‍Introduction

In the eyes of most people, Bitcoin is merely a blockchain technology of “trusting code.”

But in fact, Bitcoin’s logic goes far beyond the world of algorithms. It is a three-layered structure spanning computability, incompleteness, and game-theoretic completeness.

Only by understanding these three realms can one truly understand why Bitcoin is unique.

First Layer: The Algorithmic World of the Turing Machine (Trusting Code)

Theoretical foundation: Turing machine computability theory.

Functions implemented:

• UTXO transaction computation
• Nonce hash computation
  

Limitations: There exist two interfaces that cannot be self-proven within the Turing machine itself:

• Double-spending problem — cannot prove whether a transaction already has validity.
• Longest chain selection problem — cannot determine which chain will ultimately be accepted.
  

This layer is what the vast majority of people call “blockchain technology.” But this is only the starting point of Bitcoin.

Second Layer: The Ordinal Logic System (Gödel Boundary and Dialectical Logic)

Theoretical foundation: Gödel’s incompleteness theorem + Turing’s ordinal logic system (computable Turing machine + non-computable oracle machine).

Logical breakthroughs:

• Reducing the double-spending problem to a blockchain fork problem.
• Further reducing the fork problem to the longest chain selection problem.
  

Implementation: By combining oracle machines with transfinite iterative logic, local decisions are made at the incompleteness boundary.

However, problems still remain: Even so, the ordinal logic system still cannot guarantee the uniqueness of the longest chain.

This means the final step of decentralized convergence is not yet complete.

Third Layer: Nash Non-Cooperative Game (Decentralized Completeness)

Theoretical foundation: Nash equilibrium, non-cooperative game theory.

Core contribution: Bitcoin hands over the “longest chain uniqueness problem” left behind at the second layer to game theory to solve.

In a global network without centralization, nodes achieve consistent convergence through non-cooperative games.

Ultimately, a unique and irreversible time order is formed. This step is the true pinnacle of Bitcoin’s technology and philosophy.

It is not only a Turing machine, not only an incomplete logic system, but also a self-completing game machine.

Conclusion

The three realms of Bitcoin correspond to three types of logic:

• First Layer: The Turing machine algorithmic world (trusting code)
• Second Layer: The ordinal logic world (oracle + incompleteness decisions)
• Third Layer: The game theory world (decentralized completeness)
  

Bitcoin = a system unifying three layers of logic.

It spans computability → incompleteness → game-theoretic completeness, and finally realizes, for the first time in human history, a truly decentralized time order.