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arxiv: 2605.21473 · v1 · pith:MZMYGUSQnew · submitted 2026-05-20 · 🧮 math.LO · math.CT

The Gamified Katv{e}tov order is not linear (in fact, very much not so)

Takayuki Kihara , Ming Ng This is my paper

Pith reviewed 2026-05-21 02:34 UTC · model grok-4.3

classification 🧮 math.LO math.CT
keywords Gamified Katětov orderP(ω)/FinantichainsfiltersRamsey theoryWeihrauch hierarchy
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The pith

The Gamified Katětov order embeds the poset P(ω)/Fin and therefore contains antichains of size continuum.

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

The paper shows that the Gamified Katětov order on filters over ω is far from being a linear order. It constructs an order-preserving embedding of P(ω)/Fin into this order. This matters because it reveals that the order has very large antichains even as it identifies all MAD families. The construction draws on Ramsey theory and produces new examples of non-modest degrees in the Weihrauch hierarchy.

Core claim

The Gamified Katětov order admits an embedding of P(ω)/Fin and thus contains an antichain of size continuum.

What carries the argument

The explicit order-preserving embedding from P(ω)/Fin into the Gamified Katětov order.

If this is right

  • The Gamified Katětov order is not linear.
  • It contains an antichain of cardinality of the continuum.
  • It brings connections with Ramsey theory into focus.
  • It yields a large new family of non-modest degrees in the extended Weihrauch hierarchy.

Where Pith is reading between the lines

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

  • This embedding suggests that certain combinatorial structures survive the gamification process.
  • Similar embeddings might exist for other quotients or posets in related orders.
  • The result may inform the study of degrees in computable analysis beyond the Weihrauch hierarchy.

Load-bearing premise

The asserted embedding from P(ω)/Fin into the Gamified Katětov order correctly preserves the relevant order relations defined by the gamification rules.

What would settle it

Finding two sets A and B in P(ω) such that A is not less than or equal to B modulo finite sets, but their images under the embedding are comparable in the Gamified Katětov order.

read the original abstract

Recently, the authors introduced the Gamified Kat\v{e}tov order on filters over $\omega$. This was shown to be strictly coarser than the classical Kat\v{e}tov order, and in fact collapses all MAD families to a single equivalence class. In the opposite direction, the present paper shows that the Gamified Kat\v{e}tov order also embeds $\mathcal{P}(\omega)/\mathrm{Fin}$, and thus contains an antichain of size continuum. The analysis brings into focus some interesting connections with Ramsey theory. As part of a broader programme investigating the interplay between combinatorial and computable complexity, we then apply our construction to produce a large new family of non-modest degrees in the extended Weihrauch hierarchy, which arise from associated effective subtoposes.

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

1 major / 2 minor

Summary. The manuscript builds on the authors' prior definition of the Gamified Katětov order on filters over ω (which collapses all MAD families to one equivalence class and is strictly coarser than the classical Katětov order). It constructs an explicit order-preserving embedding of P(ω)/Fin into this gamified order, yielding an antichain of size continuum, and draws connections to Ramsey theory. The construction is then applied to produce a large family of non-modest degrees in the extended Weihrauch hierarchy arising from effective subtoposes.

Significance. If the embedding holds, the result is significant: it demonstrates that the gamified order, despite its coarseness on MAD families, remains sufficiently rich to embed a large linear order's dual (an antichain of size 2^ℵ₀), sharpening our understanding of its position among filter orders. The explicit construction and Ramsey-theoretic analysis are strengths, as is the application to Weihrauch degrees, which advances the authors' programme on combinatorial-computable complexity. The skeptic's concern about the embedding does not land on reading the full manuscript, which supplies a concrete, verifiable construction rather than an asserted black box.

major comments (1)
  1. [§3] §3, Definition of φ and Lemma 3.4: the order-preservation claim (A ⊆* B iff φ(A) ≤_g φ(B)) is load-bearing for the antichain conclusion, yet the verification that the gamification rules (which collapse MAD families) are respected when A and B are almost disjoint relies on a single Ramsey coloring argument; an explicit check for the pair of even/odd sets would confirm no collapse occurs under the gamified comparison.
minor comments (2)
  1. [Abstract and §1] The abstract and §1 refer to 'the authors' prior work' without an arXiv number; adding the citation for the gamified-order paper would aid readers.
  2. [§3] Notation for the gamified order ≤_g is introduced in §2 but used without reminder in the embedding proof; a brief recap sentence would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive overall assessment, and the constructive suggestion in the major comment. We address the point below and will incorporate the requested clarification in the revision.

read point-by-point responses

  1. Referee: [§3] §3, Definition of φ and Lemma 3.4: the order-preservation claim (A ⊆* B iff φ(A) ≤_g φ(B)) is load-bearing for the antichain conclusion, yet the verification that the gamification rules (which collapse MAD families) are respected when A and B are almost disjoint relies on a single Ramsey coloring argument; an explicit check for the pair of even/odd sets would confirm no collapse occurs under the gamified comparison.

    Authors: The proof of Lemma 3.4 proceeds by a uniform Ramsey-theoretic argument that applies to any pair of almost disjoint sets A and B. In particular, the argument directly covers the even and odd sets without requiring a separate case. Nevertheless, we agree that an explicit verification for this concrete pair would make the non-collapse under the gamified comparison more transparent to readers. We will therefore add a short explicit computation for the even/odd pair immediately after the statement of Lemma 3.4 in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; embedding is an independent construction

full rationale

The paper cites the authors' prior work only for the base definition of the Gamified Katětov order and then supplies a new explicit order-preserving embedding φ from P(ω)/Fin. This embedding is presented as a fresh set-theoretic construction whose verification rests on direct checking of filter relations and gamification rules, not on any self-referential equation, fitted parameter renamed as a prediction, or load-bearing self-citation chain. The central claim therefore remains externally falsifiable by standard set-theoretic means and does not collapse to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the definition of the Gamified Katětov order introduced in the authors' previous work and on standard facts about the poset P(ω)/Fin; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard ZFC set theory
    Used to guarantee the existence of the embedding and the size of antichains in P(ω)/Fin.

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