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The Moschovakis coding lemma is a lemma from descriptive set theory involving sets of real numbers under the axiom of determinacy. The lemma was developed and named after the mathematician Yiannis N. Moschovakis.
The lemma may be expressed generally as follows:
A proof runs as follows: suppose for contradiction θ is a minimal counterexample, and fix ≺, R, and a good universal set U ⊆ for the Γ-subsets of. Easily, θ must be a limit ordinal. For δ < θ, we say u ∈ ω codes a δ-choice set provided the property holds for α ≤ δ using A = U u and property holds for A = U u where we replace x ∈ dom with x ∈ dom ∧ |x| ≺. By minimality of θ, for all δ < θ, there are δ-choice sets.
Now, play a game where players I, II select points u,v ∈ ω and II wins when u coding a δ1-choice set for some δ1 < θ implies v codes a δ2-choice set for some δ2 > δ1. A winning strategy for I defines a Σ1 set B of reals encoding δ-choice sets for arbitrarily large δ < θ. Define then