‍I. The Three Layers of Computation

1. Consistency Layer: Turing Machine

• Essence: Pursuit of consistency
• Features: Untyped recursion (mechanical steps based on cardinality)
• Methodology: Rationalist reasoning
• Limitation: Can only guarantee consistency, unable to handle uncomputable abstract problems
  

2. Completeness Layer: Ordinal Logic System

• Essence: Pursuit of completeness
• Features: Typed recursion (constructed through transfinite iteration)
• Methodology: Empiricist reasoning (relying on external judgment, similar to “Gödelian adjudication”)
• Limitation: Relies on mathematical logic construction, unable to achieve endogenous completeness; can only tend toward completeness, but cannot prove completeness itself
  

3. Self-Driven Layer: Non-Cooperative Game (Nash Equilibrium System)

• Essence: Pursuit of self-driven introspection
• Features: Dynamic interactive relations, anonymous peer-to-peer (decentralization)
• Methodology: Intuitive constructivist proof
• Advantage: Through dynamic game equilibrium, can converge as a whole to a unique non-entity

II. Turing’s Two Mistakes

1. Error in the Choice of Logical Path

After proposing the Turing Machine (guaranteeing consistency), Turing could have continued using the dynamic computational method of the Turing Machine to construct the ordinal logic system, thus advancing the exploration of completeness.

But instead, he chose to rely on his advisor Church’s mathematical logic framework, with the results being:

• The ordinal logic system cannot achieve endogenous completeness;
• It inevitably requires “Gödelian diagonal adjudication” to continuously advance, tending toward completeness rather than achieving completeness.
  

Fundamental Problem: Mathematical logic is a static language, incapable of describing dynamic computation.

2. Limitation of Starting from Static Recursion

Turing’s thinking started from “static, untyped mechanical steps” and gradually moved toward “dynamic, typed ordinal logic.”

However, because of the static nature of mathematical logic, that system can never reach completeness, and must rely on external human judgment.

If approached from another direction—starting with dynamic interactive relations—one would enter the “domain of dynamic computability,” naturally transitioning to the dynamic equilibrium of Nash’s non-cooperative game, ultimately achieving overall unique convergence.

In other words: interactive relations better express the essence of dynamic computation than single-agent computation.

III. Summary

• First Layer: Turing Machine → Static consistency (Rationalism)
• Second Layer: Ordinal Logic → Tending toward completeness (Empiricism + External judgment)
• Third Layer: Non-Cooperative Game → Dynamic convergence (Constructivism + Endogenous game)
  

Turing’s mistake lay in failing to start from the computability of dynamic interaction, remaining instead within the static framework of mathematical logic, and thereby missing the path toward “self-driven completeness.”