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 a distributed computation completed independently by each miner, and the result is a Boolean value indicating whether the block meets the workload requirements. 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 judgment can be abstracted as a first-order predicate: IsLongestChain (Chain X), whose result is either "true" (the longest chain) or "false" (not the longest chain). 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'. When miners successfully find a nonce that meets the difficulty requirements and broadcast a new block, other nodes will independently verify the validity of the block. Only verified blocks are considered valid and may be used to extend the known longest chain. The formation of overall consensus follows a core logic: participants in the network tend to continue mining and confirming transactions on the longest valid 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. 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. The 'proof' here is not a formal logical deduction, but rather refers to the probability that the longest chain will surpass any other competing chain through the competition and accumulation of computing power. 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 is a necessary foundation for forming effective blocks, ensuring that only blocks with sufficient computing power are accepted and contributing to the growth of the chain. The consensus on the first-order predicate 'whether it is the longest chain' ensures that the network will eventually tend towards a unique and widely supported historical version with computing power. 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.