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:

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• tex

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.