I. Turing’s Breakthrough on Logical Completeness

In 1938, Alan Turing asked in his doctoral dissertation “Systems of Logic Based on Ordinals”:

“How can we go beyond Gödel’s incompleteness?”

To address this, he introduced two groundbreaking concepts:

• Ordinal Logic: Using external “ordinal induction” to enhance a system’s expressive power;
• Oracle Turing Machine: An abstract model capable of invoking “external sources of truth.”
  

This allows the system to handle problems of the form: (∀x)(∃y)R(x, y) —That is, “For all x, there exists a y such that the relation R holds.”

II. How Does Bitcoin Engineer “Oracle Logic”?

Bitcoin’s system validation structure is, in essence, a distributed implementation of “oracle behavior.”

Let’s define:

• x = a transaction tx
• y = a block block
• R(tx, block): the transaction is included in that block and on the longest chain.
  

Bitcoin’s core decision problem becomes:(∀tx)(∃block)R(tx, block). In other words: any transaction must be included in a valid block to be considered “valid.”

This mirrors Turing’s notion of “relative completeness,” but implemented as an engineering arbitration mechanism in Bitcoin:Miners act as oracle invokers, using PoW to determine the longest chain; Consensus becomes logical judgment, where all transactions ultimately settle on the chain.

III. Shared vs. Unshared: Where Are the Boundaries of UTXO?

In other words: we can use Bitcoin’s UTXO data structure, but cannot use its consensus chain to arbitrate other applications.

IV. Parallel Arbitration Systems: The Next-gen Oracle Turing Machines

We can construct a decentralized ecosystem like this:

UTXO --> BTC Transfer  
    --> BTC-Vote  
    --> BTC-ID  
    --> BTC-Copyright

They share UTXO state, but each has its own independent arbitration mechanism:

Each is a Turing-style decentralized arbitration device for solving a specific (∀x)(∃y)R(x,y) problem.

V. From Bitcoin to Complex Adaptive Systems

We call this structure:Parallel Oracle Turing Machine Systems —Each system solves its own version of the (∀x)(∃y)R(x,y) problem.

In the future, we can:

• Use BTC as the state source
• Use varied systems for decentralized arbitration
• Build a multi-semantic, multi-consensus decentralized social architecture

Conclusion

I. Theoretical Foundation: Oracle Logic in Turing’s Dissertation

In 1938, Turing in his doctoral thesis “Systems of Logic Based on Ordinals”:

• Proposed Ordinal Logic, attempting to transcend the limitations of formal systems revealed by Gödel’s incompleteness theorem;
• Introduced the Oracle Turing Machine, used to handle undecidable propositions of the following form:

(∀x)(∃y)R(x, y)

Here, R is a recursively decidable relation. This structure represents an attempt to extend formal systems at the Q₂ (double quantifier) level.

II. Bitcoin’s Engineering Realization: The Practical Landing of Oracle Logic

Bitcoin’s core issue is the double-spending problem, whose logical structure aligns with Turing’s Q₂ model:

(∀tx)(∃block) R(tx, block)

• tx: a transaction;
• block: a block containing the transaction and residing on the longest chain;
• R(tx, block): whether the transaction is validly included and confirmed.
  

Bitcoin’s miner system can be viewed as a “distributed oracle behavior system,” dynamically constructing and evolving the range of R.

III. Core Breakdown: Modular Structure of Bitcoin’s Arbitration System

‍Conclusion: UTXO is a universal state expression structure; Bitcoin’s PoW consensus mechanism can only arbitrate BTC transfers, and cannot be reused for other semantic applications.

IV. Proposing a New Paradigm: Shared UTXO + Parallel Consensus Systems

Key Idea: Share Bitcoin’s UTXO structure, but establish independent, parallel decentralized arbitration systems for each application.

Structure:

V. Application Examples: Constructing Parallel Systems

VI. Summary: UTXO as “Universal State Structure”, Parallel Systems as “Engineering Expression of Oracle Logic”

• ✅ Shareable: Bitcoin’s UTXO data structure, serving as the foundation of state and proof;
• ❌ Not Shareable: Bitcoin’s main chain consensus mechanism, semantically specialized, only applicable to transfers.

VII. Concluding Statement

Bitcoin’s consensus system is an engineering realization of Turing’s Q₂ logical structure, used to arbitrate BTC transfers. We, however, can build a series of independent decentralized arbitration systems based on UTXO sharing, each mapping to a new R(x, y), together forming a “Parallel Oracle Turing Machine Ecosystem” — providing diverse support for decentralized societal infrastructure.