TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Borel games are determined

Statement

In ZFC every Borel payoff game on a set-sized tree with terminal taboos is determined. In particular every Borel Gale–Stewart game on $N$ is determined. This theorem uses AC and is not a ZF supplier for the AD implications.

Facts & Assumptions

[F1]

Borel payoffs admit unraveling covers supplies an unraveling at any natural depth.

[F2]

Unraveling covers give determinacy descends determinacy from an unraveling.

[A1]

Assume The Axiom of Choice.

Proof

Given: A taboo tree $T$ on a set alphabet and Borel $A \subseteq [T]$ .

1.1

The set-alphabet and Borel hypotheses are exactly those of F1; A1 supplies its choice assumption. Apply it with $k = 0$ to obtain a covering whose inverse image of $A$ is clopen. By F2, with the same ZFC assumption, $G (A; T)$ is determined. This also applies when the root is terminal or there are no infinite branches, since both suppliers include finite taboo plays.

F1F2A1

2.1

For an ordinary Gale–Stewart game take $T = N^{< N}$ and no terminal taboos. This is a set-sized pruned tree and its branch space with the cylinder topology is $N^{N}$ . Thus a Borel payoff satisfies step 1.1, and its conclusion is precisely a winning strategy for one of the ordinary two players. QED.

step 1.1

Depends on

Used by

Every set of reals is Borel False statement

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Corollary 6 (standard reference, not scraped)
Theorem 2.1.9, printed p77 (standard reference, not scraped)
Corollary, printed p454 (standard reference, not scraped)