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arxiv: 1906.11757 · v1 · pith:3WOWC2PInew · submitted 2019-06-27 · 🧮 math.LO

Long Borel Games

J. P. Aguilera This is my paper

Pith reviewed 2026-05-25 13:51 UTC · model grok-4.3

classification 🧮 math.LO
keywords Borel gamesgame determinacyextender modelsWoodin cardinalsZermelo set theorylong gamesdescriptive set theoryinner model theory
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The pith

Borel games of length ω² are determined if and only if fine-structural extender models of Zermelo set theory with α-many iterated powersets exist above limits of Woodin cardinals for every countable ordinal α.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves an equivalence between the determinacy of Borel games lasting ω² many moves and a specific large-cardinal hypothesis in inner model theory. If the games are determined, then for each countable ordinal α there must exist a fine-structural countably iterable extender model of Zermelo set theory containing α iterated power sets above a limit of Woodin cardinals; the converse also holds. This characterization links questions about winning strategies in long games directly to the consistency strength of certain iterable models. A reader would care because it gives an exact calibration of what is needed to guarantee determinacy at this length rather than leaving it as an open consistency question.

Core claim

Borel games of length ω² are determined if and only if, for every countable ordinal α, there is a fine-structural, countably iterable extender model of Zermelo set theory with α-many iterated powersets above a limit of Woodin cardinals.

What carries the argument

The if-and-only-if equivalence that reduces determinacy of ω²-length Borel games to the existence of countably iterable fine-structural extender models of Zermelo set theory above limits of Woodin cardinals.

If this is right

  • Determinacy of these games yields the existence of iterable inner models with Woodin cardinals and iterated power sets.
  • The models in turn suffice to prove determinacy of the ω²-length games via the fine-structure and iterability assumptions.
  • The result extends the known pattern of determinacy-implies-inner-model implications from shorter game lengths to length ω².
  • It supplies a lower bound on the consistency strength required for determinacy statements at this specific length.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar equivalences could plausibly be sought for games of length ω^ω or other ordinals beyond ω².
  • The calibration may allow transferring techniques between descriptive set theory and inner-model constructions for other classes of games.
  • It opens the possibility of proving new determinacy results by constructing the required extender models rather than working directly with strategies.

Load-bearing premise

The background theory of fine-structural extender models and their countable iterability is sufficient to carry the determinacy proof in both directions.

What would settle it

A concrete counterexample would be a countable ordinal α for which no such extender model exists yet all Borel games of length ω² remain determined, or the converse situation in which the models exist but some Borel game of length ω² fails to be determined.

read the original abstract

It is shown that Borel games of length $\omega^2$ are determined if, and only if, for every countable ordinal $\alpha$, there is a fine-structural, countably iterable extender model of Zermelo set theory with $\alpha$-many iterated powersets above a limit of Woodin cardinals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper proves that Borel games of length ω² are determined if and only if, for every countable ordinal α, there exists a fine-structural, countably iterable extender model of Zermelo set theory with α-many iterated powersets above a limit of Woodin cardinals.

Significance. If the equivalence holds, the result supplies a sharp characterization of determinacy for Borel games of length ω² in terms of the existence of specific fine-structural inner models, advancing the program relating long-game determinacy to large-cardinal hypotheses and iterability. The bidirectional implication is a strength, as is the use of standard fine-structure and countable-iterability assumptions.

minor comments (1)
  1. [Abstract] The abstract states the main theorem cleanly but does not indicate the length or structure of the proof; a single sentence on the overall strategy (e.g., one direction via inner-model theory, the other via a direct game argument) would help readers gauge the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept. The referee's summary accurately captures the main result.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states an explicit if-and-only-if equivalence between determinacy of Borel games of length ω² and the existence, for every countable ordinal α, of fine-structural countably iterable extender models of Zermelo set theory with α iterated powersets above a limit of Woodin cardinals. Both sides of the equivalence are independently defined mathematical statements; neither is constructed from the other by definition, fitting, or renaming. The background assumptions on fine structure and iterability are the standard external framework used in such results and do not reduce the central claim to a self-citation chain or internal tautology. No load-bearing step in the abstract or stated claim exhibits any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the standard background of inner-model theory (fine structure, countable iterability, Woodin cardinals) plus the definition of Zermelo set theory; no free parameters or new entities are introduced in the abstract.

axioms (3)
  • standard math Zermelo set theory as the base theory for the extender models
    The models are explicitly of Zermelo set theory.
  • domain assumption Existence and properties of limits of Woodin cardinals inside the models
    The models are required to contain a limit of Woodin cardinals.
  • domain assumption Countable iterability of the extender models
    Iterability is part of the model description used in the equivalence.

pith-pipeline@v0.9.0 · 5552 in / 1290 out tokens · 28347 ms · 2026-05-25T13:51:31.050974+00:00 · methodology

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Reference graph

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