Variety Dynamics Case Study: The Mutual Constitution of Variety Space and Sets

Author: Terence Love

Date: March 2026 Framework: Variety Dynamics (VD)

DOI: https://doi.org/10.5281/zenodo.19144588

Summary

Variety Dynamics (VD) proposes that variety space and set theory are mutually constitutive primitives — neither is derivable from the other, neither has absolute ontological priority, and any formal system powerful enough to do mathematics requires both or their equivalent. This case study develops that claim through analysis of the definitional dependency between variety and sets, the pre-enumerative character of variety space relative to the post-enumerative character of sets, and the structural regress that results from any attempt to establish absolute foundational priority for either. The analysis establishes that variety space is the informatic structure of potential distinctions prior to actualisation, that sets are the result of actualisation operations on variety space, and that the mutual dependency between the two is not a circularity to be resolved but a genuine structural feature of the foundations of mathematics and logic. This case study builds on the foundational arguments developed in Love (2026a) and Love (2026b) and should be read in conjunction with both.

The Foundational Problem

Mathematics requires a foundation — a set of primitive concepts and axioms from which its structures are derived. Zermelo-Fraenkel set theory with Choice (ZFC) has served as the standard foundation for most of the twentieth century. Its primitives are sets and membership: everything in ZFC is built from sets, and the relationship between sets and their members is the only primitive relationship the theory requires.

This foundational programme has been enormously productive. The bulk of mathematics — number theory, analysis, topology, algebra, geometry — can be derived from ZFC's nine axioms. But the programme has a persistent difficulty at its base: it cannot fully account for its own starting point. To form a set, elements must be distinguishable. To distinguish elements, there must be variety — the structural possibility of difference. ZFC presupposes variety but does not define it. Variety is imported silently as a pre-formal primitive on which the entire set-theoretic edifice depends.

VD identifies this as a genuine foundational gap, not merely a presentational oversight. The gap is not closable within ZFC because any attempt to define distinguishability within ZFC requires already-formed sets of distinguishable elements — which reintroduces the dependency it was intended to resolve. The regress is real and irreducible.

Variety as Pre-Enumerative, Sets as Post-Enumerative

The relationship between variety space and sets is most precisely characterised through the distinction between pre-enumerative and post-enumerative structure established in Love (2026b).

Variety space is pre-enumerative: it is the structured totality of potential distinctions available before any enumeration has occurred. It characterises what could be distinguished — the informatic space of possible differences — prior to any act of collection, selection, or fixing. Variety space does not contain elements in the set-theoretic sense. It contains the structural possibility of elements — the conditions under which distinctions could be made.

A set is post-enumerative: it is the result of an enumeration operation that has already occurred. A set has specific, fixed members. Those members have been distinguished from one another and collected. The set is the recorded outcome of a selection operation performed on a variety space.

Axiom 9 defines variety as the possibility of a variable to have different values. This definition is explicitly pre-enumerative — it characterises possibility, not actuality. The number of different options that are possible is the variety of the situation. None of those options needs to be actualised for the variety to exist as a structured informatic space.

The set-theoretic operation of forming a set from elements is the operation of actualising a selection from variety space: fixing specific elements as the members of the set, distinguishing them from one another and from non-members, and recording the result as a determinate collection. The set is the post-enumerative residue of that operation. The variety space from which the selection was made remains intact and available for further selection operations — it is not consumed by the formation of the set.

This asymmetry — variety space persists through selection operations, sets are produced by them — establishes the structural relationship between the two without requiring either to be absolutely prior to the other.

The Mutual Dependency

Attempting to define variety without sets generates a regress. Variety is the possibility of a variable to have different values. A variable ranges over a domain. A domain is a collection of possible values. A collection is a set. Variety space, defined precisely, requires a set-theoretic structure to specify what the possible values are.

Attempting to define sets without variety generates the same regress from the other direction. A set is a collection of distinct elements. Distinct means distinguishable — different from one another. Distinguishability is variety. Without variety — without the structural possibility of difference — no two elements could be distinguished and no set could be formed. Set theory, defined precisely, requires variety to specify what distinguishability means.

