Agnes Scott College

Judith Roitman

The Uses of Set Theory
The Mathematical Intelligencer, Vol. 14, No. 1 (1992), 63-69.


In this article Roitman gives six examples from analysis, algebra, and algebraic topology whose results make use of modern set-theoretic techniques. These examples are

  1. The ideal of compact operators
    The purely analytic question "Is the ideal of compact operators on Hilbert space the sum of two properly smaller ideals?" is equivalent to purely set-theoretic combinatorics.
  2. A characterization of free groups
    The proof that "an Abelian group is free if and only if it has a discrete norm" exploits the use of model theory within set theory.
  3. The fundamental group
    The proof that "the fundamental group of a nice space is either finitely generated or has cardinality of the first uncountable cardinal" uses methods related to consistency results.
  4. The Hawaiian Earring
    Questions in strong homology theory are related to consistency results and the continuum hypothesis.
  5. A Banach space with few operators
    An example of a nonseparable Banach space where every linear operator is a scalar multiplication plus an operator with separable range is connected to set theory through infinite combinatorics on the first uncountable ordinal.
  6. The free left-distributive algebra on one generator
    Questions on free left-distributive algebras on one generator are connected to large cardinal theory.

In the conclusion Roitman writes:

I have presented a few theorems of mainstream mathematics that have been proved by set-theoretic techniques. In some cases we know that set theory is necessary; in other cases it has certainly proved convenient. The theorems presented are just a small percentage of such applications. One suspects that the existing applications are just a small fraction of the applications to be found in the near future. My thesis has been that set theory is an important tool of mathematics, whose use extends far outside the obvious.