Large Cardinals Part III of the Mathematical Tripos Lent Term 2022

Lecturer. Benedikt Löwe

Lectures. Monday, Wednesday 11-12.

Examples Classes. Ioannis Eleftheriadis.

The following definitions and facts should be familiar from an introductory course on set theory: a cardinal \(\kappa\) is called regular if every unbounded subset of \(\kappa\) has cardinality \(\kappa\); successor cardinals, i.e., cardinal s of the form \(\aleph_{\alpha+1}\), are always regular; the usual limit cardinals, e.g., \(\aleph_\omega\), \(\aleph_{\omega+ \omega}\), or \(\aleph_{\omega_1}\), are not.

Thus, the following is a natural question: "Are there any uncountable regular limit cardinals?''. If they exist, they must be very large, in particular, much larger than any of the mentioned limit cardinals. It turns out that this question is intricately connected with the incompleteness phenomenon in set theory: if there is an uncountable regular limit cardinal, then there is a model of \(\mathsf{ZFC}\); therefore, \(\mathsf{ZFC}\) is consistent, and hence (by Gödel's Second Incompleteness Theorem) \(\mathsf{ZFC}\) cannot prove the existence of these cardinals (unless, of course, it is inconsistent). Regular limit cardinals (a.k.a. weakly inaccessible cardinals) are the smallest examples of set-theoretic notions called large cardinals: cardinals with properties that imply that they must be very big and whose existence cannot be proved in \(\mathsf{ZFC}\). In this course, we shall get to know a number of these large cardinals, study their behaviour, observe consequences of their existence for set theory, and develop techniques to determine the logical strength of large cardinals (the so-called consistency strength hierarchy). In modern set theory, large cardinals are used as the standard way to calibrate logical strength of extensions of \(\mathsf{ZFC}\).

Literature.

1. Thomas Jech, Set Theory, The Third Millenium Edition, revised and expanded, Springer 2003 [Springer Monographs in Mathematics].
2. Akihiro Kanamori, The Higher Infinite. Large Cardinals in Set Theory from Their Beginnings. Springer 2003 [Springer Monographs in Mathematics]

Teaching mode. The teaching mode of this course will be synchronous online teaching with additional in-person office hours: all lectures will be offered synchronously via Zoom. Room MR 14 is reserved for the course during the regular lecture hours (Monday & Wednesday 11–12) and students can sit there and watch the lecture on their computers with head-phones on. This moodle page will contain the Zoom link for the lectures (in the section Lectures). Lectures will also be recorded and posted on Panopto within 24 hours of the lecture.