The Becoming of Possibility: Russell’s Paradox through the Lens of Relational Ontology: The cut that creates the impossible set

Bertrand Russell’s paradox is usually taken as a crisis at the foundations of logic — a sign that naïve set theory collapses under its own expressive power. The infamous “set of all sets that do not contain themselves” appears to force an impossible oscillation: if it contains itself, then by definition it cannot; if it does not, then by definition it must.

But beneath this familiar formulation lies a deeper philosophical assumption — one so ingrained that its consequences are rarely examined. The paradox arises not because of sets, nor even because of self-reference, but because the very notion of a set-as-thing is treated as if it could stand outside the act of construal that constitutes it.

Once we re-frame the entire issue through relational ontology, the paradox dissolves — not by restricting set formation, but by exposing how the representational assumptions behind the “naïve set” misconstrue the nature of systems, instances, and the perspectival cut that binds them.

1. The representational mistake built into “the set of all sets”

The classical paradox depends on two assumptions:

Sets exist as objects independent of perspective.
They are “there” as definite entities.
Membership is a determinate fact about those objects.
A set either contains itself or does not.

These assumptions are inherited from a metaphysics in which:

the world consists of things,
classification is a matter of labelling those things, and
the labelling remains stable independent of perspective.

Russell’s paradox only bites if we accept this framework.

But relational ontology rejects it outright.

In a relational ontology:

there are no mind-independent sets,
the system of possible classifications is itself a theory (a structured potential),
and the act of forming a class is an instantiation — a perspectival actualisation, not the naming of a pre-existing object.

Once that shift is made, the ground on which Russell built the paradox simply disappears.

2. Classes as relational potentials, not representational containers

Let us recut the terrain.

In relational ontology:

System = structured potential, a theory of what can count as an instance.
Instance = a perspectival cut through that system, actualising one possible event or configuration.
Construal = first-order phenomena, the meaning realised through that cut.

A class — any class — belongs to the systemic potential.

Its “members” belong to the domain of instances, but only within a given construal.

This means:

There is no such thing as “the set of all sets” independent of the cut that forms it.
There is no determinate question “Does this set contain itself?” until a construal is made.
And crucially: no construal can include itself without shifting the frame that enables the construal in the first place.

Which is precisely why Russell’s formulation collapses.

He attempts to form a class from outside the frame that constitutes classes, then asks a membership question that simultaneously belongs within the frame.

This is analogous to:

trying to draw a boundary around all boundaries but forgetting that drawing the boundary creates a new boundary;
or asking whether the definition of “definition” satisfies itself without specifying the perspective from which “definition” is construed.

In relational terms:

Russell’s paradox is a failed attempt to instantiate a system from a viewpoint that presupposes itself outside the system.

3. The cut that cannot be cut: why the paradox arises

The key move in Russell’s reasoning is treating the “set of all non-self-membered sets” as if:

it could be defined without choosing a perspective,
that definition could be evaluated from the same perspective, and
the evaluative step would not change the space of possible construals.

But in relational ontology:

the definition of the class is one cut,
the evaluation of membership is another cut,
and there is no frame-invariant standpoint that can perform both without altering the system/instance relation.

So the supposed paradoxical set is not a contradictory object.

It is a perspectival impossibility.

The cut that defines the class cannot be the cut that evaluates its membership conditions without collapsing the system/instance distinction on which the act of classification depends.

Or more concisely:

Russell’s paradox attempts to perform an instantiation that requires two incompatible perspectival positions at once.

The contradiction is not in logic; it is in the impossible viewpoint the paradox presupposes.

4. Why the paradox dissolves without restricting set formation

Traditional responses — type theory, ZF axioms, cumulative hierarchies — “solve” the paradox by banning the problematic formation rule.

Relational ontology solves it by showing:

the paradox does not arise in the first place
once sets are seen not as representational objects,
but as relational potentials actualised through cuts.

There is no need to forbid anything.

The “set of non-self-membered sets” can be construed — as a systemic pattern.

But the very moment one attempts to instantiate it, the frame must be chosen, and the paradoxical construction evaporates because:

the instantiation cannot evaluate its own systemic conditions without shifting frames.

This is precisely parallel to Gödel’s incompleteness results, but with a difference:

Gödel shows that no sufficiently rich formal system can internalise all truths about itself.
Russell’s paradox shows that no set-forming construal can internalise the membership conditions that presuppose a shift in perspective.

Both arise from the same relational point:

A perspective cannot fully contain itself.

Or:
No cut can simultaneously include the frame that makes the cut possible.

5. What becomes of “sets” under the relational re-cut?

They are not objects.

They are modes of organisation of potential within a theoretical system.

Every set belongs to a relational field whose boundaries are constituted through the act of construal. The moment one tries to build a “global set” — a totality that includes all sets including the one forming the totality — one is attempting to:

instantiate the entire system as one of its own instances.

This is ontologically incoherent.

You cannot cut the whole system with a cut that is itself part of the system being cut.

And thus the paradox is not a contradiction, but an artefact of trying to flatten the stratification between:

theory (system),
event (instance),
and phenomenon (construal).

Conclusion: Russell’s paradox as a lesson in relational humility

Russell’s paradox teaches nothing about contradiction in logic.

But it teaches a great deal about perspective, construal, and the impossibility of stepping outside the relational conditions of meaning.

In the relational ontology, the paradox is:

not a flaw in set theory,
not a bug in logic,
but a performative mistake —
an attempt to occupy two incompatible positions in the system/instance relation at once.

The “set of all non-self-membered sets” is not contradictory.

It is simply unconstructible under any single perspectival cut.

And that impossibility marks not a limit of mathematics,

but a deeper limit:

No act of classification can classify itself without changing what it is.

Or, stated in the idiom of your ontology:

A system cannot be instantiated from the perspective of its own totality.

The cut cannot cut the cutter.

The Becoming of Possibility

Saturday, 15 November 2025

Russell’s Paradox through the Lens of Relational Ontology: The cut that creates the impossible set