Spring 2003
Contents
Q: What is the axiom of choice?
Howard: Set theory is much like your high school geometry course in that you begin with some basic assumptions and then everything you prove has to be based on those axioms. Set theory works the same way – you have a few basic axioms and everything follows from the axioms. The axiom of choice is one of the more controversial axioms in set theory.
Q: Why is it controversial?
Howard: The axiom itself is obvious for finite sets and is generalized for infinite sets. It says that if you have a collection of non-empty sets, you can choose one thing out of each set. It’s controversial because it has some funny consequences.
Q: Can you give me an example of a funny consequence?
Howard: Well, there’s the Banach Tarski Paradox that resulted from the axiom of choice. They proved that you can take a solid ball with a 2-inch radius, divide it into a finite number of pieces and reassemble the pieces to get two balls with the same size as the original.
Q: Hmmmmm...
Howard: The pieces of the ball, when broken apart, are so irregular that they don’t have a volume so when you reassemble them you arrive at this strange conclusion and paradox.
Q: Is the axiom of choice very useful if you can arrive at such strange conclusions?
Howard: Actually it’s used in a lot of places – in all branches of math – from linear algebra to analysis. People have looked for an alternative to the axiom – maybe something a little weaker – that would not have the unpleasant consequences of the axiom of choice.
Howard and Jean E. Rubin co-authored Consequences of the Axiom of Choice, published by the American Mathematical Society in 1998. The book is a survey of the research done during the last 100 years on the axiom of choice and its consequences.
Q & A with Paul Howard
Dr. Paul Howard has worn out four bicycles. He has ridden his bike to campus every day for the last 33 years – a six mile ride from his home in the snow of winter and the heat of summer. But what most students will tell you about Dr. Howard is that he is one of their favorite teachers.
Dr. Howard joined EMU in 1970. He has done most of his research investigating implications of the axiom of choice. It was in graduate school where he was first intrigued by the axiom.