‍Abstract‍

This paper aims to establish a logical chain from the concept of “undecidability” in computer science to the security of decentralized systems. We argue that the structural security risks of traditional centralized systems are rooted in the concentration of decision-making power over “undecidable problems” at the system boundary. We then analyze the security model trade-offs in Turing-complete blockchains such as Ethereum, and finally reveal that Bitcoin’s true innovation lies not in attempting to eliminate or avoid undecidability, but in ingeniously designing a decentralized game-theoretic mechanism that continuously and probabilistically performs “reductive judgments” on undecidable problems. This design is highly aligned in thought with the “ordinal logic systems” proposed by Alan Turing in his work Systems of Logic Based on Ordinals, together pointing toward a new paradigm of trust and security.

I. The Nature of Vulnerabilities: Programming at the Boundary of Computation

The conventional view holds that system vulnerabilities are the result of coding errors or design oversights. However, a deeper perspective suggests: every successful attack is an accurate exploitation of the system’s “boundary of undecidability.”

A computational system, no matter how meticulously designed, is essentially a formal automaton. Its designers attempt to describe the system’s entire state space using deterministic rules (“Code is Law”). However, according to Gödel’s Incompleteness Theorem and Turing’s proof of the Halting Problem, any sufficiently complex (e.g., Turing-complete) formal system inevitably contains propositions that cannot be proven within the system, or problems that are undecidable.

These “undecidable” zones are precisely the “gray areas” that designers have failed to fully formalize, cannot predict, or have overlooked. The brilliance of hackers lies in their ability to:

1. Reverse Modeling: Understand the complete state space of the system more profoundly than its designers, especially the abnormal, boundary, and undefined behaviors.
2. Construct Higher-Order Logic: Build a “meta-logic” beyond the system’s preset semantics on top of these gray areas. For example, by exploiting a buffer overflow (an undefined boundary behavior), an attacker can construct shellcode (a higher-order attack instruction set).
   

Thus, we can propose the first core argument: Any automaton system that contains an undecidable problem inherently leaves space for “vulnerability programming.” The essence of attack lies in creative higher-order programming beyond the system’s decidable boundary.

II. The Achilles’ Heel of Centralized Systems

In centralized systems, how are these inevitable “undecidable problems” handled? The answer: authoritative adjudication. Whether it’s an operating system’s kernel, a website’s server administrator, or a bank’s database architect, when the system encounters a fuzzy state beyond the rules, there is always a centralized authority to arbitrate. This adjudicator (human or organization) holds the ultimate interpretive and decision-making power over all undecidable problems within the system.

This leads directly to the second core argument—the structural vulnerability of centralized systems. In a centralized system, the “adjudication of undecidable problems” is vested in a specific center. Therefore, whoever controls this center effectively controls the logical fate of the entire system. This adjudication center is both the apex of power and the system’s most fatal Single Point of Failure.

Security problems do not originate from the clear algorithm implementations, but from the boundary behaviors, exception handling, and power models that require centralized adjudication.

III. Ethereum’s Trade-off: The Double-Edged Sword of “Turing-Completeness”

Ethereum, by introducing the EVM (Ethereum Virtual Machine) to achieve Turing-complete smart contracts, greatly expanded the application scenarios of blockchains. However, this design also introduced new complexities and attack surfaces, precisely validating the arguments above.

1. EVM’s Undecidability: The EVM itself is a “Turing-complete but undecidable” execution environment. This means that in principle, it is impossible to create a universal static analysis tool that can predict 100% of any smart contract’s behaviors (equivalent to solving the Halting Problem). Attackers can exploit complex state interactions, re-entrancy, and other mechanisms within contracts to construct highly concealed “undecidable” execution paths, leading to catastrophic events like The DAO attack.
2. Consensus Layer Power Focus: Although the PoS mechanism is superior to PoW in energy consumption, from a power structure perspective, it ties the consensus adjudication right to “staked equity.” This makes large nodes or staking pools theoretically identifiable, attackable, and even regulatable “central points,” and thus the risk of power centralization remains.
3. Developer Cognitive Disconnection: Solidity developers typically think about business logic at the “Transaction Layer (TX Layer),” controlling only the internal state of contracts. However, what truly determines whether a transaction is ultimately confirmed and state transitions are accepted by the network is the “Block Layer’s” consensus meta-logic. This disconnection causes developers to often overlook underlying consensus security, staying merely at the semantic level of language without examining security from the perspective of global state transitions.
   

This illustrates a key point: Turing-completeness brings infinite creativity, but also infinite undecidable complexity, making “fully verifiable security” a theoretical luxury.

IV. Bitcoin’s Ingenuity: Using “Undecidability” Against “Undecidability”

Satoshi Nakamoto’s design is revolutionary not because it attempts to create a perfect system without undecidable problems (which is theoretically impossible), but because it accepts this reality and designs an entirely new game-theoretic process around it.

The innovation of Bitcoin lies in: It does not establish any centralized final adjudicator. Instead, it uses a decentralized, never-ending competitive process (PoW mining + longest chain rule) to continuously, probabilistically, and incrementally judge the core undecidable problem of “whether a transaction is valid.”

This process can be understood as:

1. Problem Transformation: It transforms the static, absolute yes/no question of “determining the final validity of a transaction” into the dynamic, ongoing game process of “which historical version containing the transaction gains the most computational power support in the future.”
2. Probabilistic Reduction: A transaction (TX) being written into a block does not mean it is “absolutely confirmed.” Its degree of confirmation (the probability of being accepted as fact by the network) grows exponentially with each subsequent block. Every new block is a “re-confirmation” and “re-vote” on all historical events.
3. Ordinal Adjudication in Reality: This structure is strikingly similar to Turing’s 1938 idea of “Ordinal Logics.” Turing envisioned that by continually moving to stronger, higher-order logical systems (marked by ordinals ω, ω+1, …), one could adjudicate previously undecidable propositions.
   

Bitcoin’s blockchain is an engineering realization of this “ordinal adjudication structure”:

• Block N is a logical system.
• The birth of Block N+1 is equivalent to jumping to a higher-order system. This new system can render a stronger “judgment” on the transaction history within Block N.
  

Thus, Satoshi Nakamoto’s true contribution is: By using a distributed, never-ending, physically-costed (computational power) ordinal iterative structure, Bitcoin achieves a game-theoretic reductive judgment on the system’s core undecidable problems (double-spending, transaction validity).

This is the fundamental reason why Bitcoin is “trustless.” It does not eliminate the problem but disperses the adjudication of the problem into an endless spatiotemporal competition that anyone can participate in.