Department of Mathematics
Cantor's Continuum Problem
Justin Moore
Boise State University
In the early stages of the study of sizes of infinite sets, Cantor showed that the set of reals was uncountable. Thus he showed that there are two infinite sets of real numbers which have a different "sizes" --- the set of all natural numbers and the set of all the real numbers. He asked whether it was possible to find a third set which, from the point of view of cardinality, lay strictly between these two sizes. It is now known that this problem cannot be decided within the framework of the usual axioms of mathematics. The purpose of this talk is to give an introduction to Cantor's Continuum Problem, its resolution and the modern research which relates to it. First I will present a probabilistic interpretation of Cohen's method of forcing and how he used it to solve the Continuum Problem. Next I will discuss Solovay's results on the properties (including cardinality) of definable sets of reals and contrast this with Cohen's work. Finally (time permitting) I will mention some of the work aimed at gaining a better understanding of the relationship between the size of the set of all reals and infinitary combinatorics (such as the study of uncountable graphs).
All interested persons are welcome.
The talk should be accessible to those taking upper division
math classes. Strong students in Math 187 should be able to come away with
something from the talk. This talk will, in part, be an advertisement for
the course in Set Theory and Forcing being offered in the spring (MAT
497--003/597--004). The prerequisite is successful completion of MAT314
or comparable course.