| Area of Mathematics: | Analysis (EA) | ||
| Semester: | 7ο | ||
| Course ID: | 72101 | ||
| Course Type: | Elective | ||
| Teaching hours per week: | Theory: 4 | Practice: 0 | Laboratory: 0 |
| ECTS : | 5 | ||
| Eclass: | For the course’s material, click here. | ||
| Instructors: | Nikolaos Tsirivas |
Introduction. Algebra of Sets. General Comprehension principle , naive set theory and Russell’s antinomy. Sets and Classes. Equinumerous sets, ordinal ,cardinal numbers of sets, Cantor theorem and Shroeder-Bernstein theorem.
● Axiomatic foundation of set theory I.
– axiom of extensionality (I) and equality of sets. Emptyset (II) and pairset (III) axioms, subsets (or separation axiom) (IV), powerset set axiom (V) and unionset axiom (VI). The axiom of infinity (VII).
– Applications of the axioms. Structured sets (Algebras, Topological spaces, Graphs). Natural numbers and the recursion theorem. Parametric recursion . Finite sets.
– Partial, total and well-ordered relations. The well-ordered set of natural numbers. Issues of well-ordered sets.
● Axiomatic foundation of set theory II (ZF and ZFA).
– The axiom schema of replacement (VIII). The Axiom ( principle) of Foundation (IX). Well-founded relationships. Induction into well-founded relation. Elements of set theory with Aczel’s anti-foundation axiom.
● Axiomatic foundation of set theory III (ZFC).
– The axiom of choice (X). Well-ordering principle, Zorn’s lemma and other axioms equivalent to the choice axiom. Rejection of the axiom of choice and Constructive Mathematics.
● Ordinal and cardinal numbers. Arithmetic of ordinal and cardinal numbers.
Transfinite induction.
Baire space. Continuum hypothesis, generalized continuum hypothesis
- Notes on Set Theory , Y.Moschovakis , Springer , 1994.
- Elements of Set Theory , H.Enderton , Academic Press , 1977.
