(in Polish) Aksjomatyczna teoria mnogości 1000-1S22ATM
The main goal of the seminar is to present set theory as a formal axiomatic theory, being an axiomatic foundation of all mathematics on one hand, but on the other being itself the subject of study of mathematical logic, in particular concerning its fragments and extensions. We want to enhance understanding of the role of set-theoretic axioms in standard mathematical reasoning as well as look closer at some consequences of some additional axioms.
Besides, we want to include some theorems and proofs from infinitary combinatorics which are omitted in the standard Set Theory course.
At the very least, we intend to discuss:
- Axioms of ZFC theory, relations between them, models of (fragments of) ZFC theory, absoluteness, Goedel's theorems for ZFC and related theories.
- Cumulative hierarchies, Reflection Principle, the class of well-founded sets WF, independence of the Foundation Axiom from other axioms of ZFC.
- Constructible universe L, the axiom of constructibility V=L, relative consistency of the Axiom of Choice AC and the Generalized Continuum Hypothesis GCH with theory ZF,
- the Axiom of Choice, its weaker versions, mathematics w/o AC, permutation models, independence of AC from set theory w/o the Foundation Axiom, the Banach-Tarski paradox.
- Large cardinals, relations between different kinds of them, measurable cardinals and elementary embeddings, non-existence of measurable cardinals in L,
- More advanced infinitary combinatorics, e.g. Silver's theorem, the Suslin line and the Suslin tree, diamond principle and related principles, their consistency using V=L and their consequences.
Additionally, we consider discussing the following subjects, depending on time, participants' interest and background:
- Consequences of V=L and/or large cardinal axioms for descriptive set theory,
- Additional axioms in Set Theory and their interesting consequences in topology, measure theory and real analysis: Continuum Hypothesis CH, Martin's Axiom MA, Covering Property Axiom CPA, Open Coloring Axiom OCA, Projective Determinacy PD).
- Shoenfield's absoluteness theorem.
Main fields of studies for MISMaP
Type of course
elective monographs
Mode
Prerequisites (description)
Learning outcomes
Upon completing the seminar, the student:
- knows the Zermelo-Fraenkel axioms of Set Theory (ZFC) and understands how this system can be adopted as the formal basis of all modern mathematics,
- understands the roles of specific axioms in this system, in particular knows the strength of various weaker theories (eg. Set Theory without The Axiom or Choice or without the Foundation Axiom).
- understands the general framework of relative consistency proofs in relation to subtheories and extensions of ZFC, i.e. knows what general approach can be applied to prove the possibility of extending a theory by an additional axiom, preserving its consistency.
- knows the proof of relative consistency of the Generalized Continuum Hypothesis (GCH) and the Axiom of Choice with other axioms of set theory
- knows some useful additional set-theoretic axioms along with their consequences.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: