T-Minimal Theories in Model Theory

• T-Minimal Theories are frameworks that define minimality through definable topologies, robust dimension calculus, and computational constraints across diverse logical systems.
• They enforce that a definable set has non-empty interior if and only if it is infinite, supporting tame topological properties and canonical group topologies.
• The theory spans multiple domains, linking t-minimality, visceral theories, generalized Keisler orders, and computational set theory to yield multifaceted insights.

“T-Minimal theories” designates several non-equivalent notions of minimality. In recent model theory, “t-minimal” usually means a theory equipped with a definable Hausdorff topology in which a unary definable set has non-empty interior iff it is infinite; within this setting one studies dimension, visceral theories, canonical topologies on definable groups, and largeness of definable fields (Johnson, 2024, Johnson, 7 May 2026, Castle et al., 25 Aug 2025). In different literatures, closely related language is used for theories minimal in generalized Keisler orders for $L_{\theta,\theta}$ at a compact cardinal, for the minimal computational set theory $RST_{HF}^m$, and for recursively enumerable theories that would be minimal under Turing reducibility or interpretability in the context of Gödel’s first incompleteness theorem (Shelah, 2013, Avron et al., 2018, Cheng, 2021).

1. t-minimality as a tame topological condition

In the topological model-theoretic sense, a theory $T$ is equipped with a definable Hausdorff topology on the home sort $M$. The definition used by Mathews is that $T$ is t-minimal if every definable subset $D \subseteq M$ in one variable has finite boundary. Equivalently, for every definable $D \subseteq M$,

$$int(D) \neq \varnothing \quad \Longleftrightarrow \quad D \text{ is infinite}.$$

Since the topology is Hausdorff, this is also equivalent to saying that there are no isolated points in $M$, and that for every definable $D \subseteq M$, the boundary $RST_{HF}^m$0 is finite (Johnson, 2024).

This hypothesis is explicitly described as a very weak “tame topology” assumption. It controls definable sets in one variable, but it does not imply generic continuity or cell decomposition in general. The same basic axiom is used in later work on definable groups and fields, where t-minimality is presented as broad enough to include dense o-minimal, P-minimal, C-minimal, and hensel-minimal settings (Johnson, 7 May 2026).

A visceral theory is a specialization of this setting: it is t-minimal and the definable topology comes from a definable uniformity, i.e. a definable uniform structure. If $RST_{HF}^m$1 is the family of entourages, the induced topology is generated by balls

$RST_{HF}^m$2

Examples explicitly listed as visceral include o-minimal expansions, C-minimal valued fields, $RST_{HF}^m$3-adics and $RST_{HF}^m$4-minimal expansions, and dp-minimal expansions of fields (Johnson, 2024).

2. Dimension theory in t-minimal structures

A central development is that t-minimality already supports a robust dimension theory. The construction uses broadness and $RST_{HF}^m$5-independence. For $RST_{HF}^m$6, the set $RST_{HF}^m$7 is broad if for every $RST_{HF}^m$8, there are subsets $RST_{HF}^m$9 of size at least $T$0 such that

$T$1

Otherwise it is narrow. In the model-theoretic setting, a definable $T$2 is broad iff it contains arbitrarily large Cartesian grids; in a saturated model this is equivalent to containing an $T$3 grid. The key equivalence is

$T$4

Dimension of a tuple $T$5 over parameters $T$6 is then defined as the length of any tuple $T$7 such that $T$8 and $T$9 are interalgebraic over $M$0, and $M$1 is $M$2-independent over $M$3; a technical proposition shows this is well-defined (Johnson, 2024).

For a type-definable $M$4,

$M$5

The resulting dimension satisfies a substantial rank-like calculus: $M$6

$M$7

$M$8

with equality iff $M$9 has non-empty interior,

$T$0

$T$1

It is invariant under definable bijection, monotone under definable injections, preserved by definable surjections with finite fibers, and satisfies the fiber inequality

$T$2

There is also definability in families: for a definable family $T$3, the set

$T$4

is definable (Johnson, 2024).

This dimension is related to more classical ranks. If $T$5 has the exchange property, then $T$6 agrees with $T$7-rank. If $T$8 is dp-minimal, then $T$9 agrees with dp-rank. In general,

$D \subseteq M$0

where $D \subseteq M$1 is the naive topological dimension. A major caveat is that the familiar monotonicity of dimension under arbitrary definable surjections can fail: one can have a definable function $D \subseteq M$2 with

$D \subseteq M$3

This is the main exceptional phenomenon of the theory and marks a sharp difference from dp-minimal and o-minimal settings (Johnson, 2024).

3. Visceral theories, cell decomposition, and space-filling pathologies

Within visceral theories, many of Dolich–Goodrick’s tame topology theorems survive without the additional assumptions of definable finite choice (DFC) and no space-filling functions (NSFF). In particular, DFC is not needed for cell decomposition. The cells are defined as sets of the form

$D \subseteq M$4

where $D \subseteq M$5 is non-empty open, $D \subseteq M$6 is a continuous $D \subseteq M$7-correspondence, and $D \subseteq M$8 is a coordinate permutation. Every definable set is a finite disjoint union of cells, and every definable function, and even every definable correspondence, is continuous on each cell in some finite partition (Johnson, 2024).

