‍Introduction

Bitcoin is not only a decentralized currency protocol; it is a new system that touches the boundaries of computation theory and physics.

This article will help you understand: why the longest chain principle of Bitcoin goes beyond the scope of recursive computability, becoming a “super-formal system” that fuses computation theory, game theory, and relativistic time.

I. Recursiveness and the Boundary of Turing Machines

In computation theory, “recursive” usually means “computable.” All Turing machines (whether deterministic or nondeterministic) belong to the recursive category.

However, Bitcoin’s longest chain cannot be fully described by recursive algorithms:

• The nonce collision of a single block and difficulty adjustment are local NP characteristics.
• The overall longest chain selection, however, requires external observation and adjudication, and cannot be fully endogenous to the Turing machine model.
  

In other words: Bitcoin’s longest chain cannot be generated solely by recursive algorithms.

II. A New Perspective on NP-Completeness

In traditional discussions, NP-completeness is confined to the formal boundary of Turing machines, without incorporating the factor of “time.”

But if we introduce the axioms of time from physics, the situation changes completely:

• Turing machine NP-completeness: static, confined within the framework of computability.
• Quantum physics NP-completeness: time and uncertainty are fundamental axioms, with the constancy of light speed as the boundary.
  

Bitcoin’s design clearly belongs to the latter.

III. Global Time and the Longest Chain

In Bitcoin, the longest chain = the equivalent of global time.

• Local miners: cannot perceive global time, and can only make choices based on the current chain state.
• Global consistency: is jointly achieved through PoW difficulty adjustment and the longest chain principle.
  

This is highly similar to the clocks in Einstein’s theory of relativity: We cannot surpass the speed of light, but we can freely operate within its boundary. Satoshi’s genius lies in this—he used the longest chain to capture the “God’s clock.”

IV. Nash Equilibrium and Non-centralization

Since no one can master global time information, the best strategy is: the majority of hashpower follows the longest chain.

This brings about a Nash equilibrium:

• Individual miners maximize their returns.
• The overall system remains decentralized.
• No one can “dominate the market” and control the whole system in the long run.
  

In other words, Bitcoin, through game theory, exploits the unobservability of time to instead achieve global completeness.

V. Bitcoin and Turing’s Ordinal Logic System

In his doctoral thesis, Turing proposed the Ordinal Logic System to surpass the limits of recursive computation, introducing transfinite iteration and oracle processes.

Bitcoin’s structure is precisely a real-world mapping of this idea:

• The longest chain = a proxy for global time.
• It is not recursive computation, but a non-recursive, ordinal generative process.
• Each new block added is like one step of “transfinite iteration.”
  

Therefore, Bitcoin can be regarded as: an engineered implementation of the Ordinal Logic System.

VI. Conclusion

The longest chain principle of Bitcoin is not only the core mechanism of blockchain consensus, but also a structure that transcends recursive computability.

Relying on the cosmic axiom of light-speed constancy, it embeds time into the system, so that:

• under limited local perception,
• through game theory and Nash equilibrium,
• global consistency and completeness can be achieved.
  

Bitcoin is not just a currency protocol, but a “super-formal system” fusing computation theory, game theory, and relativistic time.

Final Note

If the Turing machine is humanity’s ultimate exploration of “computability,” then Bitcoin is humanity’s first attempt in the real world to embed incomputable time into a practically functioning system.

That is where its greatness lies.