Moschovakis coding lemma

The Moschovakis coding lemma is a lemma from descriptive set theory involving sets of real numbers under the axiom of determinacy (the principle — incompatible with choice — that every two-player integer game is determined). The lemma was developed and named after the mathematician Yiannis N. Moschovakis.

The lemma may be expressed generally as follows:
 * Let $Γ$ be a non-selfdual pointclass closed under real quantification and $∧$, and $≺$ a $Γ$-well-founded relation on $ω^{ω}$ of rank $θ ∈ ON$. Let $R ⊆ dom(≺) × ω^{ω}$ be such that $(∀x∈dom(≺))(∃y)(x R y)$. Then there is a $Γ$-set $A ⊆ dom(≺) × ω^{ω}$ which is a choice set for R, that is:

A proof runs as follows: suppose for contradiction $θ$ is a minimal counterexample, and fix $(∀α<θ)(∃x∈dom(≺),y)(|x|_{≺}=α ∧ x A y)$, $(∀x,y)(x A y → x R y)$, and a good universal set $≺$ for the $Γ$-subsets of $R$. Easily, $θ$ must be a limit ordinal. For $U ⊆ (ω^{ω})^{3}$, we say $(ω^{ω})^{2}$ codes a $δ$-choice set provided the property (1) holds for $δ < θ$ using $u ∈ ω^{ω}$ and property (2) holds for $α ≤ δ$ where we replace $A = U u$ with $A = U u$. By minimality of $θ$, for all $x ∈ dom(≺)$, there are $x ∈ dom(≺) ∧ |x| ≺ [≤δ]$-choice sets.

Now, play a game where players I, II select points $δ < θ$ and II wins when $u$ coding a $δ$-choice set for some $u,v ∈ ω^{ω}$ implies $v$ codes a $δ_{1}$-choice set for some $δ_{1} < θ$. A winning strategy for I defines a $δ_{2}$ set $B$ of reals encoding $δ$-choice sets for arbitrarily large $δ_{2} > δ_{1}$. Define then

which easily works. On the other hand, suppose $τ$ is a winning strategy for II. From the s-m-n theorem, let $Σ1 1$ be continuous such that for all $ϵ$, $x$, $t$, and $w$,

By the recursion theorem, there exists $δ < θ$ such that $x A y ↔ (∃w∈B)U(w,x,y)$. A straightforward induction on $s:(ω^{ω})^{2} → ω^{ω}$ for $U(s(ϵ,x),t,w) ↔ (∃y,z)(y ≺ x ∧ U(ϵ,y,z) ∧ U(z,t,w))$ shows that

and

So let