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This course has two equally important objectives. The first one is to acquaint students with the rigorous mathematical style in which modern mathematics is thought, written and spoken about. The vehicle to convey these essential skills is the concrete construction of the foundational notions of real analysis in rigorous fashion, whose mastery is the second objective of the course.
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This course develops the foundations of the mathematical language that needs to be mastered in order to unlock the precise meaning of definitions and theorems in any branch of mathematics. This includes a systematic training in parsing and constructing proofs. These essential skills can only be learnt and appreciated by seeing them in action. To this end, this course starts from first principles in logic and develops solid foundations of real analysis in a coherent and self-contained manner. Mastering the material of this course requires a most serious effort but then rewards the successful student with a depth of understanding and skill that prepares the study of any branch of mathematics at master level.
The course runs for eight weeks, with a workload of 14 academic hours per week. It covers the following key topics, each of which builds on the previous ones.
- Propositional logic and first-order predicate logic
- Axiomatic set theory
- Relations and Maps
- Construction of natural, integer and rational numbers
- General topology
- Construction of real numbers
- Real series
- Differentiation of real functions
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