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With a proper class of supercompacts, limit cut-and-choose games of successor length are consistently determined for every complete Boolean algebra.

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2026-07-12 02:24 UTC pith:7BBZDEBO

load-bearing objection Solid consistency result answering Zapletal’s 1995 question on limit cut-and-choose games, plus cleaner undetermined examples under approachability that survive MM++.

arxiv 2607.03444 v1 pith:7BBZDEBO submitted 2026-07-03 math.LO

On Determinacy for Cut and Choose Games of Uncountable Length

Maxwell Levine This is my paper
classification math.LO MSC 03E0503E3503E5506E10
keywords cut and choose gamesBoolean algebrasstrong distributivitysupercompact cardinalsapproachability idealdeterminacyMartin's Maximum
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Cut-and-choose games on complete Boolean algebras ask whether the player who carves partitions or the player who selects pieces has a winning strategy. Zapletal already showed that the ordinary countable game can be undetermined. This paper proves that the limit version of the game of length a successor cardinal—λ many rounds with no final move—can be made determined for every complete Boolean algebra at once, provided a proper class of supercompact cardinals is consistent. The same large-cardinal hypothesis yields a global model in which every such limit game is decided. At the same time the paper shows that the version that does end with a last round remains undetermined under the comparatively mild approachability ideal, and that such undetermined examples are compatible with Martin’s Maximum++. The contrast isolates precisely when length and the presence of a final move force indeterminacy.

Core claim

Assuming the consistency of a proper class of supercompact cardinals, it is consistent that for every successor cardinal λ and every complete Boolean algebra B the limit game G^µ_<λ(B) is determined for all µ<λ. In particular this settles Zapletal’s Question 2 on the countable case after collapse. Undetermined instances of the successor-length games that do possess a final round follow from the approachability property, and such instances remain consistent with MM++.

What carries the argument

A global Easton-support iteration of Lévy collapses of successive supercompacts, combined with a lifting argument that produces <λ-complete ultrafilters from strong distributivity in the extension; the ultrafilters supply winning strategies for the chooser in every limit game of length a successor.

Load-bearing premise

That the Easton-support class iteration of successive collapses preserves enough supercompactness and GCH at every stage so the local lifting construction can be repeated for every successor cardinal.

What would settle it

A model containing a proper class of supercompacts in which some complete Boolean algebra B and some successor λ make the limit game G^c&c_<λ(B) undetermined, or a ZFC proof that approachability already forces undetermined limit games.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

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Summary. The paper studies determinacy of cut-and-choose games of uncountable length on complete Boolean algebras. The main positive result (Theorem 2.14) shows that, from a proper class of supercompacts, an Easton-support class iteration of successive Lévy collapses yields a model in which, for every successor cardinal λ and every complete Boolean algebra B, the limit game G^µ_<λ(B) is determined for all µ<λ. This answers Zapletal’s Question 2. Supporting lemmas establish the equivalence of strong (<γ,µ)-distributivity with the non-existence of a winning strategy for Cut (Lemma 2.5), the preservation of Choose’s winning strategies under <λ-closed forcing (Lemma 2.13), and the construction of <λ-complete ultrafilters via lifted embeddings. Negative results show that approachability at λ^{+} produces undetermined games of length λ (Theorem 3.2, extending Dobrinen), that such undetermined instances exist when λ^{+} is not weakly compact in L (Theorem 3.9), and that they are compatible with MM^{++} (Theorem 3.13).

Significance. The work cleanly settles a natural open question of Zapletal by producing a global model of determinacy for the limit games at all successors. The technical core—strong distributivity characterizations, strategy-preservation under closed forcing, and embedding-lifting through Easton iterations—is standard but carefully executed and of independent interest for the study of games on Boolean algebras. The negative results under approachability and the MM^{++}-compatibility argument further clarify the boundary between determined and undetermined instances, extending the earlier work of Dobrinen and Cummings. The paper therefore advances both the positive and negative sides of the subject in a balanced way.

