From Turing's' computability 'to Bitcoin's' determinacy': An analysis of the evolution of computational paradigms In the foundational stage of computer science, Alan Turing's groundbreaking paper "On computable numbers and their applications in decision problems" established a rigorous mathematical definition of computability. By introducing the abstract model of Turing machine, Turing constructed a deterministic computing tool capable of executing any algorithmic process. This model not only laid a theoretical foundation for the design of modern computers, but also opened a computing era with mechanization and deterministic logic as the core. However, the essence of Turing machines lies in their determinacy and pre-set rules, which make them inherently limited when dealing with unpredictable adaptive complex systems. G ö del's incompleteness theorem and the need for expanding computational paradigms Prior to Turing's work, Kurt G ö del's incompleteness theorem had already revealed that no formal system containing arithmetic could simultaneously satisfy consistency and completeness. This means that in a sufficiently powerful formal system, there are always true propositions that cannot be proven or falsified. G ö del's discovery posed a profound challenge for the subsequent development of computer science: how to construct systems capable of handling more complex, non deterministic problems in the presence of inherent incompleteness? It was in this context that Turing, under the guidance of his mentor Alonzo Church, shifted his research focus from pure computability to more macroscopic decidable problems during his doctoral studies at Princeton University. His doctoral thesis "Logical Systems Based on Ordinals" does not aim to directly "solve" G ö del's incompleteness, but rather to explore a new methodology for constructing logical systems that can infinitely approach completeness. Oracle Machine and Transcendence Induction: An Attempt to Transcend Certainty Turing introduced the concept of Oracle Machine in his doctoral thesis, marking an extension of the deterministic boundaries of traditional Turing machines. The oracle can be defined as an abstract component that can immediately answer membership questions for a specific set, and its operation does not rely on any internal computational processes. This' non computational 'ability enabled Turing to explore relative computability problems beyond traditional computability categories. Furthermore, Turing utilized the mathematical tool of Transfinite Induction to combine oracle machines with Transfinite Ordinals. Through this method, he constructed a hierarchical logical system that allows for over limit iterative approximation of the decidable nature of the problem. Although this method does not construct a "complete" system in the sense of G ö del, it provides a theoretical framework for extending the decidable boundary in specific contexts, that is, improving the system's "knowledge" or "certainty" through over limit iterations and possibly introducing external "sources of truth". Bitcoin: A Trustworthy Adaptive System The mechanism adopted by Satoshi Nakamoto in building the decentralized digital currency system of Bitcoin bears structural similarities to the ideas contained in Turing's oracle and transcendental methodology. The core of Bitcoin's operation is blockchain, a distributed ledger composed of continuous blocks. Each block contains a set of transactions and is connected to the previous block through a cryptographic hash value, forming an immutable chain. The 'Longest Chain Principle' as a consensus mechanism: In the Bitcoin system, the Longest Chain Principle functions as a consensus mechanism. It is not determined by a predetermined, single entity, but rather a consensus result that naturally emerges through decentralized competitive computing (proof of work) across all network nodes. When a fork occurs in the network, all nodes will ultimately choose the longest chain, which accumulates the most proof of work, as the valid chain. This dynamic and adaptive consensus mechanism provides a means to establish shared 'truth' in a trustless environment. Miners' behavior and oracle mechanism: The behavior of miners who generate the longest chain of new blocks can be understood as a decentralized oracle Turing machine. Miners do not need to pass on the calculation process or intermediate results of each attempt to other peer miners throughout the entire process of searching for * * random numbers (nonce) * * that meet the proof of work conditions. Only when a miner successfully finds a random number that meets the difficulty requirements and constructs an effective block, will the block containing a specific random number be broadcasted (i.e. "oracle" given) to the entire network. Other miners and nodes accept this block by verifying its validity, thereby extending the longest chain. This mechanism makes the exploration process of miners private and non deterministic, but the final result is verifiable and widely accepted by the network, like the output of an oracle. The 'over limit' growth of transaction confidence: The validity of transactions in each block will increase with the increase of subsequent blocks, and the system confidence will also increase. This can be seen as a pattern of excessive growth in transaction confidence. The confirmation of a single block only provides preliminary transaction validity. As the number of subsequent blocks (confirmations) continues to accumulate, the "depth" and "irreversibility" of each transaction increase nonlinearly. Although it is theoretically impossible to achieve 100% certainty in the mathematical sense (as chain recombination is always possible), in practice, when the number of confirmations reaches a certain threshold (such as 6 confirmations), the confidence level of the transaction has reached a very high level, making it considered as the "final confirmation" in economic activity. This pattern of enhancing "certainty" through over limit iteration and gradually accumulating "evidence" corresponds to Turing's idea of approximating determinacy through over limit ordinal numbers. conclusion From the determinacy and computability represented by Turing machines, to the formal system limitations revealed by G ö del's incompleteness theorem, and to Turing's exploration of beyond determinacy in oracle machines and transcendental methods, these constitute the evolutionary trajectory of computational paradigms in dealing with complex systems. Bitcoin, through its decentralized consensus mechanism, particularly the "longest chain principle" and the "over limit" accumulation of transaction confidence, demonstrates how to build highly reliable systems using distributed computing and cryptography in an unpredictable and adaptive environment. The practice of Bitcoin has provided real-world solutions to the deep computational and logical problems proposed during the Turing and G ö del eras. It indicates that in the context of incomplete formal systems, through clever design and the use of over limit iteration and dynamic consensus, a system that can cope with complexity and achieve a highly "confident" state can be constructed.