The same analysis yields generic continuity and local Euclideanity. For definable correspondences, continuity holds on a dense definable subset, and in many cases on a dense relatively open subset. For any definable set $D \subseteq M$9, there is a dense relatively open definable subset $D \subseteq M$0 such that $D \subseteq M$1 is locally Euclidean. This provides a strong replacement for older continuity arguments based on DFC (Johnson, 2024).

The strongest misconception corrected by this work is that viscerality should automatically rule out space-filling phenomena. The counterexample is explicit: there exists a visceral theory with DFC that has a definable space-filling function. The first example is the 3-sorted real closed valued field with angular component $D \subseteq M$2, with sorts $D \subseteq M$3, $D \subseteq M$4, and $D \subseteq M$5, valuation $D \subseteq M$6, angular component $D \subseteq M$7, and the definable surjection

$D \subseteq M$8

from $D \subseteq M$9 onto $$int(D) \neq \varnothing \quad \Longleftrightarrow \quad D \text{ is infinite}.$$0. The paper then converts this into a genuine 1-sorted visceral structure by taking the disjoint union of the sorts with a combined definable topology (Johnson, 2024).

This example shows that NSFF is not automatic. Under NSFF, the stronger Dolich–Goodrick-style conclusions are recovered: $$int(D) \neq \varnothing \quad \Longleftrightarrow \quad D \text{ is infinite}.$$1 agrees with $$int(D) \neq \varnothing \quad \Longleftrightarrow \quad D \text{ is infinite}.$$2, definable surjections do not increase dimension, and frontier inequalities such as

$$int(D) \neq \varnothing \quad \Longleftrightarrow \quad D \text{ is infinite}.$$3

hold, with

$$int(D) \neq \varnothing \quad \Longleftrightarrow \quad D \text{ is infinite}.$$4

under exchange. Without NSFF, the paper constructs visceral theories where $$int(D) \neq \varnothing \quad \Longleftrightarrow \quad D \text{ is infinite}.$$5 but $$int(D) \neq \varnothing \quad \Longleftrightarrow \quad D \text{ is infinite}.$$6 is arbitrarily large, and where definable surjections can increase dimension dramatically (Johnson, 2024).

4. Definable groups, fields, and topological 1-basedness

Recent work extends t-minimality from a theory of definable sets to a structural theory of definable groups and fields. If $$int(D) \neq \varnothing \quad \Longleftrightarrow \quad D \text{ is infinite}.$$7 is a definable abelian group, a subset $$int(D) \neq \varnothing \quad \Longleftrightarrow \quad D \text{ is infinite}.$$8 is called big iff $$int(D) \neq \varnothing \quad \Longleftrightarrow \quad D \text{ is infinite}.$$9, and a basic neighborhood of $M$0 is a set of the form $M$1 with $M$2 big. From this, one obtains a translation-invariant group topology $M$3 on $M$4; it is definable, and it is characterized by

$M$5

for every definable $M$6. This yields uniqueness, continuity of definable homomorphisms, and product compatibility

$M$7

In the visceral case, abelianity can be dropped: every definable group admits a unique definable topology $M$8 such that $M$9 is a definable manifold and $D \subseteq M$0 is a topological group (Johnson, 7 May 2026).

The corresponding field theory is equally rigid. If $D \subseteq M$1 is a definable field in a t-minimal theory, then $D \subseteq M$2 is finite or large in the sense of Pop: any smooth algebraic curve $D \subseteq M$3 over $D \subseteq M$4 with at least one $D \subseteq M$5-rational point has infinitely many $D \subseteq M$6-rational points. The argument proceeds via a canonical topology on $D \subseteq M$7, together with the implication

$D \subseteq M$8

For infinite definable fields, the canonical topology is st-henselian and $D \subseteq M$9 is large. A separate consequence of the dimension theory is that every definable field in a t-minimal theory is perfect (Johnson, 2024, Johnson, 7 May 2026).

A further dividing line is topological 1-basedness. In a highly saturated t-minimal structure with the independent neighborhood property, $RST_{HF}^m$00 is topologically 1-based over $RST_{HF}^m$01 if

$RST_{HF}^m$02

With exchange, this agrees with weak 1-basedness; in o-minimality, it coincides with the classical linear/non-linear divide. If the structure is non-trivial and topologically 1-based, then it admits a type-definable abelian group $RST_{HF}^m$03 with $RST_{HF}^m$04 an open subset of $RST_{HF}^m$05. Moreover, $RST_{HF}^m$06 can be chosen to be a topological group with the subspace topology inherited from $RST_{HF}^m$07, and the induced structure on $RST_{HF}^m$08 satisfies a topological analog of the Hrushovski–Pillay classification of 1-based stable groups. The resulting group geometry is locally linear, and for such groups

$RST_{HF}^m$09

The exclusion of definable infinite fields from the topologically 1-based side makes this a genuine linear/non-linear boundary (Castle et al., 25 Aug 2025).

