Department of Mathematics
Has Modern Mathematics Finally Understood The Infinite?
The Good News and the Bad News.
Paul Corazza
Boise State University
Prior to the beginning of the 20th century, there was an almost superstitious fear among mathematicians of the concept of the infinite. It was believed, for example that, although the natural numbers "go on forever", they cannot all be collected together into a single set. Such concerns were tied both to philosophical beliefs about the infinite and to paradoxes that were popular at the time.
The work of Georg Cantor showed that unless infinite sets were allowed in mathematics, it would not be possible to have a completely rigorous calculus -- even the concept of a "real number" depends on the concept of an infinite set.
Nearly single-handedly, Cantor developed the foundation for the modern theory of infinite sets, which eventually became today's axiomatic set theory, usually denoted ZFC.
ZFC was seen by its creators to be a kind of ultimate foundation for all of mathematics: ZFC not only provided a framework for Cantor's theory of infinite sets; it also provided a set of axioms from which all known theorems of mathematics could be derived.
Ironically, even as ZFC was being developed, research in other areas of mathematics was uncovering certain bizarre mathematical entities--now known as large cardinals--which would eventually be shown to lie outside the framework of ZFC.
In this talk, I'll describe what a large cardinal is, and why they are important in mathematics. I'll then describe an axiomatic framework that can provide the same kind of unifying foundation for "set theory + large cardinals" that ZFC has provided for the rest of mathematics.
All interested persons are welcome.
The talk will be accessible to upper class students.