Each concept, when pursued to its foundation, requires the other. This is not a failure of definition. It is a genuine structural feature of the foundations of formal systems. The two concepts are mutually constitutive: each is part of the foundational ground of the other.

Axiom 32 establishes that counterfactuals and varieties correspond exactly — both represent multiple possible states of a situation. The mutual constitution of variety and sets is itself a counterfactual structure: the variety space of possible foundational starting points includes both variety-first and set-first approaches, and neither can be actualised as a complete foundation without presupposing the other. The mutual dependency is a feature of the variety space of possible mathematical foundations, not a defect in any particular foundational programme.

The Turtles Argument

The mutual dependency between variety and sets is an instance of a general foundational structure that appears wherever formal systems attempt to ground themselves completely in their own primitives. Gödel's incompleteness theorems establish that any sufficiently powerful formal system contains true statements it cannot prove — the system cannot fully account for its own truth conditions. The mutual dependency between variety and sets is a pre-Gödelian version of the same structural phenomenon: any sufficiently powerful formal system contains primitive concepts it cannot fully define — the system cannot fully account for its own definitional foundations.

This is sometimes described colloquially as the turtles problem: whatever foundation is proposed, there is always a further question about what that foundation rests on. The regress has no natural stopping point within any formal system. It does not terminate at sets, or at variety, or at categories, or at any other proposed primitive. Each proposed primitive either presupposes other primitives or generates a regress when examined closely.

The correct response to this structure is not to search for a deeper foundation that terminates the regress — no such foundation exists within formal systems — but to recognise the mutual constitution of foundational primitives as the actual structure of mathematical foundations. Variety and sets are co-present at the base of the formal structure. Neither is more primitive. Neither can be eliminated in favour of the other. The regress goes down together.

This does not undermine the productivity of set theory or the validity of VD. It establishes the correct relationship between them: two mutually constitutive primitives, each required for the other, jointly providing the foundational structure from which formal mathematics is built.

Alternative Foundational Programmes and the Same Structure

The mutual constitution of variety and sets is not specific to ZFC. Alternative foundational programmes encounter the same structure from different directions.

Category theory proposes to found mathematics on objects and morphisms — structure-preserving maps between objects — rather than on sets and membership. Category theory avoids some of ZFC's difficulties but introduces others of the same kind. To define a category, objects must be distinguishable from one another. Morphisms must be distinguishable from one another. Composition of morphisms must produce distinguishable results. Distinguishability — variety — is presupposed by category theory's primitives in precisely the same way it is presupposed by ZFC's primitives.

Type theory proposes to found mathematics on types and terms — typed expressions and their reduction rules — rather than on sets. Type theory has strong connections to constructive mathematics and computer science. But to define a type, the terms of that type must be distinguishable from one another and from terms of other types. Distinguishability — variety — is again presupposed.

Higher topos theory (Lurie) provides the most general current foundational framework, incorporating homotopy theory, sheaf theory, and the full machinery of higher category theory. As established in Love (2026b), variety space corresponds to the pre-sheaf structure in a higher topos — the totality of potential local data prior to globalisation. Even at this level of generality, the pre-sheaf structure presupposes distinguishable local data: variety space appears at the foundation of the most general current mathematical framework.

The pattern is consistent across foundational programmes: variety is the unnamed pre-formal primitive that every formal foundational programme presupposes. Set theory makes it most visible because its extensionality axiom depends on it most directly. But it is present in every alternative.

What VD Contributes

VD does not claim to resolve the foundational regress. It claims to operate at the level where the regress occurs — the level of mutual constitution of foundational primitives — rather than above it in the derived mathematical structures that standard foundational programmes address.

This is a precise positional claim. VD's axioms are formulated at the level of variety space — the pre-enumerative structure of potential distinctions — rather than at the level of sets, types, or categories. They characterise the structural relationships between variety distributions before any actualisation has occurred. This places VD at the foundational level where the mutual constitution of variety and sets is visible, rather than at the post-enumerative level where set theory operates.

