Kurzus nemzetközi vendég- és részidős hallgatóknak
- Kar
- Bölcsészettudományi Kar
- Szervezet
- BTK Filozófia Intézet
- Kód
- BMI-LOTD-103E.01
- Cím
- Foundations of mathematics
- Tervezett félév
- Tavaszi
- Meghirdetve
- 2023/24/2
- ECTS
- 3
- Nyelv
- en
- Leírás
- te_kód: BMI-LOTD-103E kurzuskód: BMI-LOTD-103E/1,BMI-LOTD-103E
- Oktatás célja
- Foundations of Mathematics Amitayu Banerjee ABSTRACT In this course we will work in Zermelo Fraenkel Set Theory with Axiom of Choice (ZFC) and discuss the followings, 1. Sets are the foundation of all mathematical entities Motivation: Sets are the foundation of all abstract mathematical concepts (like functions, relations, relational structures) and all concrete mathematical objects (like 1, 2/3). 2. Construction of Number systems leading to di erent branches of mathematics Structures, Relational Structures, Algebraic Structures. Natural Numbers- Origin and structure of natural numbers, Countability. Rational Numbers- Construction of the structure of rationals from the structure of nat- ural numbers and properties of rationals. Real Numbers- Completeness Axiom and Archimedean property, Construction of real numbers that leads to the study of real analysis. Continuum and properties of reals. Complex Numbers- Construction of complex numbers that leads to complex analysis. 3. Role of Axiom of Choice in the foundation of mathematics Axioms of ZF, Origin of Axiom of Choice(AC), Consistancy and Independence of AC from other axioms, Equivalent versions of AC. Partially Ordered sets, Zorn's Lemma and their applications. Filters, Ultralters, Ideals and Prime Ideals, Ultralter theorem and it's equivalents and applications. Dependent Choice and it's equivalence and applications. . . . 4. Motivation to work in ZF+DC or ZF+CC or ZF+AD where DC, CC, AD rep- resents Dependent Choice, Countable Choice and Axiom of Determinicy respec- tively