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Department of Mathematics and Applications
"Renato Caccioppoli"


Set Theory

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Set Theory

Credits

6

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/01 Mathematical Logic.

Examination method

Passing an oral exam.

Learning
objectives

Techniques for the axiomatisation, development and modelling of a theory. Comparison between different set theories (e.g. ZF, NBG, MK). Confidence with the concepts and results of ZF theory. Familiarity with the concepts of consistency and independence. Applications to other branches of mathematics, viewed as a single discipline.

Syllabus

Language, axioms, methods and results of the axiomatic set theories ZF, NBG, MK. Classes. Axiom of foundation. Axiom of choice, its consequences, equivalent forms and generalisations. Real numbers and their properties. Boolean algebras. Ultraproducts. The continuum hypothesis and its weak forms. Ordinal and cardinal arithmetic. Cardinal exponentiation. Regular cardinals. Large cardinals. Constructible sets.

Expected learning
outcomes

At the end of the course students must demonstrate that they

  • know and understand the techniques for the axiomatisation, development and modelling of a theory;
  • are able to apply the knowledge acquired to link abstract settings readily with the corresponding concrete examples (e.g. the theories ZF, NBG, MK);
  • are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
  • are able to identify the most appropriate methods to analyse and solve a problem related to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Command of the knowledge acquired, clarity of presentation, rigour in the use of language, confidence in using the notions acquired.