Introduction

In the information age, humanity needs a decentralized and trustworthy computational system more than ever. Bitcoin is not merely digital money—it is the first engineered system in human history capable of generating global consistency and temporal order without any central authority.

How does it achieve this? The core lies in its unique logical structure.

Bitcoin is a complete system driven by an ordinal logic system and non-cooperative games.

• The ordinal logic system solves: “How does the system distinguish truth from falsehood?”
• The non-cooperative game (PoW) solves: “How does the system converge to a unique conclusion?”
• The peer-to-peer network unifies both, realizing a decentralized mechanism for proving global time.

I. The Limitations of Traditional Distributed Systems

Traditional distributed systems rely on consensus protocols (such as Paxos, Raft, PBFT), which are essentially finite-state machine replication mechanisms aimed at achieving consistent states among nodes.

The problems are:

• Trust assumptions: Depend on fixed members and access control → cannot allow open participation
• Trust units: Rely on institutional maintenance → require centralized trust
• Scalability: Cannot remain secure in open networks → unable to form global consensus

Bitcoin’s breakthrough lies in this: it does not ask “who decides,” but redefines “what is true.”

II. From Turing Consistency to Ordinal Logical Completeness

The Turing machine system is the foundation of modern computability theory, pursuing formal system consistency—but it is constrained by Gödel’s incompleteness theorem:

Any sufficiently complex formal system cannot prove its own completeness.

In his 1939 doctoral thesis “Systems of Logic Based on Ordinals,” Turing proposed an extended path: Through oracles and transfinite recursion (ordinal recursion), a formal system can break its closure and approach completeness.

The flaw of ordinal logic: it cannot guarantee convergence to a unique conclusion—the constructed proof chain may diverge infinitely.

III. Bitcoin: Making Ordinal Logic Executable

Bitcoin introduces three core structures:

More importantly, it incorporates the Nash non-cooperative game mechanism, i.e., Proof of Work (PoW):

• Maps computational cost to economic incentives
• Reaches convergence through network competition toward the unique longest chain

Thus achieving: Turing Machine (verifiability) + Ordinal Logic (pursuit of completeness) + Non-Cooperative Game (convergence)= A Decentralized Global Truth Machine

IV. The Longest Chain: The Decentralized Oracle of Time

• Each block is both a state witness and historical proof.
• The blockchain is not merely a data structure, but a logical adjudication structure.‍
• Longest chain rule = Arrow of time
• Accumulated block difficulty = Physical entropy increase

Bitcoin constructs an unprecedented system: Mathematically based on ordinal logic, Game-theoretically grounded in Nash equilibrium, Physically embedded with an irreversible time computation structure.

That is why Bitcoin requires no trusted center or institution.

V. Why Is Bitcoin a Complete System?

Bitcoin is not a “monetary application,” but a new computational structure—the first time formal logic has achieved an engineered approximation to completeness.

Conclusion

Bitcoin is a logical revolution, not a financial one.

• Truth no longer depends on authority, but on structure.
• Time can be computed—without a central clock.
• Games can serve as the convergence engine of logic systems.

Consensus needs no rulers. Order needs no sovereignty. Trust needs no trust.

For the first time, Bitcoin allows humanity to see: Structure determines truth. Logic creates order.