Mathematics 580: Topics in Set Theory: Combinatorial Set Theory.

Section 1.
Instructor: Andres Caicedo. Time: MWF 3:40-4:30 pm. Place: Education building, Room 330. Office Hours: By appointment. See this page for details.

We will cover diverse topics in combinatorial set theory, depending on time and the interests of the audience, including partition calculus (a generalization of Ramsey theory), cardinal arithmetic, and infinite trees. Time permitting, we can also cover large cardinals, determinacy and infinite games, or cardinal invariants (the study of sizes of sets of reals), among others. I’m open to suggestions for topics.

Recommended background: Knowledge of cardinals and ordinals. A basic course on set theory (like 502: Logic and Set Theory) would be ideal but is not required.

Textbook: There is no official textbook. The following suggested references may be useful, but are not required:

Set theory. By T. Jech. Springer (2006), ISBN-10: 3540440852 ISBN-13: 978-3540440857

Set theory. An introduction to independence proofs. By K. Kunen. North Holland (1983), ISBN-10: 0444868399 ISBN-13: 978-0444868398

Set theory for the working mathematician. By K. Ciesielski. Cambridge U. Press (1997), ISBN-10: 0521594650 ISBN-13: 978-0521594653

Set theory. By A. Hajnal and P. Hamburger. Cambridge U. Press (1999), ISBN-10: 052159667X ISBN-13: 978-0521596671

Discovering modern set theory. By W. Just and M. Weese. Vol I. AMS (1995), ISBN-10: 0821802666 ISBN-13: 978-0821802663. Vol II. AMS (1997), ISBN-10: 0821805282 ISBN-13: 978-0821805282

Problems and theorems in classical set theory. By P. Komjath and V. Totik. Springer (2006), ISBN-10: 038730293X ISBN-13: 978-0387302935

Notes on set theory. By Y. Moschovakis. Springer (2005), ISBN-10: 038728723X ISBN-13: 978-0387287232

Grading: Based on homework.

I will use this website to post any additional information, and encourage you to use the comments feature. If you leave a comment, please use your full name, which will simplify my life filtering spam out.

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Monday, January 12th, 2009 at 12:30 am and is filed under 580: Topics in set theory. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

I would like to cover some stuff on large cardinals (since questions concerning partition calculus and trees become questions concerning large cardinals so naturally) and infinite games/AD (since I really know so little about determinacy).

I thought about this question a while ago, while teaching a topics course. Since one can easily check that $${}|{\mathbb R}|=|{\mathcal P}({\mathbb N})|$$ by a direct construction that does not involve diagonalization, the question can be restated as: Is there a proof of Cantor's theorem that ${}|X|

First of all, note (as Monroe does in his question) that if $\mathbb P,\mathbb Q$ are ccc, then $\mathbb P\times\mathbb Q$ is $\mathfrak c^+$-cc, as an immediate consequence of the Erdős-Rado theorem $(2^{\aleph_0})^+\to(\aleph_1)^2_2$. (This is to say, if $\mathbb P$ and $\mathbb Q$ do not admit uncountable antichains, then any antichain in their product ha […]

The technique of almost disjoint forcing was introduced in MR0289291 (44 #6482). Jensen, R. B.; Solovay, R. M. Some applications of almost disjoint sets. In Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968), pp. 84–104, North-Holland, Amsterdam, 1970. Fix an almost disjoint family $X=(x_\alpha:\alpha

At the moment most of those decisions come from me, at least for computer science papers (those with a 68 class as primary). The practice of having proceedings and final versions of papers is not exclusive to computer science, but this is where it is most common. I've found more often than not that the journal version is significantly different from the […]

The answer is no in general. For instance, by what is essentially an argument of Sierpiński, if $(X,\Sigma,\nu)$ is a $\sigma$-finite continuous measure space, then no non-null subset of $X$ admits a $\nu\times\nu$-measurable well-ordering. The proof is almost verbatim the one here. It is consistent (assuming large cardinals) that there is an extension of Le […]

A notion now considered standard of primitive recursive set function is introduced in MR0281602 (43 #7317). Jensen, Ronald B.; Karp, Carol. Primitive recursive set functions. In 1971 Axiomatic Set Thoory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) pp. 143–176 Amer. Math. Soc., Providence, R.I. The concept is use […]

The power of a set is its cardinality. (As opposed to its power set, which is something else.) As you noticed in the comments, Kurepa trees are supposed to have countable levels, although just saying that a tree has size and height $\omega_1$ is not enough to conclude this, so the definition you quoted is incomplete as stated. Usually the convention is that […]

The key problem in the absence of the axiom of replacement is that there may be well-ordered sets $S$ that are too large in the sense that they are longer than any ordinal. In that case, the collection of ordinals isomorphic to an initial segment of $S$ would be the class of all ordinals, which is not a set. For example, with $\omega$ denoting as usual the f […]

R. Solovay proved that the provably $\mathbf\Delta^1_2$ sets are Lebesgue measurable (and have the property of Baire). A set $A$ is provably $\mathbf\Delta^1_2$ iff there is a real $a$, a $\Sigma^1_2$ formula $\phi(x,y)$ and a $\Pi^1_2$ formula $\psi(x,y)$ such that $A=\{t\mid \phi(t,a)\}=\{t\mid\psi(t,a)\}$, and $\mathsf{ZFC}$ proves that $\phi$ and $\psi$ […]

YEA!

My interests in loose order of preference would be:

1) infinite trees

2) partition calculus

3) determinacy and infinite games

billy

I would like to cover some stuff on large cardinals (since questions concerning partition calculus and trees become questions concerning large cardinals so naturally) and infinite games/AD (since I really know so little about determinacy).