The Axiom of Determinacy: Game Theory in Axiomatic Set Theory
In the early 20th century, several mathematical results were published that were considered so counterintuitive that some mathematicians rejected the possibility for these results to be accurate, including the ability to divide a sphere into 5 pieces and then, through translations and rotations of those pieces, create 2 spheres identical to the original (this is known as the Banach-Tarski paradox). These pieces cannot be explicitly constructed, only nonconstructively shown to exist. These mathematicians were looking for the root cause of these paradoxes and what they found became known as the Axiom of Choice (AC). This axiom essentially states we can make an arbitrary number of arbitrary choices; more formally, this is given any family (this really just means a set) of nonempty sets, there is a function on that family taking each set to one of its elements. Most modern mathematicians now accept AC, but for a good portion of the 20th century, a serious concern when evaluating the validity of a theorem was whether or not AC was used. Several alternatives to AC were proposed that seemed to resolve such paradoxes, including one from an area we have studied in networks: game theory.
The specific alternative to AC introduced here is the Axiom of Determinacy (AD). AD describes the results of a specific type of game played with two players and sequences of natural numbers. We first take a set A to be our favorite set of sequences of natural numbers. We have two players Alice and Bob who alternate choosing natural numbers. Alice chooses the first, Bob chooses the second, Alice chooses the third and so on, constructing an infinite sequence of natural numbers. Alice wins if the sequence they construct is in A and Bob wins if the sequence they construct is not in A. We might then ask if either player has a dominant strategy. With all the sets anyone will run into in the wild, it turns out that one player always has a dominant strategy for example, if A is finite, then Bob can choose an element his first turn and win, therefore having a dominant strategy. Essentially, what AD states is that for every such game, one player has a dominant strategy (in other words, every game is determined).
If we accept AD (which implies the negation of AC for reasons discussed in the second attached piece), we find that a number of the paradoxes resolved; for example, we can no longer chop a sphere into finitely many pieces and create 2 spheres out of them. However, AD creates a number of new problems which seem no more intuitive than Banach-Tarski. The first article describes one such paradox: by dividing the real numbers into certain equivalence classes, we can create a set that is strictly larger than the real numbers.
This relates to our previous discussion of game theory and shows how game theory has applications far beyond the primarily economical applications we have discussed so far. Game theory has applications even in modern mathematical research that does not have immediate applications to reality.
fhttp://stanwagon.com/public/TheDivisionParadoxTaylorWagon.pdf
http://cantorsattic.info/Axiom_of_determinacy