In the foundational stage of computer science, Alan Turing’s groundbreaking paper “On Computable Numbers, with an Application to the Entscheidungsproblem” established a rigorous mathematical definition of computability. By introducing the Turing machine as an abstract model, Turing created a deterministic computational tool capable of executing any algorithmic process. This model not only laid the theoretical foundation for modern computer design, but also ushered in an era of computation centered on mechanization and deterministic logic. However, the essence of the Turing machine lies in its determinism and preset rules, which reveal inherent limitations when facing unpredictable, adaptive complex systems.
Before Turing’s work, Kurt Gödel’s incompleteness theorems had already shown that any formal system encompassing arithmetic cannot simultaneously be consistent and complete. This means that within any sufficiently powerful formal system, there will always exist true propositions that cannot be proven or disproven. Gödel’s discovery posed a profound challenge for the future of computer science: how can we build systems capable of handling more complex, non-deterministic problems under the premise of intrinsic incompleteness?
It was within this context that Turing, under the mentorship of Alonzo Church during his doctoral studies at Princeton University, shifted his research focus from pure computability to broader questions of decidability. His doctoral dissertation, “Systems of Logic Based on Ordinals,” did not aim to directly “solve” Gödel’s incompleteness, but rather to explore a new methodology for constructing logical systems that could asymptotically approach completeness.
In his dissertation, Turing introduced the concept of the Oracle Machine, marking an extension of the deterministic boundaries of traditional Turing machines. An oracle machine can be defined as an abstract component capable of instantly answering membership questions for a specific set, independent of any internal computation process. This “non-computational” ability enabled Turing to explore problems of relative computability beyond the conventional computational domain.
Furthermore, Turing employed transfinite induction as a mathematical tool, combining oracle machines with transfinite ordinals. Through this method, he constructed a hierarchical logical system allowing for the transfinite iterative approximation of decidability. While this method did not yield a “complete” system in Gödel’s sense, it provided a theoretical framework for extending the boundaries of decidability within a specific context—that is, by iterating beyond limits and potentially introducing external “sources of truth,” the system’s “knowledge” or “believability” could be enhanced.
The mechanisms used by Satoshi Nakamoto to construct Bitcoin as a decentralized digital currency system structurally resemble the ideas embedded in Turing’s oracle machines and transfinite methodology. At its core, Bitcoin operates on a blockchain—a distributed ledger composed of consecutive blocks. Each block contains a set of transactions and is cryptographically linked to the previous block, forming an immutable chain.
The Longest Chain Rule as a Consensus Mechanism:
In the Bitcoin system, the longest chain rule functions as a consensus mechanism. It is not dictated by any predetermined, centralized entity, but instead emerges as the consensus result of decentralized competitive computation (proof of work) across the entire network. When a fork occurs, all nodes ultimately choose the longest chain—the one with the most accumulated proof of work—as the valid chain. This dynamic, adaptive consensus mechanism provides a way to establish a shared “truth” in a trustless environment.
Miner Behavior as an Oracle Mechanism:
Miners who produce new blocks that extend the longest chain can be understood as decentralized oracle Turing machines. In the process of searching for a nonce that satisfies the proof-of-work condition, miners do not need to share each individual trial or intermediate result with other peers. Only when a miner successfully finds a nonce that meets the difficulty requirement—thereby creating a valid block—is that block broadcast (i.e., “oracled”) to the entire network. Other miners and nodes verify the block’s validity and extend the longest chain accordingly. This mechanism ensures that the miners’ exploration is private and non-deterministic, yet the final result is verifiable and collectively accepted—akin to an oracle’s output.
Transfinite Growth of Transaction Believability:
The validity of a transaction within a block gains systemic believability as subsequent blocks are added. This can be seen as a transfinite growth pattern in transaction confidence. A single block confirmation provides only initial validation of a transaction. As more blocks (confirmations) accumulate, each transaction’s “depth” and “irreversibility” increase non-linearly. Although it is theoretically impossible to reach 100% certainty in the mathematical sense (since chain reorganizations are always possible), in practice, once the confirmation count reaches a certain threshold (e.g., six confirmations), the transaction’s confidence level becomes sufficiently high to be regarded as “final” in economic activities. This model of increasing “believability” through transfinite iteration and accumulating “evidence” corresponds to Turing’s concept of approximating decidability via transfinite ordinals.
From the deterministic computability represented by the Turing machine, to the limitations of formal systems revealed by Gödel’s incompleteness theorems, and then to Turing’s exploration beyond determinism through oracle machines and transfinite methods—these developments trace the evolutionary trajectory of computational paradigms in response to complex systems. Bitcoin, through its decentralized consensus mechanism—especially the longest chain rule and the “transfinite” accumulation of transaction believability—demonstrates how, in an unpredictable and adaptive environment, distributed computing and cryptography can be leveraged to construct a system with high believability.
Bitcoin provides a real-world solution to the deep computational and logical challenges posed during the era of Turing and Gödel. It shows that, even in the context of incomplete formal systems, it is possible to build systems capable of handling complexity and achieving a high degree of “believability” through clever design, transfinite iteration, and dynamic consensus.
