This article attempts to systematically explain the GEB theory structure composed of “Turing Machine + Oracle Machine + Ordinal Logic” through an integrated perspective of philosophy and computational theory, and proposes its real-world embodiment in Bitcoin's design by Satoshi Nakamoto. This structure can also be considered a form of a real-world Complex Adaptive System (CAS).

I. The Three Computational Structures of GEB Theory

We understand the GEB theory as an "organic computer" structure composed of the following three types of systems:

1. Turing Machine System: A rational deduction system based on formal logic, handling Shannon information, emphasizing determinism and computability.
2. Oracle Machine System: Originating from undecidable problems, it introduces external arbitration (e.g., physical processes), corresponding to non-formal experience-based judgments.
3. Ordinal Logic System: The "super recursive iteration" system proposed by Turing to address incompleteness, accepting contradictions and driving evolution, representing the meta-rationality of the system.

These three systems work together to form an adaptable, evolving structure with nonlinear and unpredictable characteristics — the computational foundation of a Complex Adaptive System (CAS).

II. Philosophical Lineage: From Descartes to Hegel

The three computational architectures in GEB theory also have clear correspondences in philosophical traditions:

• The Turing Machine embodies Descartes' rationalism of "I think, therefore I am" and Leibniz's ideal of a "universal deductive system."
• The Oracle Machine inherits Hume's empiricist intuitionism, emphasizing unpredictable judgments from outside the system. Bergson introduced "time" into the philosophical system, advocating for "lived time." Prigogine explicitly stated, "Systems without time structures are false."
• The Ordinal Logic System responds to Kant’s paradox of "noumenon of reason" and is sublimated in Hegel's philosophy of "contradiction as drive." Turing formalized it as the super-recursive iteration system to combat Gödel-style incompleteness in mathematical logic.

III. Bitcoin: The Real-world Projection of GEB Theory

The Bitcoin system designed by Satoshi Nakamoto is the first real-world embodiment of the GEB three-system structure.

1. Turing Machine Layer: Transaction System and Script Validation
   Bitcoin's scripting language has Turing incompleteness, which precisely limits it to the realm of "decidable formal systems," corresponding to the Turing machine structure. All transaction verification rules are formalized, locally decidable logic.
2. Oracle Machine Layer: PoW Mechanism and Timestamp Arbitration
   Double-spending attacks are undecidable problems that the Turing machine itself cannot resolve. In Nakamoto's design, an external dissipative structure arbitration mechanism is introduced through “Proof of Work + the longest chain across the network + timestamps,” which is the engineering expression of the Oracle Machine.

• Each nonce attempt is a form of physical dissipation.
• Hash collisions serve as probabilistic arbitration.
• Timestamp records ensure a globally irreversible time structure.

3. Ordinal Logic Layer: Irreversible Chain Evolution
   The birth of each new block is an irreversible evolution of the previous block’s state. The entire blockchain system is not a closed, complete system but rather a process of continual evolution when facing new contradictions (attacks, forks, time competition) — this is precisely the "contradiction-expansion-recompletion" dynamic structure described by Turing's "ordinal logic system."

IV. Philosophical Formula for Complex Adaptive Systems

We can summarize the unified structure of GEB + Bitcoin with the following philosophical formula:

• Turing Machine = Rational Computation
• Oracle Machine = Irrational Arbitration
• Ordinal Logic = System Evolution after Accepting Contradictions

Bitcoin = Rational × Oracle × Super-recursive Evolution = Organic Complex Adaptive System (CAS)
It is not a closed software system, but a real-world "philosophical machine" embedded in time, with physical structure, capable of self-adaptation and evolution.

V. Further Reading and Research Path

To deeply understand the above system, the following literature is recommended:

• Turing: On Ordinal Logic Systems
• Prigogine: Order from Chaos
• Gödel: General Theory of Formal Systems
• Hegel: Phenomenology of Spirit
• Heidegger: Being and Time
• Bergson: Creative Evolution
• Nakamoto: Bitcoin White Paper

VI. Conclusion: Bitcoin as a Living Philosophical Entity

If you want to redesign a system like Bitcoin that is indestructible, trustless, and naturally evolving, then:

• You cannot just learn programming languages.
• You cannot just copy the whitepaper.
• You must understand the boundaries of computation, the reality of time, and the inevitability of contradictions.

Because Nakamoto's craft was not simply writing code, but using the language of mathematics and physics to transform "Hegel's Phenomenology of Spirit" into a real-world, irreversible time entity.

Bitcoin is not a program; it is existence.