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arxiv: 2404.11453 · v3 · submitted 2024-04-17 · 🧮 math.LO

Visceral theories without assumptions

Will Johnson This is my paper

Pith reviewed 2026-05-24 01:43 UTC · model grok-4.3

classification 🧮 math.LO
keywords t-minimal theoriesvisceral theoriesdimension theorydefinable fieldstame topologyspace-filling curvesmodel theory
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The pith

If a theory is t-minimal then definable sets admit a dimension theory and any definable field is perfect.

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

The paper shows that t-minimality, where a definable topology makes every one-variable definable set have finite boundary, is enough to produce a dimension theory on definable sets. This dimension shares the main features of dp-rank but allows the dimension of a function's domain to drop below the dimension of its image. It proves that any field definable in a t-minimal theory must be perfect. For the subclass of visceral theories the paper shows that most tame topology results survive without the earlier requirements of definable finite choice and no space-filling functions. It also supplies an example of a visceral theory that does contain a space-filling curve.

Core claim

Let T be a theory with a definable topology. If T is t-minimal, meaning every definable set in one variable has finite boundary, then there is a good dimension theory for definable sets satisfying properties similar to dp-rank in dp-minimal theories, except that the dimension of the domain of a definable function f can be less than the dimension of its image. Using this dimension theory, any definable field in a t-minimal theory is perfect. Specializing to visceral theories, almost all tame topology theorems hold without definable finite choice and no space-filling functions. There exists a visceral theory with a space-filling curve.

What carries the argument

t-minimality of the definable topology, the condition that every definable unary set has finite boundary, which supports the dimension theory and all later results.

If this is right

  • Any definable field in a t-minimal theory is perfect.
  • Almost all tame topology theorems for visceral theories remain valid without definable finite choice or the no-space-filling-functions assumption.
  • Visceral theories can contain space-filling curves.

Where Pith is reading between the lines

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

  • The dimension theory may apply directly to other minimality notions that control boundaries in one variable.
  • More examples of visceral theories become available for study once the extra assumptions are dropped.
  • The existence of space-filling curves inside visceral theories may produce new constructions in definable geometry.

Load-bearing premise

The theory comes with a definable topology in which every definable set in one variable has only finitely many boundary points.

What would settle it

A t-minimal theory containing a definable field that is not perfect, or in which the dimension of the domain of some definable function exceeds the dimension of its image.

read the original abstract

Let $T$ be a theory with a definable topology. $T$ is t-minimal in the sense of Mathews if every definable set in one variable has finite boundary. If $T$ is t-minimal, we show that there is a good dimension theory for definable sets, satisfying properties similar to dp-rank in dp-minimal theories, with one key exception: the dimension of $\operatorname{dom}(f)$ can be less than the dimension of $\operatorname{im}(f)$ for a definable function $f$. Using the dimension theory, we show that any definable field in a t-minimal theory is perfect. We then specialize to the case where $T$ is visceral in the sense of Dolich and Goodrick, meaning that $T$ is t-minimal and the definable topology comes from a definable uniformity (i.e., a definable uniform structure). We show that almost all of Dolich and Goodrick's tame topology theorems for visceral theories hold without their additional assumptions of definable finite choice (DFC) and no space-filling functions (NSFF). Lastly, we produce an example of a visceral theory with a space-filling curve, answering a question of Dolich and Goodrick.

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 / 3 minor

Summary. The paper shows that any t-minimal theory (every definable unary set has finite boundary) admits a dimension theory on definable sets that satisfies the usual properties of dp-rank except that dim(dom(f)) may be strictly less than dim(im(f)) for definable f. It then proves that every definable field in a t-minimal theory is perfect. Specializing to visceral theories (t-minimal theories whose topology arises from a definable uniformity), the paper establishes that nearly all of the tame topology results of Dolich-Goodrick continue to hold after dropping the assumptions DFC and NSFF. Finally, it constructs an explicit visceral theory containing a space-filling curve, answering a question left open by Dolich and Goodrick.

Significance. If the derivations hold, the work supplies a dimension theory for the broader class of t-minimal structures and removes two auxiliary hypotheses from the existing tame-topology results for visceral theories. The explicit counter-example to the necessity of NSFF is a concrete contribution that directly resolves an open question. The arguments are presented as self-contained developments from the t-minimality axiom and the definable-uniformity condition.

minor comments (3)
  1. [Introduction] The dimension function is introduced in §3; a short forward reference in the introduction would help readers locate the precise list of axioms it satisfies.
  2. [§5] In the statement of the space-filling-curve example, the uniformity is defined via a specific formula; a one-sentence reminder of why this uniformity is definable would improve readability.
  3. [Table 1] The comparison table of properties with dp-rank (Table 1) lists the domain-image exception but does not indicate whether the other axioms are proved in the same order as in the dp-minimal literature; a brief remark on proof order would be useful.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the paper. The referee's summary correctly identifies the key contributions: the dimension theory for t-minimal theories (with the noted exception for functions), the result that definable fields are perfect, the extension of tame topology theorems to visceral theories without DFC and NSFF, and the explicit visceral theory with a space-filling curve resolving the open question from Dolich-Goodrick.

Circularity Check

0 steps flagged

Derivation chain is self-contained from t-minimality assumption

full rationale

The paper starts from the explicit assumption that T is t-minimal (every definable unary set has finite boundary) and derives a dimension theory for definable sets, proves definable fields are perfect, shows that visceral tame topology theorems hold without DFC/NSFF, and constructs an explicit counterexample to the space-filling question. These steps are presented as direct consequences of the t-minimality hypothesis plus the visceral uniformity condition; no parameter fitting, self-definitional loops, or load-bearing self-citations are invoked to force the conclusions. The cited prior notions (Mathews, Dolich-Goodrick) serve only as background definitions, not as unverified uniqueness theorems or ansatzes that smuggle in the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No free parameters or invented entities; the paper works within existing framework of t-minimal and visceral theories from prior literature, adding new theorems and an example.

axioms (2)
  • domain assumption The theory T has a definable topology.
    This is the setup for t-minimality and all results.
  • standard math Standard axioms and definitions of model theory for definable sets and functions.
    Used throughout for the notions of definable sets and fields.

pith-pipeline@v0.9.0 · 5739 in / 1398 out tokens · 50223 ms · 2026-05-24T01:43:54.253344+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Definable groups and fields in t-minimal theories

    math.LO 2026-05 unverdicted novelty 7.0

    Definable fields in t-minimal theories are finite or large, and definable groups are equipped with canonical topologies that form manifolds under visceral assumptions.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · cited by 1 Pith paper

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