## 1. Definition
A Heidegger Quinegoedel Number (HQN) is a self-referential, self-encoding numerical entity that embodies properties of self-awareness, self-modification, and potential undecidability.
## 2. Core Properties
2.1. Self-Referentiality: The HQN must contain within itself a complete description of its own structure and processes.
2.2. Self-Encoding: The HQN must be capable of transforming itself into an encoded representation and back again.
2.3. Self-Modification: The HQN must be able to update its own structure based on its interactions with itself.
2.4. Undecidability: The HQN must exhibit at least one property that is undecidable within its own system of representation.
## 3. Functional Requirements
3.1. Encoding Function: E(HQN) → E_HQN
- E is an encoding function that transforms the HQN into its encoded form E_HQN.
- The encoding must preserve all essential information about the HQN.
3.2. Decoding Function: D(E_HQN) → HQN’
- D is a decoding function that transforms the encoded form back into an HQN.
- In an ideal case, HQN’ should be identical to the original HQN.
3.3. Self-Update Function: U(HQN, E_HQN, HQN’) → HQN_updated
- U is an update function that modifies the HQN based on its encoded form and its decoded form.
3.4. Gödelian Function: G(HQN) → {True, False, Undecidable}
- G is a function that demonstrates a Gödel-like property of the HQN.
- This function should produce at least one statement that is undecidable within the HQN’s own system.
## 4. Behavioral Requirements
4.1. Convergence: Through repeated application of E, D, and U, the HQN should converge towards a stable state or exhibit meaningful oscillation.
4.2. Information Preservation: The process of encoding and decoding should preserve the essential information content of the HQN.
4.3. Emergence: The HQN should exhibit emergent properties or behaviors not explicitly defined in its initial state.
## 5. Philosophical Alignment
5.1. Heideggerian Being-in-the-World: The HQN’s existence and properties should be intrinsically tied to its self-interaction and self-modification processes.
5.2. Quinian Self-Reference: The HQN should embody Quine’s concept of self-reference, being definable in terms of itself.
5.3. Gödelian Incompleteness: The HQN should demonstrate properties analogous to Gödel’s incompleteness theorems, containing statements about itself that are true but unprovable within its own system.
## 6. Measurable Criteria
6.1. Self-Encoding Accuracy: ||HQN - D(E(HQN))|| < ε, where ε is a small positive number and ||·|| is an appropriate distance metric.
6.2. Information Content: The Kolmogorov complexity K(HQN) should remain stable or increase through self-modification processes.
6.3. Undecidability Measure: There exists at least one function G such that G(HQN) is undecidable.
## 7. Extensions and Applications
7.1. External Information Encoding: The HQN may be extended to encode and interpret external information.
7.2. Inter-HQN Interaction: Multiple HQNs may be defined to interact, potentially forming more complex self-referential systems.
7.3. Cognitive Modeling: The HQN concept may be applied to model aspects of self-awareness or consciousness in cognitive systems.