# Cardinal arithmetic

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M2
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

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.

Objectives and competences

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

Intended learning outcomes

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.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

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

Lecturer's references

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]