Axiom 47 proposes that VD operates at a more foundational level than physical theories through its grounding in set theory and its capacity to address discontinuities intrinsically. The analysis of this case study extends that claim: VD operates at a more foundational level than set theory itself, not by replacing set theory but by characterising the pre-enumerative variety space from which set-theoretic operations proceed. VD and set theory are mutually constitutive at the foundational level — each requires the other — and VD makes this mutual constitution explicit rather than importing variety silently as an undefined primitive.

Axiom 9 is the formal statement of VD's foundational primitive: variety is the possibility of a variable to have different values. This statement is deliberately pre-enumerative — it does not define variety in terms of sets, collections, or enumerated options. It characterises the structural possibility of difference prior to any actualisation of specific differences. This is the correct level at which to state the foundational primitive, and it is the level at which ZFC's extensionality axiom implicitly operates without acknowledgement.

The mathematician Alexander Grothendieck worked with a structurally similar foundational intuition, approached from within algebraic geometry and cohomology theory. Grothendieck's characteristic move was to work at the level of general mathematical structure — the space of all possible relationships an object could have — and allow specific results in specific fields to emerge as local expressions of that general structure. His construction of toposes as universes of possible mathematical objects, his motives programme as an attempt to reach the pre-enumerative variety space underlying all cohomological descriptions, and his relative point of view — that an object is best understood through the full variety of its possible relationships rather than through intrinsic properties — all reflect the same foundational orientation that VD makes explicit through its axioms. VD was developed independently, and awareness of Grothendieck's foundational thinking came through indirect and third-hand exposure rather than direct study of his work. The structural parallel was recognised after the fact. That the same foundational orientation emerged independently from two entirely different starting points — Grothendieck from within algebraic geometry, VD from applied mathematical encounters with hyper-complexity across multiple domains — is itself evidence for the validity of the shared structural intuition. The intellectual convergence is acknowledged here directly.

The Informatic Character of the Mutual Constitution

As established in Love (2026b), varieties are informatic descriptions — descriptions of reality in terms of distinguishable options, states, decisions, or events — not the physical things they describe. The mutual constitution of variety and sets is therefore a relationship between informatic structures, not between physical entities or abstract Platonic objects.

Variety space is the informatic structure of potential distinctions. Sets are the informatic results of actualisation operations on that structure. The mutual dependency between them is a dependency between two modes of informatic description: the pre-enumerative description of possible differences and the post-enumerative description of actualised collections.

This informatic framing has a practical consequence established in Love (2026b): representing a variety distribution in a physical system requires a physical substrate and incurs thermodynamic costs. The mutual constitution of variety and sets is not a free-floating logical relationship. It is instantiated in physical information media — in mathematical texts, in computational systems, in human cognitive structures — and those instantiations have real physical costs. The foundational relationship between variety and sets is informatically real in this sense: its representation requires physical substrate and thermodynamic commitment, regardless of the abstract logical structure

Relationship to Constructor Theory

Constructor theory (Deutsch and Marletto) proposes to reformulate physics in terms of counterfactuals — statements about which transformations are possible and which are impossible — rather than in terms of dynamical laws. As established in Love (2026b), Axiom 32 identifies counterfactuals and varieties as the same informatic entity: counterfactuals are varieties, not merely analogous to them.

The mutual constitution of variety and sets has a direct consequence for constructor theory. Constructor theory's counterfactual statements are variety descriptions — informatic descriptions of possible and impossible transformations. Those descriptions presuppose distinguishable transformation types, distinguishable initial and final states, and distinguishable constructors. Distinguishability is variety. Constructor theory therefore presupposes variety space in precisely the same way set theory does.

Additionally, constructor theory describes which transformations are possible within a given representational framework. As Axiom 32 establishes, VD operates at a higher abstraction level — examining what variety distributions enable different representational frameworks themselves. The mutual constitution of variety and sets is part of the variety space of possible foundational frameworks: constructor theory, set theory, type theory, and category theory are all actualisations from a variety space of possible mathematical foundations, each selecting different structural constraints from that space.

