Students are required to write a ten to twelve page term paper, in LaTeX, on a topic of their choice. Below is a list of suggested topics, though you may also choose your own, subject to instructor approval. In order to select a subject that you will find interesting, I suggest browsing through some of the books listed below that have been put on reserve in the library, or some of the material available online (see links below).

The purpose of this project is to learn something about a topic related to the material in the course, and to write a paper explaining some of what you have learned. You are not expected to do any original research. Rather, you will need to organize and synthesize your material and present it in a way that good undergraduate student like yourself would be able to read and understand it, and learn something new and interesting.

This is harder than it sounds, especially if you have never written a math paper before. Make sure to start early!

As an added incentive, there is the possibility, for those who produce a particularly interesting and well-written paper, of submitting your paper, after further revision, to MIT's Undergraduate Journal of Mathematics.
For more information, copies of the journal are in Hayden Library - Science Journals | QA.M679.

Some important dates

• Thursday 3/22: Choose topic, together with one or two main sources, which may be selected from the list of references below.
  Submit in class, or by email.
• Thursday 4/5: One page outline due, in LaTeX.
• Thursday 4/26: Complete draft due.
• Thursday 5/17: Final paper due.

LaTeX

You are required to write your paper using LaTeX, which is the standard method for writing mathematical text. LaTeX is not difficult to use, but it can take a little while to get used to. Here are a few links that should be helpful for both the beginner and the expert.

• Short Math Guide to LaTeX
• The Not So Short Introduction
• CTAN, the Comprehensive TeX Archive Network
• LaTeX on wikipedia, an article with many more links.

Possible topics

(Following each topic is a brief description, together with a few relevant references.)
Topics marked with a (!) are somewhat more advanced.

Axioms and axiom systems

• ZFC (Zermel-Fraenkel set theory with choice) This is the standard axiomatization of set theory, expressed in first-order logic. One possible project would be to give an introduction to first-order logic, explain the axioms of ZFC, and discuss in some detail the notion of a model of set theory. (many different references)
• NBG (von Neumann-Bernays-Godel set theory) This is an alternative axiom system for set theory. It is essentially equivalent to standard ZFC, but allows for the existence of classes in a more explicit manner. (e.g., Mendelson)
• The Axiom of Choice There is much to say about the axiom of choice. For example, there are many related, though inequivalent axioms, which can be considered. Also, applications of the axiom to other branches of mathematics. (e.g., Levy, Hrbacek and Jech, many others)
• Non well-founded set theory In non-well founded set theory, a set may be a member of itself. This is an alternative to the standard axiomatization of set theory, with applications to computer science and other subjects. (e.g., Devlin, Aczel)

New axioms, consistency, independence

• (!)Independence and relative consistency Examine what it means for an axiom to be independent of the axioms of set theory. How does one show this? Examples of such axioms. Possible arguments for adopting new axioms. (e.g., Hrbacek and Jech, Hajnal and Hamburger, Kunen. Maddy contains a more philosophical discussion of the issues involved.)
• (!)Godel's axiom V=L Godel's axiom of constructability, usually written V = L, is an axiom that can be added to the standard axioms of set theory, which makes it possible to resolve many problems, including the generalized continuum hypothesis. Thus the axiom is quite interesting, despite the fact that 'most' set theorists 'believe' it is in some sense false. (e.g., Kunen, Jech, Devlin)
• (!!)Forcing Cohen invented forcing in the 1960's in order to construct a model of set theory in which the continuum hypothesis does not hold. Since then it has become an incredibly powerful tool in set theory to prove consistency results involving new axioms. This should be an interesting, though rather difficult topic. (Kunen contains what is probably the standard introduction to forcing, though there are many others.)

Combinatorial set theory

• Partition properties One version of the finite Ramsey's theorem says that for any number n, there is another number N such that in any group of N people, there is a set of n people that either all know each other, or all do not know each other. The aim of this project would be to look at generalizations involving infinite sets. (e.g., Levy, Hrbacek and Jech, Hajnal and Hamburger)
• Infinite trees A tree is a particular kind of well-ordered set that looks like, well, a tree. They come up in lots of different contexts in set theory, and have many interesting properties. (e.g., Levy)

Descriptive set theory

• Topological background Metric spaces, Polish spaces, Baire category theorem (e.g., Srivastava, Levy, Marker)
• Borel sets Borel sets generalize the open and closed intervals of R, and have many interesting properties. (e.g., Srivastava, Marker, Jech)

Other

• Ordinals and/or cardinals There is much more to say about ordinals and cardinals than we will cover in class. (e.g., Levy, Jech, many others)
• Boolean algebras A Boolean algebra is a kind of algebraic object that comes up in different areas of set theory. The typical example is the power set of a given set, on which one has the operations of intersection and union. (e.g., Cori and Lascar, Levy)
• (!)The Banach-Tarski paradox This result states that, using the axiom of choice, a sphere in 3-dimensions may be cut into finitely many pieces, which may then reassembled into two spheres with the same radius as the original one. Requires some knowledge of group theory. (e.g., Wagon)

The following references have been put on reserve in the library. Details about some of these books, including tables of contents, can be found on amazon. One can also find reviews of these books online at the AMS's MathSciNet, which is an invaluable resource. (This is a subscription service that will work from the MIT network.)

1. A. Levy, Basic Set Theory
A very nicely written introduction to the subject. Somewhat more sophisticated than an undergraduate textbook, but still very accessible. (And only $19 on amazon.com)
2. K. Devlin, The Joy of Sets: Fundamentals of Contemporary Set Theory
Undergraduate text, which includes a discussion of non well-founded set theory.
3. K. Hrbacek and T. Jech, Introduction to Set Theory
Undergraduate text containing some material not in Moschovakis.
4. A. Hajnal and P. Hamburger, Set Theory
5. R. Cori and D. Lascar, Mathematical Logic: A course with exercises. Parts I and II
Contains a chapter on boolean algebras, as well as chapters on first-order logic and ZFC.
6. E. Mendelson, Introduction to Mathematical Logic
Includes treatment of NBG set theory.
7. S. Wagon, The Banach-Tarski Paradox
8. K. Kunen, Set Theory: An Introduction to Independence Proofs
The standard graduate level introduction to the subject.
9. T. Jech, Set Theory
An encyclopedic reference.
10. S. Srivastava, A Course on Borel Sets
A very nice, accessible introduction to the subject.

• The Continuum Hypothesis, Part I, by W. Hugh Woodin. There is also a Part II. A survey article in the Notices of the AMS by one of the world's top set theorists. He argues that the continuum hypothesis, though independent of the axioms of set theory, may indeed be solvable.
• How Godel Transformed Set Theory, by Juliet Floyd and Akihiro Kanamori. Another, more historical article in the Notices of the AMS on Godel's famous theorem that it is consistent with the axioms of set theory that the continuum hypothesis is true.
  Later, in 1963, Paul Cohen proved that it is also consistent with the axioms of set theory that the continuum hypothesis is false. (For this work, he was awarded the Field's Medal, the so-called Nobel Prize for mathematics.)
• The introduction to Kanamori's book, The Higher Infinite, about large cardinals. A very accessible and interesting discussion of some of the major themes and currents in set theory.
• Course homepage and online lecture notes on descriptive set theory, by David Marker.
• Peter Aczel's book on non well-founded set theory is online here.
• Penelope Maddy's "Believing the Axioms, Parts I and II". Available online at JSTOR, through MIT. A philosophical and mathematical discussion about new axioms for set theory.

Last updated March 18 2007