Incompleteness, Perspective, and Hybrid Systems: 2 Gödel’s System as a Theory of Its Instances

In Episode 1, we reframed incompleteness: it is not a flaw or epistemic gap, but a structural inevitability arising from the perspectival limits of any system attempting to capture its own instances.

Episode 2 situates Gödel’s formal construction inside relational ontology, showing that his “undecidable sentence” is not an obstacle but a feature—an illustration of what it means for a system to be a theory of its own instances.

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1. Systems as Structured Potentials

Consider a formal system, $S$:

• $S$ has axioms, rules, and derivations.

• Each derivation is an actualisation of the system’s potential.

• $S$ does not enumerate truths; it defines the space of possible instantiations.

In other words, $S$ is a structured potential, not a repository of pre-existing truths. Proofs are not discoveries; they are actualisations within the system’s relational structure.

Gödel’s genius was to make this structure explicit. The “Gödel sentence” is simply a potential within that structured space—one that the system cannot exhaust from its own internal perspective.

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2. The Undecidable Sentence as Instance

The classic formulation of Gödel’s sentence, $G$, says roughly:

“This sentence is not provable in system $S$.”

But relationally, $G$ is nothing more than:

• An instance in the system’s structured potential that cannot be captured by the system’s own operations.

• A necessary consequence of $S$ being both self-referential and consistent.

Key insight:

• $G$ is not a “truth outside the system” that the system fails to reach.

• $G$ is a perspectival inevitability: the system cannot act on it internally without violating its own rules.

No paradox exists. No “failure of proof” occurs. $G$ simply illustrates the boundary between instantiation and system-relative capture.

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3. Perspective and Relational Constraint

What makes $G$ undecidable is not a quirk of logic, but a fundamental relational constraint:

1. Internal perspective: $S$ operates using its own axioms and rules.

2. Instance-space: $G$ is part of the structured potential that (S\ defines.

3. Actualisation vs. system capture: $S$ cannot simultaneously perform all potential derivations and maintain self-consistency.

This is exactly the same structural form we saw in coordination and semiotics:

• Value systems cannot fully account for construal.

• Scaffolds can host semiotic operations, but cannot instantiate them alone.

• Hybrid systems require structural openness while maintaining local closure.

Gödel’s formal system is simply the most precise articulation of this general relational principle.

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4. Why Gödel Is Not About Knowledge

Relationally, the theorem is ontological, not epistemic:

• It does not tell us what humans can or cannot know.

• It does not expose limitations of our reasoning.

• It describes what a system can be, given its own relational constraints.

The “undecidable” is not a truth beyond reach—it is an uninstantiable potential from the system’s own viewpoint. Its existence is inevitable and explanatory, not problematic.

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5. Bridging to Perspective and Construal

Gödel’s theorem foreshadows two key relational concepts:

1. The perspectival cut: any system can only see some actualisations of its potential, not all.

2. Structural openness: the “outside” is not metaphysical, but relationally necessary for instantiation.

These are exactly the mechanisms that underlie:

• Construal in semiotic systems

• Hosting via scaffolds

• Hybrid systems where coordination and semiotics co-exist

In short, Gödel’s system provides a formal template for thinking about relational incompleteness, long before any semiotic or biological interpretation.

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6. Looking Ahead

Episode 3 will extend this insight:

• We will argue that incompleteness is ontological, not epistemic.

• Even idealised reasoners cannot “complete” their systems.

• The relational limits revealed by Gödel are necessary for all actualisation, including meaning, coordination, and hybrid systems.

The key takeaway of Episode 2 is simple:

Gödel sentences are not obstacles, paradoxes, or failures.

They are instances that any system cannot exhaust from its own internal perspective, demonstrating the fundamental relational structure of incompleteness.