Turing, Penrose, and Bitcoin: A Bridge of Ordinal Logic between Intuition and Trust abstract This article starts from Turing's 1939 proposal of "ordinal logic" and "oracle Turing machine", and combines Penrose's reinterpretation of "cautious oracle" to explore in depth the tension between formal systems and human cognition. We point out that Turing broke down mathematical reasoning into two abilities: "ingenuity" (rule deduction) and "intuition" (non mechanical judgment), while Penrose further proposed the "cautious oracle" to simulate the credibility of human judgment. By analogy between the "counting" behavior and the mechanism of Bitcoin miners, the article reveals how ordinal logic serves as a bridge between computable judgments and verifiable processes. 1、 Turing's insight: the split between ingenuity and intuition In his doctoral thesis in 1939, Turing first broke down mathematical reasoning into two essential abilities: Ingenuity: refers to the ability to rely on formal rules for reasoning, manifested as the gradual deduction of Turing machines. Intuition: refers to the non mechanical ability in the human mind to directly determine whether a statement is true without relying on the chain of reasoning. Turing believed that true intelligent systems cannot be limited to "ingenuity". When humans engage in mathematical activities, they often need to break out of the rule system and make judgments such as' this can be used as the next axiom 'or' this is an acceptable construct '. These judgments cannot be derived from existing rules, but require a higher-order cognitive ability - intuition. 2、 Ordinal logic: leaving space for intuition Faced with the limitations of G ö del's incompleteness theorem on formal systems, Turing proposed the attempt of * * ordinal logic * *: Arrange the formal system in ordinal order α \ alpha, and construct a series of increasingly strong systems S α S \ alpha; The construction of each layer S α S \ alpha depends on identifying the formula ϕ α \ phi_ \ alpha that is "suitable for promotion" from the previous layer; And the action of 'recognition' cannot be completed by existing rules, it must be done by intuition. In other words, every transition point in ordinal logic is the point where intuition intervenes. This is where Turing was profound: he did not deny the role of formal systems, but acknowledged that to surpass them, external non mechanical judgments must be relied upon for intervention. 3、 Oracle Turing Machine: Formal Expression of "Intuition" To model this judgment behavior beyond the capabilities of Turing machines, Turing introduced the famous Oracle Turing Machine: During its operation, it can call an "oracle" and ask it a question that a regular Turing machine cannot solve (such as a shutdown problem); Oracle is like a "black box judge" who can directly return "yes" or "no"; In terms of the model, the oracle provides relative computability, which means that we can still perform certain calculations relative to an uncomputable set. This setting has been misunderstood by many as a 'universal machine', but Turing's intention was not to create omnipotence, but to allow formal systems to 'assume' the existence of some judgment in order to continue constructing more complex systems. 4、 Penrose and the Oracle of Caution: Incorporating Trust into the Scope of Discussion Penrose reinterpreted the idea of oracle machines in "The Once and Future Turing" and proposed a model that is closer to human behavior patterns - Cautious Oracle: Cautious oracles may not always provide answers; When faced with uncertainty, it can remain silent or fall into a state of 'continuous effort'; Once it outputs an answer, the answer must be trustworthy and genuine. Penrose defined the three possible actions of the oracle as follows: D (g)=true \ mathcal {D} (g)=\ text {true} D (g)=False \ mathcal {D} (g)=\ text {False} D(g)=? \mathcal{D}(g) = ? , or enter an infinite "loop" to find the answer This' waitable but trustworthy 'mechanism simulates the cautious strategy of human experts when faced with complex judgments, and also echoes the attitude of' inconclusive 'in scientific exploration. It does not rely on omniscience and omnipotence, but on sustained effort and the establishment of limited trust. 5、 Bitcoin miners: the 'cautious oracle' in reality In the Bitcoin system, miners play a role that is very similar to the Oracle of Caution: Miners judge the validity of each transaction (such as whether it is double spending); Continuously attempt to find acceptable solutions through Proof of Work; Once the solution is found, it is broadcasted to the entire network and verified and trusted by other nodes. This is a consensus model for informal system internal judgment mechanisms: The Turing Penrose model sets the Bitcoin system as a cautious oracle, Cautious Oracle miner, and miner judgment mechanism that allows for "thinking", silence, experimentation, output of proof of work, broadcasting, and verification of trustworthiness standards. The output must be "trusted to be true". The blockchain has the longest chain and node confirmation, and the reasoning foundation is relatively computable and ordinal logic. Distributed computing and incentive mechanisms are not easily output in the face of incompleteness; The output is acceptable but unpredictable, and continuous attempts are made to approach the correct value Bitcoin miners are exactly the kind of imperfect yet reliable judgmental mechanism in reality, which is a distributed engineering implementation of the "cautious oracle". 6、 Intuition as an Example in Counting This kind of 'intuitive judgment' is not just philosophical talk. Give a common example: When we say 'counting the fifth apple', it may seem like a mechanical process, but in reality it involves non mechanical judgment - we must first determine that 'this is an apple' before we can include it in the counting range. This judgment does not come from a strict definition, but from the ability of experiential perception and conceptual induction. This intuition for recognizing objects of the same kind is precisely what Turing believed could not be mechanized. Turing had already realized that formal systems could not cover all the sources of "judgments" in the real world. This is his effort through ordinal logic and oracle construction - to leave a bridge for human judgment in a purely rule-based world. 7、 Conclusion: From uncomputable to verifiable trust path Turing, Penrose, and the Bitcoin system have depicted a common structure from the perspectives of logic, cognitive science, and engineering practice, respectively: We do not deny incompleteness and incomputability; But we can construct a mechanism that strikes a balance between trust and verifiability in judging behavior; The core of this mechanism is to provide space for formal models of intuition and introduce informal judgment capabilities through external devices such as oracles and miners; Ultimately, establish a judgment path that transcends the rule system - not absolute truth, but credible truth. This is the most profound echo of Turing's ideas extending to contemporary times.