Authenticity, completeness, and Bitcoin: a practical example of Turing ordinal logic 1、 The goal of axiomatization shifts towards the issue of completeness Hilbert's formal plan aims not only to construct formal logic, but also to achieve "axiomatization": any proposition that is logically true can be deduced through a first-order system. This goal ultimately boils down to the 'decision problem', which is how to determine whether a proposition is provable within the system. However, G ö del's incompleteness theorem states that in any sufficiently strong first-order system, there exist propositions that cannot be determined as true or false by the system itself. These propositions are neither decidable nor fully implementable. Therefore, the goal of achieving completeness based on judgment has been proven to be unattainable. The question then turns to: how to handle undecidable propositions? Is there a higher-order reasoning framework to handle these logical blind spots? 2、 The completeness depends on the basis of determinacy The completeness must be based on the basis of judgment. Deterministic problems can be seen as solving the question of whether they are valid through some mechanism; And completeness requires that all true propositions can be systematically derived. Without the ability to effectively judge truth and falsehood, completeness cannot be discussed. The implementation of completeness does not rely on computability itself, but on the structure of "super poor iteration": continuously introducing stronger rules (or oracles) on top of decidability, merging computability and decidability, gradually approaching completeness. Turing proposed the oracle Turing machine and ordinal based systems precisely to solve this problem: to recursively enhance the decidability of formal systems to a stronger level. 3、 The logical foundation of Satoshi Nakamoto's structure: from binary logic to super poor combination The Bitcoin system can be seen structurally as an evolutionary system based on determinism and approaching completeness. It includes two basic subsystems: Turing computable parts (such as transaction scripts and execution logic) The decidable part (such as the longest chain selection rule determined by the proof of work mechanism) These two subsystems are organized into a super poor logical structure based on natural ordinal numbers through continuous operation and verification over time. Its combination form is not accidental, but a logical construction method dominated by ordinal numbers. This system reflects the true structure of an informal system: the evolution process of the system that continuously approaches completeness through iterative judgment and calculation. Its logical structure can be expressed by the following formula: (∀x)(∃y)R(x,y) Among them, x represents input (such as transactions), y represents response (such as mining), and R is a recursively verifiable relationship. 4、 The Conversion Relationship between Authenticity and Probabilistic Safety In the Bitcoin system, 'authenticity' does not rely on absolute judgment, but rather on converting certainty through cumulative probability security. This transformation transforms "absolute safety" into "relative safety under system rules, where the probability approaches 1 as time and computing power advance. Authenticity is built on the ordinal structure of logical systems, and its stability is supported by continuous logical structures and evolutionary mechanisms. Authenticity can be seen as an approximation of completeness, but its essence is different from the "repeatable truth" in logical form. The meaning of authenticity here is closer to the inherent consistency between nature and system structure - it is the structural stability that evolves from the system, is self consistent, and does not rely on the central judge. This authenticity cannot be simply replicated, nor can it be simplified or approximated. 5、 Engineering Implementation of Logical Structure: An Example of Bitcoin Bitcoin can be seen as an engineering implementation example of Turing ordinal logic. This system combines "Turing computability" with "oracle determinacy" and completes super poor iterations through ordinal logic structure, thus achieving an "almost complete" evolutionary system at the engineering level. The system has the following characteristics: No need for central arbitration: rely on distributed consensus to achieve state confirmation. Structural evolution: adaptability is achieved through chain structure and difficulty adjustment. Authenticity determination ability: the practical implementation of relying on probabilistic security mechanisms to solve the determination problem. Natural order: Recursive consistency in structure, reflecting logical characteristics guided by ordinal numbers. This system does not reflect the traditional "truth system", but rather a "authenticity mechanism" aimed at structural stability and continuous evolution. 6、 Summary Judgment is a prerequisite for completeness. Completion can be seen as the result of approximation guided by oracles and ordinal numbers. Turing provided an evolutionary logic for dealing with undecidable problems through oracle machines and ordinal logic. The Bitcoin system is a successful implementation example of this logical system in the fields of engineering and economics. This structure demonstrates the possibility of a logical physical dual system, where the undecidability of mathematics and the self-organizing behavior of real systems are unified in the same structure.