Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

eyond the finite realm 95Set Theorist’s Response: The development of set theory, after Cohen, has led to therealization that there is a robust hierarchy of strong axioms of infinity.Elaborating further, it has been discovered that, in many cases, very different lines ofinvestigation have led to problems whose degree of unsolvability is exactly calibratedby a notion of infinity. Thus, the hierarchy of large cardinal axioms emerges as anintrinsic, fundamental conception within set theory. To illustrate this, I discuss anexample from modern set theory that concerns infinite games.Suppose A ⊂ P(N), where P(N) denotes the set of all sets σ ⊆ N and N is the setof all natural numbers: N = {1, 2,...,k,...}.Associated to the set A is an infinite game involving two players, Player I and PlayerII. The players alternate declaring at stage k whether k ∈ σ or k/∈ σ :Stage 1: Player I declares 1 ∈ σ or declares 1 /∈ σ ;Stage 2: Player II declares 2 ∈ σ or declares 2 /∈ σ ;Stage 3: Player I declares 3 ∈ σ or declares 3 /∈ σ ; ...After infinitely many stages, a set σ ⊆ N is specified. Player I wins this run of the gameif σ ∈ A; otherwise, Player II wins. (Note: Player I has control of which odd numbersare in σ , and Player II has control of which even numbers are in σ .)A strategy is simply a function that provides moves for the players given just thecurrent state of the game. More formally, a strategy is a functionτ :[N]