5. Minimality in $RST_{HF}^m$10 at a compact cardinal

A distinct use of “minimal theory” appears in infinitary model theory for a compact uncountable cardinal $RST_{HF}^m$11. Here $RST_{HF}^m$12 is a complete theory in $RST_{HF}^m$13, and minimality is defined relative to generalized Keisler-style orders built from $RST_{HF}^m$14-complete ultrafilters and saturation of ultrapowers. The key ultrafilters are the $RST_{HF}^m$15-complete $RST_{HF}^m$16-regular ultrafilters $RST_{HF}^m$17, where regularity is witnessed by sets $RST_{HF}^m$18 such that for every $RST_{HF}^m$19,

$RST_{HF}^m$20

This is the combinatorial framework used to analyze local and full saturation of $RST_{HF}^m$21 in $RST_{HF}^m$22 (Shelah, 2013).

The central characterization theorems isolate exactly which theories are minimal. If

$RST_{HF}^m$23

then

$RST_{HF}^m$24

For $RST_{HF}^m$25, one also has

$RST_{HF}^m$26

and, for

$RST_{HF}^m$27

$RST_{HF}^m$28

These results generalize the bottom of Keisler’s order from first-order logic to the compact-cardinal $RST_{HF}^m$29 setting (Shelah, 2013).

The need for separate local and full characterizations reflects a major departure from first-order stability theory: in $RST_{HF}^m$30, notions such as 1-stability, definable stability, and higher instability patterns no longer collapse. This suggests that “minimality” here is not a smallness condition on axioms, but a precise saturation-theoretic position in a generalized order on theories (Shelah, 2013).

6. Minimal computational theories and the failure of minimality for incompleteness

In foundational set theory, “T-minimal” appears in yet another sense. The theory $RST_{HF}^m$31 is presented as the minimal predicatively acceptable theory in a static framework that proves the existence of infinite sets. Its defining feature is computationality: a theory $RST_{HF}^m$32 is computational if the set $RST_{HF}^m$33 induced by its closed terms is a transitive model of $RST_{HF}^m$34, and for every transitive model $RST_{HF}^m$35 of $RST_{HF}^m$36,

$RST_{HF}^m$37

For $RST_{HF}^m$38, the minimal transitive model is $RST_{HF}^m$39, with

$RST_{HF}^m$40

and every element of $RST_{HF}^m$41 is denoted by some closed term of the language. Within this minimal computational universe, the real line is treated as a proper class of Dedekind cuts rather than as a set, yet the theory still supports least upper bounds for separable subclasses of $RST_{HF}^m$42, connectedness as intervalhood, continuity, uniform limits, the Intermediate Value Theorem, the Extreme Value Theorem, inverse functions for strictly monotone continuous functions, and standard functions such as polynomials, $RST_{HF}^m$43, $RST_{HF}^m$44, $RST_{HF}^m$45, and $RST_{HF}^m$46 (Avron et al., 2018).

In recursion-theoretic and interpretability-theoretic work on Gödel’s first incompleteness theorem, minimality again has a different meaning. Here one asks whether there is a consistent recursively enumerable theory for which $RST_{HF}^m$47 holds and which is minimal under Turing reducibility or interpretability. The basic definition is that $RST_{HF}^m$48 holds for an RE theory $RST_{HF}^m$49 if any consistent RE theory that interprets $RST_{HF}^m$50 is incomplete; for consistent RE theories this is equivalent to being essentially incomplete and to being essentially undecidable. The main conclusion is negative: there are no minimal RE theories w.r.t. Turing reducibility for which $RST_{HF}^m$51 holds, and there are no minimal elements in the interpretability structure $RST_{HF}^m$52 below $RST_{HF}^m$53 (Cheng, 2021).

The same negative phenomenon persists for effectively inseparable theories. The theory-version tEI is introduced by requiring a recursive separator for disjoint RE filter/ideal supersets of the nuclei $RST_{HF}^m$54 and $RST_{HF}^m$55. One proves that tEI implies essential undecidability, that $RST_{HF}^m$56 tEI implies $RST_{HF}^m$57 tEI, and that there are no minimal tEI theories w.r.t. interpretability. Since

$RST_{HF}^m$58

it follows that there are no minimal EI theories w.r.t. interpretability, and even no minimal finitely axiomatizable EI theories w.r.t. interpretability (Cheng, 2022). This usage turns “T-minimality” into a question about degree structures, and the answer is that no bottom element exists.

These literatures are conceptually independent, but together they show that “T-Minimal Theories” is not a single doctrine. In one direction, t-minimality is a weak topological tameness axiom that nonetheless supports dimension theory, cell decomposition in visceral settings, canonical group topologies, and strong algebraic consequences for definable fields. In others, minimality names extremal positions in generalized Keisler orders, the smallest computational set-theoretic environment sustaining substantial mathematics, or the unattainable lower bound sought in incompleteness and interpretability theory (Johnson, 2024, Shelah, 2013, Avron et al., 2018, Cheng, 2021).