Spooky inference and the axiom of choice

A large crowd had gathered in Harvard Square, and I was curious what all the cheering and gasping was about.  HarvardSquareWorking my way through the crowd, I saw a street performer who (according to the handwritten red letters on his cardboard sign) went by the name “Zorn the Magnificent”.  He displayed a large orange, borrowed an extremely sharp knife from his assistant, and proceeded to chop the orange into five exotic-looking pieces while standing on one hand.  Working with almost unfathomably deft precision, he rearranged the pieces into two oranges, each the same size as the original one.  The oranges were given out for inspection and the crowd cheered wildly.  I clapped as well — even though I was familiar with the old Banach-Tarski paradox — since it was nevertheless an impressive display of skill and I had never seen it done one-handed before.  I heard a man with a long white beard whisper to the woman next to him “He hides it well, but I know that he’s secretly using the Axiom of Choice.” Continue reading

Real Numbers and Infinite Games, Part I

Georg Cantor

Georg Cantor

In this post I’d like to illustrate how one can use infinite games to prove theorems about the real numbers.  I’ll begin with a game-theoretic proof that the set of real numbers is uncountable, following the exposition in this paper of mine.  This will lead us somewhat unexpectedly into the realm of descriptive set theory, where we will discuss how games are used in cutting-edge explorations of the Axiom of Choice, the Continuum Hypothesis, and the foundations of second-order arithmetic.   In a sequel post I will discuss how infinite games can be used to study Diophantine approximation, with applications to complex dynamics.

Countable versus uncountable infinities

When my daughter was 5 years old, she asked me if there is just one infinity.  I proudly kissed her on the forehead and told her what an excellent question that was.  I told her no, infinity comes in many different flavors.  I pretty much left it at that, but since she’s 10 now, here are some more details for her.  (The reader familiar with the basics of set theory can move on to the next section.) Continue reading