In his doctoral dissertation, Turing proposed the concept of Ordinal Logic in response to the limitations revealed by Gödel’s incompleteness theorem: for any consistent formal system S, its own consistency statement (e.g., \text{Con}(S)) cannot be proven within the system itself.

Turing hoped to overcome this incompleteness by constructing stronger logical systems, thereby extending the expressive power and decision range of formal logic.

To this end, he introduced the Oracle Turing Machine, a theoretical model capable of querying an “oracle” to solve certain undecidable problems. This machine could help formal systems handle propositions of the following form:

where R is a recursive relation. Turing aimed to construct a system in which such Q₂ (two-quantifier) structured problems could achieve a form of “partial completeness.”

Mapping Bitcoin’s Consensus Mechanism to Q₂ Structures

We can map Turing’s logical structure to Bitcoin’s decentralized consensus design as follows:

• x = \text{tx}: a transaction;
• y = \text{block}: a block that includes the transaction;
• R(x, y): denotes “transaction tx is included in block y, and block y is part of the longest chain.”
  

Thus, Bitcoin’s “double-spending problem” can be formalized as:

where R is a recursive relation based on the longest chain rule.

Oracle Turing Machine Relativity and Decentralized Arbitration

Miners in Bitcoin can be seen as relative Oracle Turing Machines:

• Each miner, based on their locally visible blockchain state, uses the proof-of-work (PoW) mechanism to evaluate and extend the longest chain;
• These judgments are relative and asymmetric, yet network-wide cooperation converges toward a consistent chain state;
• The resulting “longest chain” becomes the factual standard for transaction validity across the network.
  

This is equivalent to a transfinite recursive, dynamically evolving consensus logic, which is precisely the kind of “external oracle” Turing envisioned as an engineering construct.

Summary: From Logical Completeness to Engineering System Design

The transaction verification and chain selection processes in the Bitcoin protocol essentially follow the logical structure:

to determine whether a transaction is valid and confirmed in the longest chain.

Where:

• x = \text{tx}
• y = \text{block}
• R: recursive verification mechanism based on the longest chain
  

Ultimately, under the condition of no centralized trust source, Bitcoin achieves valid arbitration of arbitrary transactions through distributed “oracle behaviors” (miners). This can be regarded as a “relative” and “engineered” realization of Turing’s logical completeness concept for Q₂ systems.