I. The Shift from Axiomatization to the Problem of Completeness

Hilbert’s formalist program aimed not only at constructing formal logic but also at achieving axiomatization—the idea that every logically true proposition could be derived from a first-order system. This goal ultimately distilled into the decision problem: how to determine whether a given proposition is provable within the system.

However, Gödel’s incompleteness theorem demonstrated that in any sufficiently powerful first-order system, there exist propositions that cannot be judged true or false by the system itself. These propositions are undecidable and make completeness unachievable. Thus, the original goal—achieving completeness based on decidability—was proven impossible.

The question then shifted: How can undecidable propositions be handled? Is there a higher-order reasoning framework to address these blind spots in logic?

II. Completeness Depends on the Foundation of Decidability

Completeness must be built upon the foundation of decidability. The decision problem can be seen as resolving whether a proposition “holds” using some mechanism; completeness, in contrast, requires that all true propositions be derivable within the system. Without effective means to judge truth or falsity, completeness becomes meaningless.

The realization of completeness does not rely solely on computability, but rather on the structure of transfinite iteration: continually introducing stronger rules (or oracles) atop decidability, merging computability with decidability to incrementally approximate completeness. Turing’s introduction of oracle machines and ordinal-based systems directly addressed this issue: recursively elevating the decidability of formal systems to stronger levels.

III. The Logical Foundation of Nakamoto’s Structure: From Binary Logic to Transfinite Composition

The Bitcoin system can structurally be seen as an evolutionary system based on decidability, approaching completeness. It contains two fundamental subsystems:

• Turing-computable components (e.g., transaction scripts and execution logic)
• Decidable components (e.g., the longest chain rule governed by proof-of-work)
  

These two subsystems, through ongoing operation and validation over time, are organized into a transfinite logical structure based on natural ordinals. This composition is not accidental but rather a method of logical construction guided by ordinal logic.

This system reflects a non-formal structure of reality: a process of iterative judgment and computation continuously approximating completeness. Its logical structure can be expressed with the formula: (∀x)(∃y)R(x,y)

where x denotes input (e.g., a transaction), y denotes response (e.g., mining), and R is a recursively verifiable relation.

IV. The Transformational Relationship Between Authenticity and Probabilistic Security

In the Bitcoin system, “authenticity” does not rely on absolute judgment but converts determinism into probabilistic security through accumulation. This transformation reframes “absolute safety” into relative safety that approaches probability 1, under the rules of the system as time and computational power progress.

Authenticity is built upon the ordinal structure of the logical system, and its stability is supported by continuous logical architecture and evolutionary mechanisms. Authenticity can be viewed as an approximation of completeness, but its essence differs from the logical concept of “reproducible truth.”

Here, authenticity refers more to the intrinsic consistency between nature and system structure—a system-emergent, self-consistent, decentralized structural stability. This authenticity cannot be merely replicated or simplified into an approximation.

V. The Engineering Realization of Logical Structure: The Bitcoin Example

Bitcoin can be seen as a practical implementation of Turing’s ordinal logic. The system combines Turing computability with oracle-based decidability, and through an ordinal logic structure, performs transfinite iteration, thereby realizing an evolutionary system that approximates completeness in engineering terms.

This system exhibits the following characteristics:

• No centralized arbitration: State confirmation is achieved through distributed consensus.
• Structural adaptability: Achieved via chained structure and difficulty adjustment.
• Truth evaluation capability: Practical resolution of decision problems via probabilistic security.
• Natural order: Recursive consistency in structure reflects ordinal logic properties.
  

What this system embodies is not a traditional “truth-value system,” but a mechanism of authenticity aiming at structural stability and continuous evolution.

VI. Summary

• Decidability is the prerequisite for completeness.
• Completeness can be seen as an approximation process guided by oracles and ordinals.
• Turing, through oracle machines and ordinal logic, provided an evolutionary framework to handle undecidable problems.
• Bitcoin is a successful practical realization of this logical system in the engineering and economic domains.
  

This structure reveals a dual logic–physical system possibility, where mathematical undecidability and real-world self-organization are unified within a single structure.