# Couting to Infinity and Beyond

This is the title of a talk I gave to undergraduates at the CMU Math Club on Sep. 26, 2012.

## Abstract

How high can you count ? In elementary school, we are taught
that there are no good answer to that question. For every
natural number n, there is a larger number, n + 1. An answer one
often hears is "infinity". What would that mean, and would then
"infinity + 1" be a larger number ? What about "infinity +
infinity" ? In this talk, I will sketch the development of a
rigorous theory of "infinitary numbers", called ordinals, that
go beyond the natural numbers while keeping many of their
properties. I will also try to make sense of the notion of
"size" of an infinite set, and show that infinite sets can in a
very precise sense have different sizes. This will lead to very
interesting questions, such as Cantor's Continuum Hypothesis.

## Pointers

- A good place to learn about basic set theory is the 21-329 Set Theory course at CMU, which covers transfinite induction, ordinals, cardinals, and much more.