Barry Garelick writes Confessions of a Math Major.

The amazement I felt at the age of seven when realizing that counting to one hundred twice is the same as counting to two hundred once was no less than when as a sophomore in college I discovered I could prove that the set of non-intersecting circles in a plane is countable.

He wanted to be a writer, became an environmental policy analyst and is now ready to start his next career as a high school math teacher.

Welcome to the club of 2nd career teachers Barry. Hey if there really is a club, can someone tell me how to join it?

Huh? The set of non-intersecting circles in a plane is not countable. Consider merely the circles centered at a point A; for every positive real number there is a unique circle with that radius centered at A, and these circles don’t intersect. (Maybe he meant the set of circles with a given radius r.)

He might have meant circles which don’t have any common area.

No, he means “countable”, as in has a one-to-one relationship with the set of integers.

No, I’m not a mathematician. I just get a few of the concepts, and that’s one of them.

(Garelick): “…the set of non-intersecting circles in a plane is countable.”

(ed): “The set of non-intersecting circles in a plane is not countable. Consider merely the circles centered at a point A; for every positive real number there is a unique circle with that radius centered at A, and these circles don’t intersect.”

(Engineer): “No, he means ‘countable’, as in has a one-to-one relationship with the set of integers. No, I’m not a mathematician. I just get a few of the concepts, and that’s one of them.”

Ed’s disproof-by-contradiction is correst. The set of points on the interval [(p,q),(r,s)] is not countable. Therefore, the set of circles with center (p,q) and radius less than [(p-r)^2+(q-s)^2]^1/2 is not countable.

The set of points on an interval is not countable; that is correct. But we are not counting points in an interval, we are counting intervals–or in this case, circles.

Consider intervals on a line. In the infinite set of intervals on a line, there is at least one rational number contained within each interval. (This is because the rational numbers are “dense” in the reals.)

Pick one rational number from each interval Call the set of all such rational numbers “A”. “A” is a subset of the set of all rational numbers. The set of rational numbers is itself countable and since a subset of a countable set is countable, “A” is countable. This principle can be extended to the infinite set of non-intersecting circles in the plane, since the set of all ordered pairs of rational numbers is countable.

This problem is contained in a book called “Stories about Sets” (http://www.amazon.com/Stories-About-Sets-Ya-Vilenkin/dp/012721951X). This problem is frequently given in undergraduate math courses in real analysis. There was even a recent discussion of this problem in Yahoo Answers: http://answers.yahoo.com/question/index?qid=20081126044702AA5Iwyq.

I appreciate all the interest in this proof.

I’m missing something, I guess. Why are you comparing the order of the set of all circles with center (p, q) passing through the interval [(p,q), (s,t)] to the order of the set of ordered pairs of rationals? That second set is countable, but why the correspondence?