I. From Gödel’s Incompleteness to Nash’s Reflection

In 1931, Gödel’s incompleteness theorem revealed a shocking fact: In any sufficiently strong formal system, there always exist propositions that are true but unprovable. This means:The ideal completeness of formal logic is inherently self-contradictory.

In 1939, Turing proposed a groundbreaking idea in his doctoral thesis “Systems of Logic Based on Ordinals”: By constructing a “tower of logic systems” based on higher-order ordinal logic,an upper-level system could prove the Gödel propositions of the lower level, thus externalizing consistency.

This became the prototype of what is later known as transfinite iterative logic. However, Turing’s system was based on recursively definable ordinal sets— all levels had to be defined by computable sequences indexed by natural numbers.

To Nash, this definition was too narrow— it could not encompass those ordinals that exist only through definition itself.

II. Nash’s Key Revision: Definitional Ordinals and the ORDDEF Function

In his later manuscript “Hierarchical Introspective Logics,” Nash proposed a fundamental shift: The hierarchy of logic should not be determined by recursive enumeration,

but by the “definition of ordinal definitions.”

To achieve this, he introduced a crucial function: ORDDEF(δ) — true if and only if δ is a definition within the base logical system that provably defines a unique ordinal.‍

This function has three key characteristics:

• Non-recursiveness: ORDDEF cannot be computed by any finite algorithm—it transcends computability.
• Recursive definitional layering: the system can expand recursively through “definitions of definitions.”
• Logical anchoring: it provides a uniquely determined definitional coordinate, ensuring the uniqueness of hierarchical expansion.

This allowed logic to generate itself internally through definitions, without relying on external axiomatic extensions.

III. Hierarchical Introspective Logic: A System Observing Itself

Nash called this system Hierarchical Introspective Logic.

His core idea: A logical system cannot prove its own consistency internally,but a higher-level system can prove the consistency of the lower one.

Thus, a multi-layered structure of logical self-observation emerges: Each level acts as the observer of the one below it.

In the process of “seeing itself,” the logical system generates its own structure.

This is Nash’s “introspective hierarchy” — a mechanism for self-evolution and self-reflection within logic itself.

IV. The Logical Equivalence Between ORDDEF and Bitcoin’s Heaviest Chain Rule

Turing’s system had a problem: The same ordinal could correspond to multiple definitions, resulting in non-uniqueness.

Nash corrected this via the ORDDEF function, which ensures unique correspondence at the definitional level, eliminating arbitrariness and divergence.

ORDDEF serves as the deterministic anchor of Nash’s system— logically analogous to Bitcoin’s Heaviest Chain Rule.

In Bitcoin:

The “heaviest chain” is the only legitimate history under global hashpower consensus.

In Nash’s logic:

The “uniquely defined ordinal” established by ORDDEF is the only extendable axiomatic base under system evolution.

The former establishes the direction of time, the latter establishes the order of logic. Together, they embody: an irreversible order derived from global dissipative behavior.

In other words: ORDDEF is the formalized Proof-of-Work of logic — by determining definitional uniqueness, it grants the logical system irreversible evolution and global consistency.

V. The Intrinsic Game Structure of Logic: Layered Equilibrium and Non-Cooperation

Nash’s hierarchical logic is not merely a formal system— it is a non-cooperative game structure: Each level verifies the consistency of the level below, yet cannot be verified by itself.

Each level gains its own rationality through observing the lower level. The system’s equilibrium point is a state where all layers act as mutual oracles and constraints for one another.

This is the Nash equilibrium at the level of logic itself: In an uncomputable hierarchical game, the stability of each level depends on verifying the consistency of the one below.

VI. From Formal Systems to Self-Generating Systems

From Gödel to Turing to Nash, logic evolved from a closed system of proof into a self-reflective, evolving cognitive system.

VII. Epilogue: The Evolution of Logic and the Temporal Order of Civilization

At the end of his manuscript, Nash wrote:

“Logic itself is a language that evolves.

Like mathematics to culture, it continually translates its own order into the future.”

He believed that the evolution of human logic does not depend on new axioms or injected truths, but on the system’s continual ascent through its own definitions. This marks the transformation of civilization from formalization to self-awareness.

Bitcoin, through the Heaviest Chain Rule, realizes a decentralized temporal order. In computational terms, it is the physical realization of Nash’s hierarchical logic: A computational entity maintained by energy consensus and driven by global game dynamics— one that can not only prove its own consistency, but also generate determinism within an uncomputable world.

References

Gödel, K. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. (1931)

Turing, A. M. Systems of Logic Based on Ordinals. Proc. London Math. Soc. (1939)

Feferman, S. Turing in the Land of O(z): The Mathematical Foundations of Ordinal Logic. (1988)

Nash, J. F. Hierarchical Introspective Logics. (Unpublished Manuscript, Fermat’s Library Annotated Version)