Turing machine, oracle Turing machine and Bitcoin: a unified logical structure from computability to decidability introduction There is a profound logical channel between G ö del's incompleteness theorem and Turing's computability theory, connecting the computational boundaries of formal systems with the judgment limits of real systems. This article attempts to explore the computability of the UTXO model in the Bitcoin system from the perspectives of Turing machines and oracle Turing machines, and unify the "decidability problem" and "P/NP problem" into a broader logical framework based on ordinal numbers. 1、 Computability and Turing machine: the boundary of formal computation Turing's Turing machine, proposed in his 1936 paper, was the first clearly defined computability model that could formally describe all 'effectively computable' processes. In this sense: Turing machine=computable formal system; The UTXO construction process of Bitcoin, such as input references, signature verification, etc., is a mechanical calculation process that can be completed by Turing machines; The execution of scripting languages and the construction of transaction input-output relationships are all instances of 'Turing computability'. Turing machines handle problems that can be fully computed within formal systems. 2、 Ordinal numbers and oracle Turing machines: relative computability and decidability Turing introduced the concepts of "ordinal numbers" and "Oracle Machine" in his doctoral thesis in 1938. Unlike ordinary Turing machines, oracle Turing machines can make judgment requests to external "oracles", thereby solving the decidability problem that ordinary Turing machines cannot solve. The ordinal number provides a "judgment hierarchy" for the oracle Turing machine, allowing it to handle problems between different forms of systems; Oracle Turing Machine=Relatively computable system; Judgment problems (such as the double flower problem and the longest chain selection) fall into this category. The oracle Turing machine deals with the issue of how to relatively determine authenticity between multiple systems. 3、 The Logical Structure of Bitcoin System: Intersection of Calculation and Judgment The Bitcoin UTXO model can be seen as a formal system at the Turing machine level, and its operation is fully computable. However, the double flower judgment and longest chain induction that appear in the consensus mechanism belong to the problem of "informal systems", involving the comparison of multiple nodes and historical states, and require a higher-level "judgment mechanism": Problem type description corresponds to machine construction UTXO computability problem, Turing machine verification script computability problem, Turing machine dual flower conflict determination, relative determinability problem, oracle Turing machine longest chain induction judgment, induction selection of informal systems beyond the scope of Turing machines Therefore, the Bitcoin system is a "composite operating structure" of Turing machines and oracle Turing machines: Turing machines support the trading layer; The oracle Turing machine supports the consensus layer. 4、 From Ordinals to P/NP Problems: A Unified Perspective of Complexity Theory In classical complexity theory, P/NP problems have always been regarded as open problems. This article proposes a unified perspective: P/NP problems can be seen as complexity versions of decidable problems across different systems. Certainty issue: Is there an algorithm that can determine the authenticity of all inputs; P/NP problem: Is there a polynomial time Turing machine solution; Both essentially belong to the problems that can be solved by oracle spirit machines. Ordinal provides a logical tool for complexity stratification, unifying decidability and complexity judgments. 5、 Philosophical Reflection on G ö del, Sets, and Fundamental Problems Why did G ö del study set theory? Its goal is to unify reality and logic, that is, to construct the foundation of the world through a logicable "set". Set=conceptualized basic unit; Cardinality=represents the number of elements in a set and is the "unit" of a computable system; Ordinal numbers represent hierarchical relationships and are the "units" that can determine the system. For example: The cardinality of the set {apples, chestnuts, bananas} is 3, but its determination in the logical system (can it be fully consumed? Is it in order?) requires ordinal support. Conclusion: The Unified Landscape of the Logical World This article distinguishes between Turing machines and oracle Turing machines, unifying computational problems, decision problems, consensus problems, and complexity problems into a logical structure. This not only helps us understand the logical operating boundaries of real-world systems such as Bitcoin, but also provides a new path to re-examine P/NP, G ö del ordinals, and formal systems. Computable problems belong to Turing machines It can be determined that the problem belongs to the oracle Turing machine The P/NP problem is the specific expansion of the decision problem under complexity Bitcoin is a 'natural computing platform' that integrates Turing system and decision system