The Logical Core of Bitcoin (II): Evolution Mechanism of Ultra Poor Iteration and Decentralized Trust introduction In the previous article, we revealed the "dual layer logic" of the Bitcoin system: The transaction verification layer handles computable problems, similar to Turing machines; The consensus reaching layer needs to determine the problem, similar to the oracle Turing machine. This article will explore a deeper question: how does this system steadily evolve over time? Why can it keep moving forward instead of falling into logical paradoxes or dead loops? The answer is hidden in formal logic through a concept called 'super poor iteration'. 1、 G ö del incompleteness and the limits of formal systems G ö del's incompleteness theorem tells us: In any sufficiently powerful uniform formal system, there always exist propositions that cannot be proven or falsified That is to say, formal systems cannot solve all problems on their own, they always have "blind spots". The solution is to expand the system. This is exactly the research direction of Turing's doctoral thesis in 1938: If a proposition cannot be judged in system S, can we add it as a new axiom to the system and generate S'? Can we construct a 'supersystem' by constantly repeating this process? Turing's answer was yes, and he proposed a transfinite iterative model that uses * * ordinal * * to label the evolutionary hierarchy of the system. 2、 Ordinal logic and super poor evolution path The core idea of ordinal logic is: We can follow the natural number order of 0, 1, 2 System extension version numbers: S, S1, S2; After reaching a certain limit (such as ω), we can define S ω as the "union" of all previous systems; Continuing construction: S ω+1, S ω+2,..., S ω · 2,..., S α (where α is any recursive ordinal). This process can theoretically continue infinitely, representing the constantly increasing ability of human logical systems. 3、 How does the blockchain structure of Bitcoin correspond to this logic? Returning to Bitcoin, we will find that the growth structure of blockchain happens to be an infinite process of "system judgment → new state addition → entering the next round of deduction": Initial system: including initial UTXO set and genesis block; Event determination: The network selects a new block to join through the PoW mechanism; System extension: The transactions contained in this block become irreversible chain history; Start a new round: Based on the current chain head, the next round of transaction collection and mining will begin. This is a concrete engineering implementation of a super poor logical structure. 4、 Q2 structure: a unified computable and decidable evolutionary framework From this perspective, we can establish the following structural model for the Bitcoin system: Q1: Turing structure - deterministic logic for processing transactions; Q ω: Oracle structure - handling uncertain judgments of consensus; Q α (α>ω): Chain evolution structure - each new block is a "reincarnation" of the system, sealing the past state as a "new rule basis" and entering the next iteration. We refer to this structure as the 'Q2 structure': a logical system formed through infinite iteration by combining the Turing machine (deduction) with the oracle machine (judgment). ** This model not only explains how Bitcoin constructs trust mechanisms from basic computing, but also why it can evolve over time and continue to operate without relying on external referees. conclusion We use formal logic tools to gradually reveal the deep structure behind Bitcoin: Computable problems are processed by Turing machines (transaction verification); The judgment problem is handled by the oracle mechanism (consensus reached); The overall system is iteratively extended through a super poor ordinal structure (blockchain evolution). This is a system self growth logic that does not require a central referee. It explains the root of Bitcoin's "decentralized trust" and provides a logical paradigm for us to understand other distributed systems in the future. Bitcoin is not only a financial innovation, but also a great experiment in computational philosophy. Note: This series of articles is based on Turing's doctoral thesis "Systems of Logic Based on Ordinals" and supplemented by Solomon Feferman et al.'s research on formal system evolution, attempting to establish a decentralized logical framework that integrates computation, determinism, and evolution.