Have you ever noticed how relational mathematics and physics are? In mathematics, calculus can be rebuilt from limits, limits from real numbers, real numbers from set-theoretic constructions, and those constructions from logic, membership, and axioms. In physics, objects appear relational in a similar way. A tree depends on wood molecules, wood molecules depend on atoms, atoms depend on electrons and nuclei, nuclei depend on quarks bound by fields, and in modern physics particles themselves are often understood as excitations or roles within quantum fields. What about consciousness? If we keep following these chains of dependence, we seem to expect a bottom on which everything else rests. But perhaps the lesson is stranger: perhaps the bottom is not a thing at all. Maybe relations, rather than things, are at the core of reality.
A set-axiomatic model can represent the universe as a structure of relations rather than as a collection of self-standing objects. However, no such model can serve as an exhaustive, closed description of reality if it is consistent, axiomatizable, and powerful enough to express arithmetic. The path to that conclusion begins with the empty set, because the empty set reveals how mathematics can build definite structures from minimal formal assumptions. The same question then returns in physics: perhaps the natural universe, too, is not made of independent things but of relations that make things appear. A complete account of reality must eventually face not only particles, fields, and laws, but also the subject for whom anything appears at all. Consciousness therefore enters the argument not as a side-topic, but as the hardest test of formal description. Subjective experience appears undeniably true from the first-person point of view, but may resist complete derivation from within the formal syntax of a physical model.
Begin with the empty set, written as $\varnothing$ or $\lbrace\rbrace$. Its cardinality is zero because it contains no elements. However the empty set can still serve as an element of another set, so $\lbrace\varnothing\rbrace$ has cardinality one. In the von Neumann construction, that distinction becomes the basis of counting: $0=\varnothing$, $1=\lbrace\varnothing\rbrace$, $2=\lbrace 0,1\rbrace$, and $3=\lbrace 0,1,2\rbrace$. Each natural number is the set of all smaller natural numbers. The empty set therefore does not “produce something from nothing” in a metaphysical sense. Mathematics begins with a formally defined object that has no elements, then uses axioms and operations to generate further structures. Numbers are not rearrangements of the empty set; they are relational constructions built from the empty set within a formal system.
The same point extends to the rest of mathematics. Integers can be constructed from pairs of natural numbers, rational numbers from pairs of integers, real numbers from rational numbers, and calculus from real numbers, functions, limits, derivatives, and integrals. On that basis, calculus can in principle be reduced to set-theoretic constructions. But the empty set alone is not enough. The foundation of standard set theory is not simply $\varnothing$, but the logical framework, the primitive membership relation $\in$, and the axioms that govern membership. The empty set is not the deepest irreducible primitive; it is the first distinguished object whose existence is guaranteed by the axioms.
From mathematics, the question moves to the natural universe. The universe may be understood as fully determinate physical reality. Perfection, in the strongest sense, would be the totality of all truths about that reality. A theory is an axiomatizable formal representation, while a model is a mathematical structure satisfying such a theory. A set-theoretic model may preserve the structure of the universe without making the universe literally a set. A map can preserve the structure of a city without being made of streets and buildings.
Structural realism provides the philosophical bridge. Under an ontic structuralist view, the universe is not fundamentally a collection of objects that later enter into relations. Rather, the universe is the network of relations itself, while “objects” are stable positions or placeholders within that relational structure. An electron, for example, is not an independent substance hidden behind its properties. It is identified by its role in a web of relations: charge, spin, mass, lepton number, coupling strengths, symmetry behavior, and interactions with fields. If all these relations were removed, physics would have no remaining criterion by which to identify the electron. The electron’s identity is exhausted by the role it plays in the whole structure.
A hypothetical lepton-antilepton universe helps test the limits of the relational view. A universe containing exactly one lepton and one antilepton may have globally balanced quantum numbers, such as zero total electric charge or zero total lepton number. But global cancellation does not mean absence of structure. The particles exist as distinct physical states because of their relational asymmetry: their positions, states, and behavior relative to one another within the total physical structure. Localization and distinguishability are not isolated essences hidden inside the particles themselves. They are features of the relational network in which the particles appear. Objects exist relationally within the universe; the universe exists as the total relational structure.
The relational picture raises the question of what remains invariant across different successful representations. A set-theoretic model uses membership as its primitive relation, but a category-theoretic or type-theoretic model may use different primitives. If all these formalisms describe the same physical universe, then the true content cannot be the specific symbol $\in$ or the empty set itself. The better candidate for reality is whatever remains unchanged across all adequate representations: symmetries, conservation laws, causal structures, dynamical relations, and transformation behaviors. Such features are not merely labels used by one formalism. They are structural relationships that any successful representation must encode.
Gödel’s incompleteness theorem introduces a limit when such invariant structure is captured in a formal theory. If a mathematical theory of the universe is consistent, recursively axiomatizable, and powerful enough to express basic arithmetic, then it cannot prove every truth expressible in its own language. That limitation does not mean physical reality itself is unstable, incomplete, or broken. The universe may remain fully determinate. The limitation belongs to the formal description, not necessarily to reality. An axiomatizable model cannot be identical to perfection if perfection means the totality of all truths. Truth in the universe and provability within a specific formal system remain distinct.
The gap between syntax and truth matters most where reality is not merely described, but experienced. Suppose the relational structure of the universe reaches a threshold of complexity where subjective, first-person experience emerges. A physical theory may chart every relation, every state transition, and every functional interaction within the system. Even then, such a theory may fail to capture the qualitative fact of what it is like to be that structure. Gödel’s theorem does not prove that consciousness is unprovable. It only shows that truth and proof can come apart in any sufficiently powerful formal system. Consciousness may be one of the places where the difference between formal description and lived reality becomes impossible to ignore.
A structural variant of the hard problem follows from that possibility. The difficulty may not be merely that current science lacks the right instruments or enough information. The difficulty may be that a formal representation of physical relations does not automatically yield the first-person reality of experience. From the first-person point of view, experience is given before theory begins. A model may describe the physical conditions under which consciousness appears, but describing those conditions may not be the same as deriving the subjective truth of experience from formal primitives. The hard problem may therefore mark a boundary between structural representation and qualitative presence.
The entire argument turns on a layered distinction. Mathematical structures can be reduced to formal primitives, but those primitives are not ordinary objects inside the theory. Physical objects can be treated as relational positions, but the universe as a whole is not an object inside a larger container. A complete reality may be fully determinate, but any consistent, arithmetic-capable axiomatizable model of that reality will leave some truths unreached by proof. Consciousness may be true even if a formal model cannot exhaust what that truth means from within its own syntax. The resulting view is neither simple formalism nor simple realism. It is a structural view constrained by incompleteness: the universe may be a pattern of relations, but no formal representation of that pattern can automatically claim to be its final perfection.