Logical self consistency of Bitcoin's longest chain consensus: completeness based on first-order predicates The core security mechanism of the Bitcoin blockchain lies in its longest chain consensus. The generation of new blocks relies on miners performing Proof of Work (PoW), which involves finding nonce values that satisfy a specific difficulty level, such that the double SHA-256 hash value of the block header is smaller than the target difficulty level (double_stha256 (block [once])<difficulty). This verification process is like employee A of the "stupid boss" working hard to calculate the nonce that meets the difficulty requirements, creating a new block. The calculation process is complex, but the result is simple for the verifier. However, this local verification process itself is not a logical judgment that can be directly described by first-order predicates. It involves calculations and numerical comparisons. What can truly be described by first-order predicates is the overall longest chain consensus of the Bitcoin network. For any given blockchain, nodes in the network will evaluate whether it is currently the longest valid chain known. This is like the job done by employee B of the 'stupid boss': verifying whether a block is valid, the answer is only' yes' or 'no'. The judgment of whether it is the longest valid chain can be abstracted as a first-order predicate: IsLongestChain (Chain X), whose result is "true" or "false". The longest chain of Bitcoin is not determined by centralized authority, but is formed through the common dependence of network participants on the 'longest effective chain'. After miners (such as A) broadcast a new block, other nodes (such as countless "Bs") will independently verify the validity of the block. Only verified blocks are considered valid and may be used to extend the known longest chain. This is like a foolish boss relying on B's "yes" or "no" verification results to determine whether to record new blocks in his "ledger". The formation of overall consensus follows a core logic: participants in the network (such as the foolish boss) tend to continue mining and confirming transactions on the longest effective chain they know. When a temporary fork occurs, honest miners occupy the vast majority of computing power and continue to add new valid blocks on the longest chain, causing the chain to continue to grow in probability and ultimately be regarded as a canonical chain by the entire network. Therefore, "whether it is the longest chain" has become a key logical judgment for the network to reach consensus, just like a foolish boss always chooses to continue recording on the effective "ledger" that he believes is the longest. From the perspective of logical completeness, a system is considered complete if it can prove all propositions that are true in its semantics. In the longest chain consensus of Bitcoin, the proposition that "the current valid chain with the highest cumulative workload is the only canonical history" can be "proven" probabilistically through continuous block generation and expansion in a network with sufficient honest computing power. This is like the "longest effective ledger" of a foolish boss, which can continuously record and represent the effective work results of the team under the premise of reliable verification by B. Any malicious attacker attempting to overthrow the longest chain requires sustained investment of resources exceeding the entire network's computing power, which is extremely difficult both economically and technically. Double_Sha256 (block [once])<difficult distributed verification (B's work) is a necessary foundation for forming effective blocks, ensuring that only blocks with sufficient computing power will be accepted and contribute to the growth of the chain. The consensus on the first-order predicate 'whether it is the longest chain' (the strategy of the foolish boss) ensures that the network will eventually tend towards a unique and widely supported historical version. This dynamic mechanism based on simple logical judgment and computing power competition constitutes the core of the security and reliability of the Bitcoin blockchain, and demonstrates its completeness in maintaining a single, trustworthy transaction history in practice.