The discourse on Bitcoin by GEB is rooted in deep principles of theoretical computer science, particularly Gödel’s incompleteness theorems, Turing’s theory of computation, and the P ≠ NP problem in complexity theory. Its core ideas are as follows:

1. Critique of the limitations of single formal systems:

Traditional “single computable formal systems” (such as blockchain under the universal Turing machine model) are considered inherently incomplete. This means they cannot fully capture or express the richness and dynamism inherent in “complex adaptive systems” in the real world. If blockchain technology is viewed merely as a simple computable system, it inherently lacks the capacity for perception and interaction with reality. This incompleteness is a key reason why it struggles to fully “land” in practical applications.

2. Bitcoin’s transcendent architecture:

Bitcoin is portrayed as a system that transcends the traditional blockchain paradigm. It is not confined to a single computable model, but rather is constructed as a “PH (Polynomial Hierarchy) three-layer transfinite iterative system” based on the P ≠ NP conjecture. The key to this multilayered structure is that each layer exhibits the P ≠ NP characteristic—solutions can be verified in polynomial time (P-class problems), but finding those solutions is difficult (NP-hard problems). More importantly, this three-layer structure is considered “non-collapsing,” maintaining its inherent complexity and layered independence.

These three layers correspond specifically to:

• UTXO (Unspent Transaction Output) layer: The private key management and signing process of UTXOs is viewed as an NP-hard problem, with its security grounded in the complexity of asymmetric cryptography.

• POW (Proof-of-Work) layer: The process of miners computing a new block nonce that meets the difficulty requirement is computationally NP-hard, involving extensive trial and error and verification.

• Longest chain selection layer: In a decentralized network, the process of judging and selecting the “longest chain,” especially under conditions of forks and asynchronous propagation, is viewed as a complex, non-trivial NP-hard decision problem.
  

3. Oracle Turing machines and the emergence of complexity:

To handle these NP-hard problems (or the intertwining of “incomputable problems” and “computable problems”), this theory introduces the concept of Turing’s “oracle Turing machine.” The oracle here plays the role of providing solutions to NP-hard problems in a non-computational manner (like a “black box”) when needed, thereby enabling the system to verify solutions on the P side.

Crucially, the three types of oracle Turing machines in Bitcoin are not centralized absolute oracles, but are “relative and decentralized”:

• UTXO oracle: Users’ private key operations and signing behavior constitute the resolution of NP-hard problems related to UTXO states.

• Miner oracle: Miners who successfully discover a valid nonce effectively serve as oracles providing solutions to the POW challenge.

• Longest chain oracle: The synchronization and broadcasting of different forked chains across network nodes ultimately converge into a consensus on the “longest chain,” functioning as a form of decentralized oracle judgment.
  

Through these three distinct NP-hard formal systems, Bitcoin achieves “emergence of complexity” via asymmetric, relatively adaptive interactions, leveraging the role of these decentralized oracle Turing machines. The core of this system is maintaining the “UTXO → Block → Longest Chain” three-layer structure in a “non-collapsing” state.

4. Transfinite iteration to maintain system stability:

To maintain this non-collapsing complex structure and enable the system to continuously adapt and evolve, Bitcoin employs a mechanism of “transfinite iteration.” This concept originates from Turing’s 1938 doctoral thesis Systems of Logic Based on Ordinals, which implies an iterative process that surpasses finite steps and continues indefinitely, ensuring the persistence of the longest chain and the overall robustness of the system.

In summary, in the GEB @BitAgere view, Bitcoin transcends the category of simple technology and is elevated to a complex adaptive system grounded in cutting-edge concepts of computation theory. Its design philosophy aims to overcome the inherent limitations of single formal systems, thereby achieving a deeper capacity for “perception” and interaction with the real world.