# Ultraproducts

A course to be given, summer 2012, at the Nesin Matematik Köyü.

Notes that I wrote in preparing for the course are available: they are subject to revision, and they are not to be considered as a textbook for the course:

## Catalogue information

The following is taken from the 2012 summer school webpage.

- Title of the course:
- Ultraproducts and their consequences
- Instructor:
- Assoc. Prof. David Pierce
- Institution:
- Mimar Sinan GSÜ
- Dates:
- 13–19 August 2012
- Prerequisites:
- Some knowledge of algebra, including the theorem that a quotient of a ring by an ideal is a field if and only if the ideal is maximal.
- Level:
- Advanced undergraduate and graduate
- Abstract:
- An ultraproduct is a kind of average of infinitely many structures.
The construction is usually traced to a 1955 paper of Jerzy Los;
however, the idea of an ultraproduct can be found in Kurt Goedel's 1930
proof (from his doctoral dissertation) of the Completeness Theorem for
first-order logic. Non-standard analysis, developed in the 1960s by
Abraham Robinson, can be seen as taking place in an ultraproduct of the
ordered field of real numbers: more precisely, in an ultrapower. Indeed,
for each integer, the ‘average’ real number is greater than that
integer; therefore an ultrapower of the ordered field of real numbers is
an ordered field with infinite elements and therefore infinitesimal
elements. Perhaps the first textbook of model theory is Bell and
Slomson's
*Models and Ultraproducts*of 1969: the title suggests the usefulness of ultraproducts in the development various model-theoretic ideas. Our course will investigate ultraproducts, starting from one of the simplest interesting examples: the quotient of the cartesian product of an infinite collection of fields by a maximal ideal that has nontrivial projection onto each coordinate. No particular knowledge of logic is assumed.