Logic and Foundations of Mathematics II (MA5220)

Lecturer
The lecturer is Frank Stephan from the Departments of Mathematics and Computer Science of the National University of Singapore.

Frank Stephan's addresses are:

  (1) Department of Mathematics, National University of Singapore
      10 Lower Kent Ridge Road, Block S17, Singapore 119076
      Primary Office: S17#07-04

  (2) School of Computing, National University of Singpore
      Computing Drive, Computing 1 (COM1), Singapore 117590
      Secondary Office: COM1#03-11

When logged into and working at a computer, Frank Stephan is best reachable
under telephone +65 65164246.

The email address is fstephan@comp.nus.edu.sg

Textbook
The textbook is the book of Kenneth Kunen, "Set Theory", Studies in Logic 34, College Publications 2011.

Time and Place
Tuesday and Thursday from 12:00 to 14:00 hrs
The room is S16#04-36 in the building next to the Science canteen.

Assessment
Assessment information will be placed late December 2025 or early January 2025.

Lecture and Tutorial
There will be a tutorial in the first part of the lecture on Thursday each week from Week 2 onwards. Lectures will always be 3 hours per week unless in weeks where Tuesday or Thursday is a public holiday; in those weeks there is one hour tutorial followed by one hour lecture. The homework file will be added and continuously expanded later. The homeworks with the number k.h should be presented in Week k, for example homework 3.4 should be presented in Week 3. The file will during the lecture be continuously maintained and updated. Everyone should do per week one homework, write it up in the Discussion Forum and present it in the tutorial. Each student should choose an own homework and not do the same homework as a classmate; reserve a homework by putting a post with the homework number into the Discussion Forum and then work on the homework by editing this post until it is good; present the homework in the tutorial on black board or with slides in the week when it is due (or, in exceptional cases, also in a later week).
All homeworks should be the own work and using ChatGPT and other AI tools is not allowed. It is, however, allowed to write on a page of paper or produce with a text-programme a word file and to upload a scan or pdf-file of the homework.

Content
The course will give an overview of set theory including background material (axioms, ordinals, cardinal arithmetic, models of set theory) and Consistency proofs, infinitary combinatorics and Forcing. In set theory, there are many interesting questions like that which level in the aleph-hierarchy of infinite cardinals is the cardinal of the real numbers (Cantor's Continuum problem). Goedel and Cohen showed that this question cannot be answered from the axioms ZFC and that many answers are possible. Thus set theorists consider a Zoo of possible models of set theory and investigate what connections are still provable and which are not. For example, Koenig showed in 1905 that the reals cannot have the cardinal aleph_{omega}. Goedel showed that it is consistent with ZFC that the reals have the cardinal aleph_1 and Cohen invented forcing to show that many other cardinals are also consistent to be the cardinal of the real numbers. Subsequent research therefore often showed that various versions of set theory are equiconsistent: If there is a model of set theory of version A then there is also a model of set theory with option B and vice versa. This course aims at giving an introduction to the most famous results in set theory (including those mentioned in this abstract and beyond) and provides the basic proof techniques used at many equiconsistency proofs.