Is Gold-Putnam diagonalization complete?

Abstract

Diagonalization is a proof technique that formal learning theorists use to show that inductive problems are unsolvable. The technique intuitively requires the construction of the mathematical equivalent of a “Cartesian demon” that fools the scientist no matter how he proceeds. A natural question that arises is whether diagonalization iscomplete. That is, given an arbitrary unsolvable inductive problem, does an invincible demon exist?

The answer to that question tunas out to depend upon what axioms of set theory we adopt. The two main results of the paper show that if we assume Zermelo-Fraenkel set theory plus AC and CH, there exist undetermined inductive games. The existence of such games entails that diagonalization is incomplete. On the other hand, if we assume the Axiom of Determinacy, or even a weaker axiom known as Wadge Determinacy, then diagonalization is complete.

In order to prove the results, inductive inquiry is viewed as an infinitary game played between the scientist and nature. Such games have been studied extensively by descriptive set theorists. Analogues to the results above are mentioned, in which both the scientist and the demon are restricted to computable strategies.

The results exhibit a surprising connection between inductive methodology and the foundations of set theory.