In the early 20th century, Hilbert proposed the “metamathematics program,” aiming to construct a complete, consistent, and decidable axiomatic system for all of mathematics. However, Gödel’s incompleteness theorems and Turing’s halting problem shattered this vision, revealing that any sufficiently powerful formal system is inherently incomplete and cannot be fully computable.
Yet these negative findings did not spell the end of rational construction. On the contrary, they led us to awaken from the illusion of closed systems and turn toward building open, dynamic, and adaptive computational structures. This article begins with Gödel’s transfinite construction and Turing’s concept of oracle machines to analyze a new path toward “relative completeness” in decentralized environments, and uses Bitcoin as a real-world case to show how mathematical logic can take root in engineering practice, shaping complex systems capable of maintaining order without central arbitration.
Gödel’s incompleteness theorems, proposed in 1931, state that in any sufficiently powerful and consistent formal system, there exist propositions that cannot be proven or disproven within the system itself. This result ended Hilbert’s program of “completeness,” but opened up deeper philosophical reflection.
Gödel believed that mathematical truth does not entirely stem from formal axiomatic derivation, but should be rooted in a kind of a priori “mathematical intuition.” He proposed a “conceptual realism,” asserting that mathematical objects possess a kind of existence beyond formal systems. This viewpoint emphasizes that we should approach holistic truth progressively through “finite steps approximating infinite structures.”
In proving the consistency of the continuum hypothesis, Gödel used ordinals and the constructible universe (L) to build a “hierarchical transfinite system expansion method.” This technique is not just a mathematical tool but also reflects his “holistic philosophy”: a system cannot be self-complete, but through progressive extensions, can approximate wholeness.
Turing’s work provided a formal computational foundation for Gödel’s ideas. Especially in his doctoral thesis, he introduced the concept of Oracle Turing Machines (OTMs) and relative computability. This model extends beyond the limits of traditional Turing machines: the machine can access an external “oracle” to solve problems that are undecidable for standard machines.
In this framework, “computability” is no longer absolute, but relative to the information source (oracle) invoked. In other words, the capability of each formal system can be extended through “externally injected” means, forming a composable and hierarchical dynamic computational structure. This structure can be seen as a computational interpretation of Gödel’s idea of “local incompleteness, holistic approximation.”
In Gödel and Turing’s theories, “transfinite induction” and “relative computation” form a unified conceptual axis: by hierarchically introducing new judgment powers, systems continuously evolve from locally closed to more open structures.
Ordinals in mathematics play a key role here. They not only provide an indexing system for transfinite recursion but also imply a model logic capable of constructing “infinitely growing hierarchies.” A primitive formal system (such as a Turing machine or ZFC set theory) can continually expand its computable space by introducing new oracle layers indexed by ordinals, thus forming an open and evolvable knowledge system.
This logical foundation provides profound theoretical support for understanding how “decentralized consensus” emerges in modern distributed systems.
Bitcoin can be viewed as an engineering realization of the above ideas. It is not merely a cryptocurrency, but a “computational philosophy experiment” capable of dynamically reaching consensus without a trusted center.
In the Bitcoin network:
Although the Bitcoin system is not “complete” in a mathematical sense, it finds a balance among economic game theory, computational difficulty, and systemic openness, making it exponentially costly to maliciously rewrite history—thereby practically approaching irreversibility, “finality,” and “factual stability.”
This mechanism is essentially a transfinite growth process driven by finite rules, approaching consensus and certainty through dynamic evolution—an engineering embodiment of the Gödel-Turing path.
Gödel once envisioned that “philosophy will one day become as precise as mathematics.” Bitcoin and the decentralized systems behind it embody this shift—from the metaphysical pursuit of completeness to the acceptance of systemic limitations and the design of dynamic, game-theoretic, relatively complete structures.
Bitcoin does not rely on any single axiom or central arbitrator. Instead, it generates verifiable truth paths through dynamic interaction. This offers a new paradigm for building trustworthy systems:
Conclusion: Building the Future Philosophy-Engineering Synthesis
Gödel and Turing’s work was once seen as a denial of formal rationality, but from today’s perspective, they revealed a stronger possibility: open systems, relative computation, and transfinite recursion are not only directions in mathematical logic but are becoming foundational in designing the next generation of engineering systems (like Bitcoin and blockchain).
This is a new path from “incompleteness” to “constructible order,” opened by the precision of logic and the patience of engineering, jointly forging a world where collective consensus is possible without central control.