Limitations and Open Questions

The argument that variety and sets are mutually constitutive is developed here at a structural and philosophical level. The formal mathematical demonstration — establishing precisely what variety space is as a mathematical object, how it relates to ZFC's primitives in formal terms, and what the mutual constitution relationship implies for the consistency and completeness of formal systems — requires mathematical development beyond the scope of this case study. That development is identified as the primary task for the standalone mathematical paper in preparation, which will draw on the categorical and higher topos framing developed in Love (2026b).

The turtles argument establishes that the regress has no natural stopping point within formal systems. This is correct but raises a question about the status of VD's own axioms. If variety space is pre-enumerative and its definition requires set-theoretic structure to be fully precise, then Axiom 9's definition of variety as the possibility of a variable to have different values is itself subject to the same mutual dependency. VD does not escape the turtles structure — it operates within it, at the level where the mutual constitution is visible. This is the correct position but it needs to be stated explicitly rather than implied.

The relationship between the mutual constitution of variety and sets and Gödel's incompleteness theorems is noted here but not formally developed. The parallel is suggestive — both identify structural limits on what formal systems can account for from within — but establishing whether the mutual constitution result is derivable from or independent of Gödel's theorems requires formal treatment.

Conclusion

Variety space and sets are mutually constitutive primitives. Variety space is pre-enumerative: the informatic structure of potential distinctions prior to any actualisation. Sets are post-enumerative: the results of actualisation operations on variety space. Each concept requires the other when pursued to its foundational basis. The regress generated by this mutual dependency — the turtles structure — is not a problem to be solved but a genuine structural feature of the foundations of formal systems.

VD operates at the level where this mutual constitution is visible, characterising the pre-enumerative variety space from which set-theoretic and other formal operations proceed. This places VD at a more foundational level than ZFC, not by replacing set theory but by making explicit the variety structure that set theory presupposes. The same structure — variety as unnamed pre-formal primitive — appears in every alternative foundational programme: category theory, type theory, and higher topos theory all presuppose distinguishability and therefore variety at their foundational level.

The mutual constitution of variety and sets is informatically real: its representation requires physical substrate and thermodynamic commitment. It is not a free-floating logical relationship but an informatic structure instantiated in the physical media that carry mathematical thought and practice.

Constructor theory, which describes possible and impossible transformations in counterfactual terms, presupposes variety space in the same way set theory does. Since counterfactuals are varieties (Axiom 32), constructor theory is a variety-processing system operating within a variety space of possible representational frameworks. VD operates at the meta-level above that framework, examining what variety distributions enable different foundational approaches and what could alter the space of possible mathematical foundations.

Axioms Applied

Axiom 9 (Variety definition), Axiom 25 (Variety dynamics and information systems), Axiom 26 (Variety dynamics, information systems, and thermodynamic constraints), Axiom 32 (Variety dynamics, counterfactuals, and constructors), Axiom 47 (Variety dynamics and fundamental physics — tentative), Axiom 57 (VD subsuming special cases — in development).

All axioms: Love, T. (2025). Variety Dynamics: Formal Statements of Axioms 1–50. Love Services Pty Ltd. https://doi.org/10.5281/zenodo.17571975

References

Deutsch, D. (2011). The Beginning of Infinity: Explanations that Transform the World. Allen Lane.

Love, T. (2025). Variety Dynamics: Formal Statements of Axioms 1–50. Love Services Pty Ltd. https://doi.org/10.5281/zenodo.17571975

Love, T. (2026a). Variety Dynamics Case Study: An Alternative Structural Explanation of the Quantum Computational Ceiling Proposed by Rational Quantum Mechanics. Zenodo. https://doi.org/10.5281/zenodo.19140950

Love, T. (2026b). Variety Dynamics Case Study: Variety Space as Foundational to Physics and Mathematics. Zenodo. https://doi.org/10.5281/zenodo.19141638

Methodology Note

This analysis applies the Variety Dynamics framework through iterative human-AI collaboration. VD axioms and analytical framework specified by human expert (T. Love); structural mapping, argument development, and initial drafting generated by Claude Sonnet 4.6 (Anthropic); reviewed, verified, edited and refined by T. Love through multiple iterations. Final analysis reflects human expert judgment of variety distributions, structural dynamics, and analytical sufficiency.