Monthly Archives: July 2014

Real Numbers and Infinite Games, Part II

In my last post, I wrote about two infinite games whose analysis leads to interesting questions about subsets of the real numbers.  In this post, I will talk about two more infinite games, one related to the Baire Category Theorem and one to Diophantine approximation.  I’ll then talk about the role which such Diophantine approximation questions play in the theory of dynamical systems.

The Choquet game and the Baire Category Theorem

The Cantor game from Part I of this post can be used to prove that every perfect subset of {\mathbf R} is uncountable.  There is a similar game which can be used to prove the Baire Category Theorem.  Let X be a metric space.   In Choquet’s game, Alice moves first by choosing a non-empty open set U_1 in X.  Then Bob moves by choosing a non-empty open set V_1 \subseteq U_1.  Alice then chooses a non-empty open set U_2 \subseteq V_1, and so on, yielding two decreasing sequences U_n and V_n of non-empty open sets with U_n \supseteq V_n \supseteq U_{n+1} for all n.  Note that \bigcap U_n = \bigcap V_n; we denote this set by U.  Alice wins if U is empty, and Bob wins if U is non-empty. 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 →