Here is the syllabus of the first part of the course, together with the updated lecture notes.
Lecture 1 (2013-Analysis-01)
- First encounters with infinity
- Formal language of mathematics: sets, operations, functions, logical constructions, quantifiers…
- Infinite operations (union, intersection, …)
- Enumeration. Comparison of sets.
- Sequences: stabilization, accumulation.
Lecture 2 (2013-Analysis-02)
- Numbers: how we discover them and how we “build” them.
- Arithmetic operations. Equations.
- Adjunction missing solutions of equations that we want to be solvable.
- Rational numbers: the structure of a field.
- Why can’t one divide by zero and what is the price to pay if we allow such division
- Irrational algebraic numbers: how to operate with them
Lecture 3 (2013-Analysis-03)
- Order on rational numbers.
- Holes between rational numbers
- Infinite systems of two-sided inequalities
- Dedekind cuts and operations on them.
- Completeness of the new “number system”
- Uncountability of the set of real numbers .
See you tomorrow to talk about convergence and limits.