## BOISE EXTRAVAGANZA IN SET THEORY (BEST) – Announcement 1

December 21, 2009

The 19-th annual meeting of BEST will be hosted at Boise State University during the weekend of March 27 (Saturday) – March 29 (Monday), 2010.

It is organized by Liljana Babinkostova, Andrés E. Caicedo, Masaru Kada, and Marion Scheepers (scientific committee), and Billy Hudson (social committee).

Contributed and invited talks will be held on Saturday, Sunday and Monday at the Department of Mathematics, Boise State University. The invited speakers are:

The conference webpage is available here. Anyone interested in participating should contact the organizers as soon as possible by sending an email to best@math.boisestate.edu

There are three important deadlines regarding the conference:

• Lodging: The Hampton Inn & Suites is providing rooms at a reduced rate for BEST participants. To take advantage of the reduced rate, reservations must be made online by MARCH 12. After March 12 rooms will be available at prevailing rates.
• Financial support: Limited financial support is available to partially offset lodging expenses of up to eight participants, and to partially offset lodging plus airfare expenses of up to two participants. Please see the conference website for details on applying for support. The deadline for applying for financial support is MARCH 3.
• Abstracts: Atlas Conferences, Inc. is providing abstract services for BEST 19. The deadline for submitting an abstract for invited or contributed talk is MARCH 25. Abstracts can be submitted here, and viewed here.
The conference is supported by a grant from the National Science Foundation. Abstract services are provided by Atlas Conferences, Inc. Reduced lodging
rate is provided by The Hampton Inn & Suites. Support from these entities is gratefully acknowledged.

## 502 – The Banach-Tarski paradox

December 17, 2009

1. Non-measurable sets

In these notes I want to present a proof of the Banach-Tarski paradox, a consequence of the axiom of choice that shows us that a naive understanding of the concept of volume can lead to contradictions. A good reference for this topic is the very nice book The Banach-Tarski paradox by Stan Wagon.

## 175 – Final exam

December 15, 2009

Here is the final exam, and here are the solutions.

## 502 – Exponentiation

December 9, 2009

This is the last homework assignment of the term: Assume ${\sf CH}.$ Evaluate the cardinal number $\aleph_3^{\aleph_0},$ the size of the set of all  functions $f:\omega\to\omega_3.$

## 502 – The constructible universe

December 9, 2009

In this set of notes I want to sketch Gödel’s proof that ${{\sf CH}}$ is consistent with the other axioms of set theory. Gödel’s argument goes well beyond this result; his identification of the class ${L}$ of constructible sets eventually led to the development of inner model theory, one of the main areas of active research within set theory nowadays.

A good additional reference for the material in these notes is Constructibility by Keith Devlin.

1. Definability

The idea behind the constructible universe is to only allow those sets that one must necessarily include. In effect, we are trying to find the smallest possible transitive class model of set theory.

${L}$ is defined as

$\displaystyle L=\bigcup_{\alpha\in{\sf ORD}} L_\alpha,$

where ${L_0=\emptyset,}$ ${L_\lambda=\bigcup_{\alpha<\lambda}L_\alpha}$ for ${\lambda}$ limit, and ${L_{\alpha+1}={\rm D{}ef}(L_\alpha),}$ where

$\displaystyle \begin{array}{rcl} {\rm D{}ef}(X)=\{a\subseteq X&\mid&\exists \varphi\,\exists\vec b\in X\\ && a=\{c\in X\mid(X,\in)\models\varphi(\vec b,c)\}\}. \end{array}$

The first question that comes to mind is whether this definition even makes sense. In order to formalize this, we need to begin by coding a bit of logic inside set theory. The recursive constructions that we did at the beginning of the term now prove useful.

December 5, 2009

Here is quiz 8.

## 598 – Upcoming talk: Laurie Cavey

December 4, 2009

Laurie Cavey, Wed. December 9, 2:40-3:30 pm, MG 120.

Developing Students’ Understanding of Mathematical Definitions: Why Bother?

Definitions are a fundamental part of doing mathematics, yet studies indicate that many students struggle to learn and apply definitions. In fact, many instructors wonder (myself included) how students can misapply definitions that are so clearly stated. Part of the issue is that a student’s previous mathematical experiences influence how she thinks, even when encountering a new idea that is seemingly unrelated. Not knowing what these experiences might entail, it can be difficult to know how to help students develop a better understanding of a particular definition. So, why bother? I will provide a brief overview of the research in this area including an instructional strategy (student generated examples) that may influence the way we think about developing students’ understanding of definitions.