‍Introduction

In Satoshi Nakamoto’s whitepaper, Bitcoin is not merely a financial innovation, but a reorganization of computational logic structure. Its core problem is not “currency issuance,” but rather— How to establish a reliable temporal order under the condition of no trusted third party.

This means that Satoshi had to construct, within a peer-to-peer system, a time structure that requires no central authority for synchronization yet can prove itself. And the answer he found was the dual fusion of two open theories: Turing’s ordinal logic system and Nash’s non-cooperative game.

I. Starting Point: The Gap in Peer-to-Peer Systems

Satoshi’s starting point is extremely simple: Without a trusted third party, a peer-to-peer transaction system has only one unresolved gap — the double-spending problem.

“Double spending” refers to the possibility that the same funds could be used twice in the absence of a global temporal order. Thus, the essence of the problem is not account security, but the verifiability of time order: How can each transaction possess a unique temporal position in the global system?

II. Inspiration: The Temporal Framework of Timestamp

For this purpose, Satoshi referred to the timestamp system (Trusted Timestamping) proposed by Haber and Stornetta in the 1990s. He cited four of their papers in the whitepaper — these four works constitute a complete theoretical framework for trusted logging systems.

The timestamp system proposed two anti-forgery schemes:

• ‍Absolutely Trusted Time Source (Trusted Time Authority)‍
  
  • Using authoritative time servers (such as NIST or Surety) to provide unified time.
  • This is practical in engineering but still essentially relies on centralized trust.

• ‍Decentralized Distributed Time Proof (Distributed Timestamping)‍
  
  • Theoretically feasible, but there was no reliable implementation at the time.
  • Satoshi cited these papers not to reuse their centralized engineering implementations, but to borrow their formal logical framework.

Timestamping provides the prototype of anti-forgery and sequential verification in formal logic, but it lacks a decentralized driving mechanism, and that became the direction of Satoshi’s breakthrough.

III. Satoshi’s Breakthrough: Decentralized Time Witnessing

Satoshi transformed the timestamp framework into an “open system of temporal logic”, completing the leap to decentralization through two key ideas:

• ‍Irreversible Hash Computation (Hash + Nonce)‍
  
  • Time is no longer given by an authority; it is generated through computation itself.
  • Each successful Proof of Work represents an irreversible “time event.”

• ‍Open Network with Free Entry and Exit‍
  
  • Any node can participate in PoW; the system spontaneously forms competition.
  • Each node’s computational behavior is individually rational, yet globally converges into a unique longest chain.

This mechanism creates a self-proving temporal order:

Time is no longer granted—it is computed.

In other words, the Bitcoin system makes time a computable existence. This is the logical revolution Satoshi achieved on top of the timestamp framework.

IV. The Fundamental Logic of Time Structure: Turing × Nash

The time system Satoshi created is not merely an engineering design, but the fusion of two open logical systems: Turing’s Ordinal Logic System × Nash’s Non-Cooperative Game System

The two are logically dual: one provides formal structure, the other provides dynamic evolution.

1. Turing’s Ordinal Logic System: The Formal Generation of Time

In his 1938 doctoral thesis “Ordinal Logics,” Turing proposed that:

The completeness of a formal system can be extended step by step through transfinite iteration.

Each iteration appends new proof witnesses on the basis of the previous level’s derivations, thus forming a computable time structure expanded along ordinals.

Bitcoin’s blockchain directly corresponds to this logical form:

• Block height = ordinal number;
• Each block is a logical witness to the previous state;
• The blockchain is an open iterative sequence system with no final halting point;
• It continuously generates new “completeness.”

Thus, Turing’s logic endows time with formal computability.

2. Nash’s Non-Cooperative Game System: The Dynamic Mechanism of Time

If Turing’s system defines the logical structure of time, then Nash’s system defines the driving dynamics of time. In Nash’s non-cooperative game, each participant seeks their own optimum, but the entire system naturally converges toward a stable equilibrium (Nash Equilibrium).

In Bitcoin:

• Miners independently compete for nonces;
• Each computation is a self-interested act;
• Yet the whole network jointly forms the globally unique sequence of time (the longest chain).

This means:

The evolution of time is an entropy-reducing process driven by competitive games.

The non-cooperative game provides the kinetic foundation for temporal flow, enabling the entire system to self-organize toward a unique consensus.

3. Duality and Fusion: The Logical Life of Time

In Satoshi’s design, these two systems achieve perfect coupling:

From the logical perspective: Turing provides the form of time, while Nash provides the dynamics of time. When the two meet, Bitcoin’s time system becomes a self-organizing formal logic — a computational time unfolded by ordinals and driven by games.

V. Summary: A Formally Complete Solution to the Double-Spending Problem

Thus, Satoshi provided a formally complete solution to the core question of the whitepaper — double spending:

Temporal Order = Turing-style Logical Sequence × Nash-style Game Dynamics.

It satisfies the timestamp’s requirement for verifiable time, while achieving dynamic evolution under decentralized conditions — self-generated time. In other words:

Bitcoin’s anti-forgery mechanism relies not on trust, but on time; and time relies not on authority, but on the interplay of computation and game.

VI. Extension: From Currency to Credit

This framework of decentralized time witnessing is not limited to monetary systems. Any system requiring temporal order and anti-forgery recording — such as peer-to-peer credit scoring, reputation systems, or scientific provenance — can be constructed on the same logical framework.

Because credit, in essence, is also a temporal trust structure: each behavioral event is witnessed in time and forms a global consensus through logical sequencing.

Epilogue

From this perspective, Satoshi’s innovation does not lie in the word “blockchain,” but in the fact that:

He made time into a computable logical entity.

This chain of time co-constructed by Turing and Nash, is the first decentralized logical system ever realized in engineering by humanity— it organizes order in ordinal form,

maintains flow through the energy of games, and for the first time in history, generates a verifiable and irreversible time in our world.