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
There will be a midterm exam (date to be fixed in Week 1 in discussion with all attending students) and a Final Exam on 6 May 2026 at 9:00 hrs, Duration 2 1/2 hours. The midterm exam counts 30 points, homeworks count 20 points and the final exam counts 50 points; the overall amount of points for the course is 100.

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. Once the tutorial is done, the lecture continues on those days. The maximum number of homeworks needed is bounded by 10, the exact number will depend on the enrolement. The homework file exists in pdf and ps and will be continuously expanded.
The homeworks with the number k.h should be presented in Week k, for example homework 3.4 should be presented in Week 3. Everyone should do up to one homework per week, 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 ℵ-hierarchy of infinite cardinals is the cardinal of the real numbers (Cantor's Continuum problem). Gödel 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, König showed in 1905 that the reals cannot have the cardinal ℵω. Gödel showed that it is consistent with ZFC that the reals have the cardinal aleph1 and Cohen invented forcing to show that many other cardinals are also consistent to be the cardinal of the real numbers. A further question is how ZF (ZFC without axiom of choice) and ZFC related. 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. For example, it is equiconsistent whether the real numbers have cardinality ℵ1 or ℵ2 or ℵω+1 to name some options. Furthermore, ZF and ZFC (Zermelo Fraenkel set theory axioms without and with choice, respectively) are equiconsistent. 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. Whether ZFC itself is consistent is unknown; to prove this, one would require a proof system which is strictly stronger than ZFC and so one would make the problem only more difficult. However, no inconsistency of ZFC has been found so far during almost one hundred years of set theory research.