. Basic cardinal arithmetic. Well-orderings. Induction and transfinite recursion. Transitive sets. Von-Neumann ordinals. Collaps of a well ordering. Basic ordinal arithmetic. Equivalent formulations of the axiom of choice. Tarski''s least fixed point theorem. Cardinals as initial ordinals. Aleph numbers. Continuum Hypothesis. The cumulative universe. Overview of independence and consistency results.
Bibliography
A. J. Franco de Oliveira, Teoria dos Conjuntos, Intuitiva e Axiomática (ZFC), Escolar Editora, 1982.
K. Hrbacek and T. Jech, Introduction to Set Theory, Marcel Dekker, 1999.
Y. Moschovakis, Notes on Set Theory, Springer, 2nd edition, 2005.