Timeline

  • First check-in, choose topic by 3/15

  • Second check-in by 4/15

  • Draft of paper due 4/29

  • Presentations 5/6 and 5/8

  • Final paper submission 5/13

Grading Requirements

There are 3 possible components for the project: a paper, a presentation, and a poster. To get a high pass you must do 2 of the 3. To get a pass you must do 1 of the 3. For all components, what you are aiming to do is to present a proof of a theorem. Give enough background + definitions for your classmates to follow, then state the theorem, then give the proof.

  • The paper should be at least 2 full pages (modulo header space on the first page). You should give full proofs, as in the standard for the portfolio, and cite your sources.

  • The presentations will each be allotted 25 minutes. You can use slides or chalk as you wish. I would recommend chalk, as slides are timely to make. For the presentation, you don’t need to give every detail of the proof. It’s better to give the big ideas, and leave the details to the paper.

  • For the poster remember that your audience is mostly people from outside this class. As such, you can’t assume they know concepts from this class. So spend more space on background and definitions. And rather than try to fit in proofs, it’s better to highlight key concepts. You can give more detail to anyone who’s interested.

Project Ideas

Here’s some suggested topics for the projects, in broad categories. For your project you will have a more narrow focus. (This is because these topics are simply too large to be a single project.) A model project is to take a single theorem, explain the definitions behind it, and give a proof.

  • Ordinal notations. How do we describe and write down names for larger and larger countable ordinals?

  • Applications of the continuum hypothesis. There’s a number of interesting theorems you can prove by assuming CH, especially about building funky objects on the reals.

  • Cardinal characteristics of the continuum. There are cardinal numbers, such as the bounding number or the dominating number, which meausure something about the continuum. And you can prove one is smaller/larger than another, even if akin to CH their exact value is not provable.

  • Infinite games. Set theorists study games with infinitely many innings—whether variants of real games like infinite chess or purely mathematical games.

  • Ultrafilters. These are abstract measures of largeness used in mathematics and other areas. Possible connections include Arrow’s impossibility theorem and Ramsey’s theorem.

  • Constructions of $\mathbb R$. In class we’ll see how to construct a copy of $\mathbb N$ using just sets. You can also construct a copy of $\mathbb R$ using sets. The two most popular options are Dedekind cuts and Cauchy sequences.

  • Independence of axioms. In class we’ll talk a little about what it means for one axiom to be independent from others. You might look at some set-theoretic axioms, or you might look in a different area of maths.

  • The axiom of choice. There’s a hierarchy of “choice principles” which are weaker than the full axiom of choice, or you could focus on an equivalent form of the axiom.