| Unit name | Axiomatic Set Theory |
|---|---|
| Unit code | MATHM1300 |
| Credit points | 20 |
| Level of study | M/7 |
| Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |
| Unit director | Professor. Welch |
| Open unit status | Not open |
| Pre-requisites | |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
A natural heirarchy of sets, the constructible sets, first defined by Godel, is obtained by a transfinite recursion through all the ordinal numbers. The heirarchy has a very smooth character, and the uniformity of its presentation enables one to see that various questions that had been found to be unanswerable in Godel's day, have solutions in the resulting model of Zermelo-Fraenkel set theory, such as the Axiom of Choice (AC) and the Continuum Hypothesis (CH). It is now known that the AC and CH are neither provable nor disprovable from the other axioms. The universe of constructible sets is a model of these statements and hence we see that they are at least non-contradictory.
Aims
To develop the theory of Gödel's universe of constructible sets; to use this model to prove the consistency of various statements of mathematics with the currently accepted axioms of set theory.
Syllabus
The Axioms of Zermelo-Fraenkel Set Theory with Choice
Class terms, relativisations to modesl; absoluteness
Consistency proofs, reflection theorem
Closed and unbounded sets, stationary sets;
Regular and singular cardinals, cofinality; inaccessible cardinals;
Goedel's Def function, and the definition of the constructible hierarchy L
The Consistency of AC and GCH with ZFC
Relation to Other Units
This is the only unit which develops further the concepts in the Level 3 units Logic and Set Theory.
It is particular;ly pertinent to those interest in, or taking courses in mathematics and philosophy.
After taking this unit, students should:
Transferable Skills:
Assimilation and use of novel and abstract ideas.
Lectures and exercises to be done by students.
The assessment mark for Axiomatic Set Theory is calculated from a 2½-hour written examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted in this examination.
A full text will be handed out.
Alternative & Further Reading:
