The three Liora stories — The Puzzle That Couldn’t Finish Itself, The Infinite Caterpillar, and The Both-Path Tortoise — operate as a narrative enactment of Gödel’s Incompleteness Theorem, reframed through the lens of relational ontology. They provide a concrete, semiotic terrain in which the abstract relational principles of system, instance, and perspectival cut can be experienced intuitively.
1. System and Instance in Narrative Form
Across the stories, we encounter structured potentials — puzzles, caterpillars, branching paths — each corresponding to a system in relational terms. These are networks of possibility, capable of producing instances but never reducible to the sum of their instances.
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The Puzzle demonstrates that any system leaves a residual element outside its own structure. The “missing piece” enacts Gödel’s undecidable statement: a potential that cannot be internalised within the system.
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The Infinite Caterpillar shows that each act of interaction with the system (counting its segments) produces an actualisation, an instance that extends the system. The caterpillar’s elongation mirrors the relational principle that actualisation is perspectival and cannot exhaust potential.
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The Both-Path Tortoise embodies undecidability at a structural level: multiple branches exist simultaneously, but any given perspective can only traverse one. The system of paths exceeds the capacity of any single instance.
In each case, the story-world demonstrates the relational distinction between system-as-potential and instance-as-actualisation — the very distinction that Gödel’s theorem makes mathematically.
2. Perspectival Cuts and the Unfinishable
A key insight from the Liora stories is that the system cannot contain all its own instances: it is always perspectival. This is enacted narratively:
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The MirrorFox’s shadow cannot be stepped into, enacting the impossibility of a system fully containing its own self-reference.
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The Infinite Caterpillar grows as it is counted, showing that any act of “observation” or “instantiation” produces new potential outside the prior system.
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The Both-Path Tortoise exists in multiple paths, each of which can only be actualised from a perspective, ensuring incompleteness is inherent to the system itself.
The “unfinishable” or “incomplete” nature of the systems in the stories reflects Gödel’s diagonal cut, where a system confronted with its own potential must leave some truths undecided, external to itself.
3. Potential and Actualisation
Relational ontology interprets the stories as follows:
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Potential (system): the structured possibilities, the network of relationships that define what can occur.
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Actualisation (instance): the perspectival event that emerges when a system is enacted or observed.
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Instantiation vs. Realisation: the system-instance relation is one of instantiation, not mere representation; it is perspectival, not formal or symbolic.
In narrative terms:
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The puzzle is a potential landscape; each piece placed is an instance.
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The caterpillar is a potential being; counting it actualises new segments.
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The tortoise is a branching network; traversing one path produces an instance while leaving others unactualised.
The stories operationalise the relational principle that no potential can be fully actualised within itself, and every system necessarily exceeds its instances.
4. Gödelian Structures in Child-Friendly Form
The narrative strategies used in these stories encode Gödelian relational structures without explicit reference to mathematics:
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Residual Elements: always one piece outside the puzzle, a segment beyond counting, an untraversed path — expressing incompleteness.
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Self-Reference: the MirrorFox, the caterpillar’s counting, and the tortoise’s coexisting paths enact the system confronting its own potential.
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Open Potential: the stories maintain a sense of becoming; the world is never closed, mirroring the relational ontology’s commitment to the impossibility of unconstrued completeness.
These narrative devices translate formal incompleteness into lived experience: the reader (or child) sees, feels, and imagines the relational principles at work.
5. Conclusion: Stories as Relational Demonstrations
The three Liora tales provide a semiotic mirror of Gödel’s theorem:
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They show that systems are structured potentials, not repositories of facts.
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They demonstrate that actualisation is perspectival and cannot exhaust potential.
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They reveal that incompleteness is not a flaw, but the condition for ongoing possibility and becoming.
In doing so, the stories accomplish what mathematical exposition alone cannot: they give the intuition of relational incompleteness, rendering abstract ontology palpable and magical. Liora becomes a guide through the landscapes of system, instance, and cut — inviting readers to perceive possibility as ever-emergent, never fully closed, and always generative.
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