Cardinal arithmetic

2022/2023

Programme:

Mathematics, Second Cycle

Year:

1 ali 2 year

Semester:

first or second

Kind:

optional

Group:

ECTS:

Language:

slovenian, english

Course director:

Prof. Dr. Andrej Bauer, Prof. Dr. Alexander Keith Simpson

Hours per week – 1. or 2. semester:

Lectures

Seminar

Tutorial

Lab

There are no prerequisites.

Sets and classes. Axioms of set theory. Axiom of choice, Zorn lemma and its applications, well ordering, transfinite induction, ordinal numbers and their arithmetic, Schröder-Bernstein theorem, cardinal numbers and their arithmetic. If time permits: filters and ultrafilters, large cardinal numbers.

W. Just, M. Weese: Discovering Modern Set Theory I. AMS, 1991.
P. R. Halmos: Naive set theory, Springer-Verlag, New York, 1974.
H. Ebbinghaus et al.: Numbers, Springer-Verlag, New York, 1990.
N. Prijatelj: Matematične strukture I, DMFA-založništvo, Ljubljana, 1996.

Improvement of knowledge of axiomatic set theory and acquaintance with the basics of ordinal and cardinal arithmetic.

Knowledge and understanding:
Understanding and application of axiomatic set theory and ordinal and cardinal arihtmetic.
Application:
Set theory is a fundamental branch of mathematics that provides the common language of mathematics. The Zorn lemma, ordinal and cardinal numbers are thus basic tools that find applications everywhere in mathematics. They are also interesting for philosophers.
Reflection:
Set theory provides a unifying approach to mathatics.
Transferable skills:
As no specific technical knowledge is necessary to follow the course, it is generally useful for development of mathematical technique and practice of mathematical thinking.

Lectures, exercises, homeworks, consultations

2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Andrej Bauer:
AWODEY, Steve, BAUER, Andrej. Propositions as [Types]. Journal of logic and computation, ISSN 0955-792X, 2004, vol. 14, no. 4, str. 447-471. [COBISS-SI-ID 13374809]
BAUER, Andrej, SIMPSON, Alex. Two constructive embedding-extension theorems with applications to continuity principles and to Banach-Mazur computability. Mathematical logic quarterly, ISSN 0942-5616, 2004, vol. 50, no. 4/5, str. 351-369. [COBISS-SI-ID 13378649]
BAUER, Andrej. A ralationship between equilogical spaces and Type Two Effectivity. Mathematical logic quarterly, ISSN 0942-5616, 2002, vol. 48, suppl. 1, str. 1-15. [COBISS-SI-ID 12033369]
Simpson Alexander Keith:
AWODEY, Steve, BUTZ, Carsten, SIMPSON, Alex, STREICHER, Thomas. Relating first-order set theories, toposes and categories of classes. Annals of pure and applied Logic. [Print ed.]. 2014, vol. 165, iss. 2, str. 428-502. [COBISS-SI-ID 17089881]
SIMPSON, Alex. Measure, randomness and sublocales. Annals of pure and applied Logic. [Print ed.]. 2012, vol. 163, iss. 11, str. 1642-1659. [COBISS-SI-ID 17091161]
SIMPSON, Alex, STREICHER, Thomas. Constructive toposes with countable sums as models of constructive set theory. Annals of pure and applied Logic. [Print ed.]. 2012, vol. 163, iss. 10, str. 1419-1436. [COBISS-SI-ID 17091417]