In the eyes of many, Bitcoin is a decentralized monetary system. But if we delve deeper into its structure, we find it is far more than that.

Bitcoin is a system that generates time through computation and uses time to decide undecidable problems. It does not operate within time, but instead constructs trustworthy time itself. This design is derived from the core ideas proposed by Dr. Turing in his paper “Systems of Logic Based on Ordinals”, and is implemented in engineering as a triple-system structure: Formal System → Turing’s Ordinal Logic System → Axiomatized Time System

1. Formal System: Precise, but with a Single Undecidable Problem

Bitcoin’s transaction layer (TX) is a Turing-computable formal system. It can handle all user business logic with 100% precision:

• Signature verification
• Script execution
• Balance checking
• UTXO tracking
  

These operations are fully computable under the Turing machine model, without any ambiguity. However, the formal system has one problem that cannot be solved internally:‍

The Double-Spending Problem: When the same UTXO is used in two transactions, the system must determine which one was broadcast first and which one is valid. But due to asynchronous networks and local information, this determination is undecidable within the formal system.

2. Ordinal Logic System: Converting Double-Spending into Forking Problem

Bitcoin does not resolve double spending at the transaction layer. Instead, it adopts a structural strategy: converting undecidable transaction conflicts into blockchain-level forks. Two miners may each package conflicting transactions into different blocks, causing a chain fork. The problem becomes: “How does the system determine which chain is real?”

This judgment is also undecidable. However, Bitcoin introduces a second system to handle this structural uncertainty: Turing’s Ordinal Logic System.

In his doctoral thesis, Turing proposed:

• When facing undecidable problems, formal systems must introduce oracles;
• Oracles enhance system capacity via “intuitive judgments”;
• To improve system stability, transfinite iteration must be used to approach completeness;
• The evolution of difficulty can be seen as an increase in Turing degree.
  

In Bitcoin:

• The “longest chain” becomes the de facto oracle;
• Mining difficulty adjustments reflect transfinite iteration;
• The entire chain’s evolution is a continuous process of axiomatic progression.

3. Axiomatized Time System: Physically Anchoring the Oracle

But the oracle itself needs a temporal foundation—otherwise, the “longest chain” is just a heuristic and cannot serve as a basis for judgment. To this end, Bitcoin introduces a third system: a dissipative-structure-based “Axiomatized Time System.”

Specific implementation:

• Each block must complete an irreversible double-hash (SHA256) computation;
• Each block must meet a difficulty target, requiring brute-force attempts by real computing power;
• The timestamp of every new block is “time” bought through the computational cost of the hash’s dissipative structure;
• This is the “natural time” generated by the system’s self-evolution.
  

This time has the following characteristics:

• Irreversibility (hash computation is irreversible)
• Precision (difficulty adjustment ensures 10-minute block intervals)
  

It forms Bitcoin’s “physical foundation” — a verifiable, axiomatizable, globally consistent time system.

4. The Closed-Loop Structure of the Three-Layer System

The entire Bitcoin system forms a nested structure of a “decision machine”:

Decision path: Double-spending problem (undecidable) → Transformed into chain fork selection (undecidable) → Judged by the time system (axiomatizable) → Reverse confirmation of transaction validity

5. Key Term Mapping Table

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6. Conclusion: Bitcoin Is a Machine That Uses Time to Resolve Undecidable Problems

We can redefine the essence of Bitcoin: Bitcoin is not a Turing-complete language system, but a Turing-axiomatized time system. It is not a system that runs in time, but one that constructs time through computation—and uses time as the ultimate arbiter. Time is no longer a background condition, but a causal order constantly “written” by computing power.

Bitcoin has created three things:

1. A formal system that is undecidable in only one problem;
2. An ordinal logic system that continually approaches completeness;
3. An irreversible time system axiomatized from physical evolution.
   

These three form the ontological structure of Bitcoin as a Meta-Formal System.