Last two blogs, I introduced the axiom of choice. I will now try to present some history behind the axiom.

The story begins in 1870, when Cantor first invented transfinite numbers and the rest of set theory. This set theory makes use of the concept of well-ordered set. A set is called well-ordered if it is equiped with a natural order < such that every subset has a minimum. The standard example of a well-ordered set are the natural numbers. It is intuitively obvious that there...

Last time I gave an informal introduction to the axiom of choice by telling a little story. The idea of this blog is to be a bit more rigorous. But, don't worry, we won't be too technical.

First, I want to introduce the formal statement of the axiom of choice. This may look a bit awkward, but it's actually very easy. first we let [tex]\{S_i~\vert~i\in I\}[/tex] be a collection of non-empty sets. If I want to make an analog to my story of last blog, then I would say that each S_i represents...

To people who don't know, I've always been extremely interested in the axiom of choice and it's various consequences. So I would like to post a series on the axiom of choice and the problems that arise from it.

Let me begin by telling you all a little story about an Indian emperor. I don't recall where I heard this story for the first time, but I considered it a nice introduction to the axiom of choice:

Long ago, there lived an Indian emperor who ruled over many kingdoms....