Russell’s paradox reveals a contradiction embedded within the first set-theoretical axiom of math.

The unrestricted axiom of comprehension in set theory states that to every condition there corresponds a set of things meeting the condition: (∃y) (y={x : Fx}). The axiom needs restriction, since Russell's paradox shows that in this form it will lead to contradiction.

In mathematical logic, Russell's paradox is a set-theoretic paradox published by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. Wikipedia

Let R = {x | x ∉ x}, then R ∈ R ⇔ R ∉ R

This OP will argue that the contradiction embedded in the unrestricted axiom of comprehension is not a fatal flaw disfiguring the relationship between math and set theory, and that unrestricted comprehension of set membership is achievable, albeit in stages.

Premise – paradox equals inconsistent equality

Premise – a paradox is a signpost pointing towards higher-order dimensional expansion

Premise – unrestricted comprehension is conditionally limited within a series of well-defined contexts

Premise – every complex of inter-connected dimensions generates paradoxes

For my argument by example, I will use our everyday world of three spatial dimensions.

Unspecifiable existence, in its effort to get itself into measurability, expands. At the level
of zero dimensions – the singularity – axiomatic disequilibrium sparks time, energy, motion and space into expansion beyond the unspecifiably-existent singularity.

Unspecifiable existence, in its effort to get itself into measurability, disappears. This is the first paradox within our universe of three empirical spatial dimensions.

Singularity-as-point. ⇔ Unspecifiable existence. Zero dimensions.

First Paradox: Point-as-line ⇔ The singularity is there and not there. One dimension.

Second Paradox: Line-as-arc converges with 360 degree sphere. ⇔The line as circle. Two dimensions.

Third Paradox: Parallelogram-as-hemisphere converges with diameter. ⇔ The circle as sphere. Three dimensions.

Fourth Paradox: Cube as time-zero expanding universe converges with hypercubic space. ⇔ The sphere as hypercube (Russell’s Paradox). Four dimensions.

The dimensionally-expanded universe is an open, bounded infinity that graduates in steps with paradox binders acting as the stitching.

In the context of open, bounded infinity configured in graduated steps with paradox binders, logic is a continuity of motion-as-empirical-narrative that evolves toward the self-reference of a borderline with next upward step of expanded dimensionality.

Paradox tells us we’ve reached a logical limit wherein a higher-order dimension in collapsed configuration populates the paradox.

Expansion of the higher-order dimension resolves the paradox.

When Frege and Russell confronted the paradox inherent within the unrestricted-comprehension axiom of set theory, they were gazing upon the infra-structure of an expandable universe configured in steps.

• Let R = {x | x ∉ x}, then R ∈ R ⇔ R ∉ R (The set of all sets not members of themselves)

• With upward dimensional expansion to hypercubic space, the paradox is resolved because hypercubic space, being bounded by cubic space, logically occupies with consistency at 4D what is paradoxical at 3D. In other words, hypercubic space, when viewed through a 3D lens, occupies two places at once.

They were looking at metaphysics itself.

The metaphysics of a step-hierarchical universe is physical transcendence in steps bounded by paradox.

Russell’s paradox understood as paradox-to-be-articulated as expansion of a collapsed higher-order dimension becomes a defense of physical transcendence and therefore of transcendental metaphysics.

Transcendental Logic ⇔ Transcendental Metaphysics