Gdelian Ontology of Interaction: Toward a Mathematical Framework for Existence as Relational Verification
Abstract
This paper proposes a novel ontological framework that synthesizes Gdel's incompleteness theorems with relational theories of existence. Traditional metaphysics often conceives "being" as self-subsistent, whereas this study argues that existence is inherently incomplete when understood as a closed formal system. Drawing on Gdel's insight---that no sufficiently expressive formal system can prove all truths within itself---we argue that "being" likewise contains undecidable existential propositions when isolated. We propose that interaction functions as the mechanism by which these gaps are closed: an entity attains actual existence only through relational verification by another entity. Using a formal mathematical schema, we define existence as a system SSS, its Gdelian gap G(S)G(S)G(S), and interaction I(S,S)I(S, S')I(S,S) as the process of extending systems to reduce incompleteness. This approach reconceptualizes ontology as a dynamic, intersubjective, and mathematically modelable field, offering implications for both metaphysics and the philosophy of science.
Executive Summary
This article develops a rigorous theoretical framework where existence is incomplete in isolation but becomes actualized through interaction. The key contributions are:
1. Reframing Ontology via Gdel:
Being-as-a-system (SSS) mirrors formal systems in mathematics, inherently incomplete.
Every entity harbors undecidable truths (G(S)G(S)G(S)) that cannot be resolved internally.
2. Interactivity as Resolution:
Interaction (I(S,S)I(S, S')I(S,S)) between entities allows for a partial closure of incompleteness.
Existence is thus relational and intersubjective, verified by others.
3. Mathematical Formalization:
Formal definitions of systems, Gdelian gaps, and interactions are introduced.
Existence Actualization Condition:
E(S)=1S such that G(S)<G(S)E(S) = 1 \iff \exists S' \text{ such that } G(S^*) < G(S)E(S)=1S such that G(S)<G(S)
4. Philosophical Significance:
Moves beyond Heideggerian and Buberian relational ontology by introducing Gdelian rigor.
Provides a bridge between analytic formalism and continental phenomenology.
Opens potential applications to systems theory, artificial intelligence, and cosmology.
This framework---Gdelian Ontology of Interaction---offers a new direction for contemporary metaphysics, reframing existence not as static substance but as dynamic relational verification.
OutlineÂ
1. Introduction
Background: Ontology, relational being, and Gdel's incompleteness.
Research gap: lack of rigorous formalization of relational ontology.
Objective: to propose a mathematical ontology of interactive existence.
2. Theoretical Foundations
Review of Gdel's incompleteness theorems.
Survey of relational ontology (Buber, Heidegger, Levinas, Whitehead).
Motivation for a synthesis.
3. Framework of Gdelian Ontology of Interaction
Assumptions: Being as closed system.
Gdelian gaps in existence.
Postulates of interactive ontology.
4. Mathematical Formalism
Definitions: system SSS, truth set R(S)\mathcal{R}(S)R(S), theorem set T(S)\mathcal{T}(S)T(S).
Gdelian gap G(S)=R(S)T(S)G(S) = \mathcal{R}(S) \setminus \mathcal{T}(S)G(S)=R(S)T(S).
Interaction as function I(S,S)SI(S, S') \mapsto S^*I(S,S)S.
Existence Actualization Condition.
5. Philosophical Implications
Ontology as relational verification.
The incompleteness of solitary being.
Inter-subjectivity and transcendence.
6. Discussion
Comparisons with phenomenology, process philosophy, and analytic metaphysics.
Relevance to AI, systems theory, and cosmology.
7. Conclusion and Future Directions
Summary of contributions.
Open questions for further research.
Toward a general theory of interactive metaphysics.
I. Introduction
A. Background: Ontology, Relational Being, and Gdel's Incompleteness
