Abstract

The difference between Bitcoin and Ethereum is not merely about consensus mechanisms or performance; at a deeper level, it lies in their completely different philosophical approaches to “trust” and “determinacy.” Bitcoin, through deliberate Turing incompleteness and reliance on physical time-driven “transfinite iteration,” defers undecidable problems to time and computation, thereby achieving a near-infinite logical finality. This article will explore key concepts like Turing completeness, the halting problem, and transfinite iteration to explain why Bitcoin represents a system structure that is closer to “irrefutable trust.”

I. Turing Completeness and the “Decidability Boundary” of a System

1. Turing Completeness: Limit of Functionality, Start of Risk

Turing completeness is a core concept in computational theory. It refers to a computational system (like a programming language or virtual machine) having the ability to simulate any Turing machine. In simpler terms, it can execute any computation task describable by an algorithm. While this expressive power fosters innovation, it also introduces fundamental risks from a logical perspective—most notably, the Halting Problem, an undecidable problem. That is, it’s impossible to create a general algorithm that can determine whether any given program will halt on a specific input. This represents the manifestation of “self-reference” and “incompleteness” in computer science.

• Ethereum’s EVM is Turing complete. This results in every smart contract deployed on it being intrinsically bound to human-designed rules and judgment to address its own “halting problems.” Consequently, the value and trust basis of tokens issued via such contracts are entirely human-determined—leading to token centralization. These tokens, in essence, offer little more than transparency over traditional game credits like Q-coins.
• Bitcoin’s Script language, by contrast, is intentionally Turing incomplete. It lacks support for loops, recursion, and other complex control structures. This seemingly “crippled” design actually brings a major advantage: every Bitcoin script’s execution path and result are entirely predictable and decidable. It eliminates the risk of “contract halting” at the protocol level. Any application logic must remain internal to the system, while the undecidable problem of its external validity is deferred to transfinite iteration.
  

In summary:

• Turing Complete = Powerful Functionality + Difficult to Decide
• Turing Incomplete = Limited Functionality + Easy to Decide
  

2. The Halting Problem: A Breakpoint in System Trust

In a decentralized consensus system, decidability is not just a technical issue, but the cornerstone of trust. If the core logic of a system (e.g., smart contract execution) is unpredictable, then its finality is undermined. Users and nodes must trust that contracts won’t behave erratically—a trust derived from external audits and developer reputations, not from native protocol guarantees.

II. Bitcoin’s Design Philosophy: From “Formal Finitude” to “Physical Infinity”

Satoshi’s design wasn’t a technical compromise, but a philosophical insight into security and determinacy. By combining Turing incompleteness, Proof of Work (PoW), and the longest chain rule, Bitcoin forms a structure that transcends pure formal systems—a meta-formal system.

1. Turing Incompleteness = Internal Logical Determinacy

Bitcoin Script’s strict limitations guarantee absolute determinacy within its formal system. Every transaction’s validity is clear, finite, and independently verifiable by all nodes. This means:

• No infinite loops: script execution has step limits.
• No state explosion: transaction outcomes are finite.
• No complex external dependencies: validation relies only on current transaction and referenced UTXOs.
  

This constitutes Bitcoin’s rock-solid internal determinacy.

2. “Transfinite Iteration” = External Physical Determinacy

However, Bitcoin still faces a fundamental undecidable problem: Which chain is the real and final one in the event of a fork? Satoshi did not attempt to “decide” this within the formal system. Instead, he offloaded it to the physical world, relying on an endless process of transfinite iteration to approach determinacy over time.

This process is powered by two mechanisms:

• Proof of Work (PoW): Anchors block production to real-world energy and time costs. Faking history requires enormous irreversible physical expense.
• Longest Chain Rule: All nodes follow and recognize the chain with the most accumulated work.
  

This mirrors transfinite iteration in set theory:

• Each new block is an iteration on the previous state.
• Iteration progresses with physical time (about every 10 minutes), theoretically unending.
• Each new block exponentially increases the difficulty of rewriting history.
  

Therefore, Bitcoin’s finality isn’t “decided” at a moment—it’s asymptotically approached over time. A block covered by six confirmations is considered “safe” not by absolute determination, but by a widely accepted probabilistic threshold. Its real security and certainty grow continuously over time, approaching 100%.

III. Ethereum by Contrast: From Computability to Limited Trust

1. EVM Complexity: Computable ≠ Decidable

Ethereum’s Turing completeness makes it a powerful “world computer” but also shifts the trust model:

• Arbitrary logic: enables complex applications like DeFi and DAOs.
• Unpredictable states: contracts can behave in unintended ways (e.g., The DAO incident).
• Trust transfer: due to undecidability, trust moves from the protocol to developers, auditors, and mechanisms like gas.
  

2. PoS and Its Bounded Trust Model

Ethereum’s shift to Proof of Stake (PoS) introduces a fundamentally different trust structure. PoS finality relies on a finite set of validators reaching consensus via voting.

• Economic incentives: staking tokens encourage honest validator behavior.
• Finite governance set: consensus is achieved through a countable group (e.g., 2/3 majority).
• Reversibility risk: if enough validators (e.g., 1/3 or 2/3 depending on implementation) act maliciously or go offline, consensus can be reversed or stalled.
  

Ethereum’s finality is decided within its formal system via a limited governance set. Bitcoin’s finality, on the other hand, is approached externally through an infinite physical evolution process.

IV. Why Is Bitcoin’s Determinacy More “Ontological”?

At the heart of this contrast is the source of trust—a philosophical question that enters the realm of ontology (the nature of being and reality).

PoS relies on an internal decision process; Bitcoin relies on an external convergence process.

This has deep implications:

• Different arbiters: Ethereum’s finality is decided by a countable validator group. Bitcoin’s finality is “proven” by the universe itself—through irreversible time and energy expenditure.
• Gödel’s Incompleteness as Metaphor: Just as Gödel showed that a formal system cannot prove its own consistency, Satoshi didn’t try to make Bitcoin “prove itself” from within. Instead, he outsourced the proof of its uniqueness and existence to physical reality.
• Foundations of trust differ: PoS trusts in economically rational humans. Bitcoin trusts in the inviolability of physical laws. The former is social science; the latter, natural science.

Conclusion: The Root of Trust Lies Not in “Who Judges,” but in “Who Cannot Overturn”

PoS-based chains, no matter how sophisticated their governance, ultimately ask: “Who should we trust to judge finality?”

Bitcoin, by contrast, sidesteps this question entirely. Instead of asking “who,” it provides a structure where all participants can independently verify an unoverturnable record of facts. It doesn’t seek an “absolute truth” bestowed by authority—but rather an evolving truth, self-reinforced over time.

This is why Bitcoin is a “Meta-Formal System”: It fuses a logically sound formal system (transaction scripts) with the impartial, irreversible nature of physical reality (PoW, time iteration). In this system, finality is not a one-time decision but an eternal process of asymptotic convergence.

Keywords Summary (Appendix)