In this section we will introduce another logical tool that sheds new light on Borel, analytic, and coanalytic sets, and is indispensable in the study of higher levels of the projective hierarchy.
Let X be any nonempty set and let A ⊆ XN. We define an infinite two player gameG(A). Players I and II alternate playing elements of X. Player I playsx0, Player II replies withx1, Player I then playsx3. . . . A full play of the game looks like this.
Player I Player II x0 x1 x2 x3 x4 x5 .. . ...
Together they play x = (x0, x1, x2, . . .) ∈ XN. Player I wins this play of the game ifx∈A. Otherwise Player II wins.
Definition 6.1 Astrategyfor Player I is a function τ:X<N→X.
Player I uses the strategy by opening with τ(∅). If Player II responds with x0, then Player I repliesτ(x0). If Player II next plays,x1, then Player II replies τ(x0, x1). . . .
The full play looks like:
Player I Player II τ(∅) x0 τ(x0) x1 τ(x0, x1) x2 τ(x0, x1, x2) .. . ...
Definition 6.2 We say that τ is a winning strategy for Player I if Player I wins any game played using the strategyτ, i.e., for anyx0, x1, x2, . . .∈X, the sequence
τ(∅), x0, τ(x0), x1, τ(x0, x1), x2, τ(x0, x1, x2), . . . is inA.
There are analogous definitions of strategies and winning strategies for Player II.
Definition 6.3 We say that the gameG(A) isdeterminedif either Player I or Player II has a winning strategy.
We first show that ifAis not too complicated, thenG(A) is determined. We considerX with the discrete topology andXNwith the product topology.
Theorem 6.4 (Gale-Stewart Theorem) If A⊆XN is closed, thenG(A)is determined.
Proof LetT be a tree such thatA= [T]. Suppose Player II has no winning strategy. We will show that Player I has a winning strategy. Supposeσ∈N<ω
and|σ| is even. We consider the gameGσ(A) where Players I and II alternate
playing elements ofNto buildx∈ N and Player I wins if σbx∈A.
LetP ={σ:|σ|is even and Player II has a winning strategy inGσ(A)}. If
σ6∈T, then Player II has already wonGσ(A). In particular, always playing 0
is a winning strategy for Player II. ThusN<ω\T ⊆P.
Claim Suppose that for alln∈Nthere is m∈Nsuch thatσbnbm∈P. Then σ∈P.
Player II has a winning strategy in Gσ(A); namely if Player I playsn and
Player II plays the leastmsuch that Player II has a winning strategy inGσ n m,
and then uses the strategy in this game.
We describe a winning strategy for Player I. This strategy can be sumarized as “avoid losing postions”.
Since Player II does not have a winning strategy∅ 6∈P. Player I’s strategy is to avoidP. If we are in positionσwhere σ6∈P and|σ|is even, then by the claim there is a least n such thatσbnbm 6∈P for allm. Player I playsn. No matter what m Player II now plays the new position is not in P. If Player I continues Playing playing this way they will play x ∈ N such that x|2n6∈ P for alln. In particularx|2n∈T for all n. Thusx∈[T] and this is a winning strategy for Player I.
Exercise 6.5 Show that ifA⊆XNis open, thenG(A) is determined
Exercise 6.6 Show that ifA, B⊆XN,Ais open andBis closed, thenG(A∩B) is determined.
Exercise 6.7SupposeX0, X1, . . .are discrete topological spaces. If A⊆QXi
we can consider a modified game where Player I playsx0∈X0, Player II plays x1 ∈ X1, Player I playsx2 ∈ X2,. . . . Player I wins if (x0, x1, . . .)∈A. Show that ifAis closed this game is determined.
What other games are determined? Under the axiom of choice there are undetermined games.
Exercise 6.8 Use the axiom of choice to construct A ⊆ N such that no player has a winning strategy inG(A). [Hint: Use AC to give a well-ordered enumeration of all strategies and diagonalize against them.]
Theorem 6.9 (Borel Determinacy) IfA⊆ N is Borel, then G(A)is deter- mined.
For a proof see [6] II§20.
This is the best result provable in ZFC. The results of the next subsection, for example, show that if all analytic games are determined, then every uncountable
Σ1
2-set contains a perfect subset and this is false ifV=L. For Γ =Σ1
n or Π1n we let Det(Γ) be the assertion that if A∈Γ, thenG(A)
is determined. Projective determinacy PD is the assertion that all projective games are determined.
Exercise 6.10 Show that Det(Σ1
n) if and only if Det(Π1n).
The determinacy of projective games is intimately tied to the existence of large cardinals.
Theorem 6.11 (Martin/Harrington) i) If there is a measurable cardinal, then Det(Σ1
1)holds.
ii) Det(Σ1
1) holds if and only if x# exists for allx∈ N. For a proof see [9] Theorem 105.
More recently Martin and Steel [13] have found reasonable large cardinal hypotheses that imply PD.