minor comments (5)
  1. In the proof of Theorem 2.14 the inductive claim that κ_α = ℵ_{α+2} in V[G_α] is stated clearly, but a one-sentence reminder that the tail forcing is sufficiently closed to preserve this equality into the full extension would improve readability.
  2. Lemma 2.13 constructs an assignment on <λΘ; the notation for the associated partial plays ⃗s_t is introduced only after the inductive construction begins. Moving the definition of ⃗s_t earlier would make the induction easier to follow.
  3. Fact 3.4 and Fact 3.5 are cited from Dobrinen; a parenthetical remark that the original statements assume full completeness while the present applications need only <λ^{+}-completeness (as noted later) would help the reader.
  4. In Proposition 3.10 the appeal to Zapletal’s theorem on semiproperness is correct, but the parenthetical observation that countable completeness suffices could be elevated to a short remark for clarity.
  5. Typographical inconsistencies appear in a few places (e.g., “G¨odel”, “Veliˇckovi´c”, occasional missing spaces around math mode). A final proof-reading pass would remove them.

Circularity Check

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No significant circularity; ordinary relative-consistency and combinatorial arguments from large-cardinal and approachability hypotheses.

full rationale

The paper's central claims (Theorems 2.11–2.14 on consistent determinacy of the limit games G^µ_<λ(B) for successor λ, and Theorems 3.2/3.9/3.13 on undetermined successor-length games) are proved by standard forcing constructions: Easton-support class iterations of Lévy collapses of successive supercompacts, lifting of elementary embeddings through closed quotients (via absorption), definition of <λ-complete ultrafilters from generic lower bounds, and a preservation lemma (Lemma 2.13) that transfers winning strategies for choose from a <λ-closed extension back to the ground model by building an explicit tree of conditions. Equivalences between non-existence of cut strategies and strong distributivity (Lemma 2.5) are proved by direct tree constructions and induction, not by definitional identification. Undetermined instances are obtained from approachability or □(κ) by building the regular-open algebra of a club-shooting poset and verifying (via elementary submodels and non-reflection) that neither player has a winning strategy. All citations (Zapletal, Dobrinen, Foreman, Jech, etc.) are to external results used as black boxes; there are no self-citations, no fitted parameters, no uniqueness theorems imported from the author, and no renaming of known patterns. The derivations are self-contained relative to the stated large-cardinal or combinatorial hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper works entirely inside ZFC plus large-cardinal hypotheses. No numerical free parameters are fitted. The only non-standard background is the existence of a proper class of supercompacts (used for the global consistency) and the approachability ideal (used for the undetermined examples). All other notions (complete Boolean algebras, strategic closure, elementary embeddings) are standard.

axioms (4)
  • domain assumption Consistency of a proper class of supercompact cardinals
    Invoked for the global Easton-support iteration that yields Theorem 2.14; without it only local determinacy at a single successor is obtained.
  • domain assumption ZFC + existence of a single supercompact cardinal
    Used for the local collapse argument of Theorems 2.11–2.12 that answers Zapletal’s question.
  • domain assumption Approachability ideal I[κ] at successors κ=λ+
    Hypothesis of Theorem 3.2 that produces undetermined games of length λ; weaker than the arithmetic assumptions of earlier Dobrinen papers.
  • standard math Standard facts about elementary embeddings, Lévy collapse, and strategic closure of Boolean algebras
    Background material taken from Jech, Cummings handbook, Foreman, etc., and used throughout Sections 2 and 3.

pith-pipeline@v1.1.0-grok45 · 22606 in / 2381 out tokens · 17017 ms · 2026-07-12T02:24:55.080065+00:00 · methodology

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read the original abstract

We obtain results on cut and choose games for complete Boolean algebras. Zapletal proved that there is a Boolean algebra $\mathbb{B}$ such that $\mathcal{G}_\omega^\textsf{candc}(\mathbb{B})$, the version of the game which ends on the $\omega$'th round, is undetermined. We prove that, assuming the consistency of a proper class of supercompact cardinals, the limit version $\mathcal{G}^{\textsf{candc}}_{< \lambda}(\mathbb{B})$, in which there are $\lambda$-many rounds but no concluding round, is consistently determined for all complete Boolean algebras $\mathbb{B}$ and all successor cardinals $\lambda$. In particular, this answers a question of Zapletal \cite[Question 2]{Zapletal1995}. We also show that undetermined instances of the game $\mathcal{G}^\textsf{candc}_\lambda(\mathbb{B})$ follow from the approachability property, extending results of Dobrinen, and we prove that undetermined instances are compatible with $\textsf{MM}^{++}$.

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