The current characteristics demonstrated by artificial intelligence, formalized blockchain technology, and Bitcoin seem to converge on a profound issue: the “incompleteness” of complex systems and the potential to transcend it. Starting from Gödel’s incompleteness theorems in mathematics, we can observe that these diverse fields all face inherent limitations, yet they also contain the potential for breakthroughs through specific mechanisms.

I. From Mathematical Incompleteness to AI as a Tool: Intelligence Without a “Self”

Just as Gödel’s incompleteness theorems reveal that any consistent formal system containing basic arithmetic contains propositions that cannot be proven or disproven within the system, today’s rapidly developing artificial intelligence—especially foundation models represented by large language models—also exposes a similar “incompleteness.” While these models perform excellently in specific tasks, their understanding and reasoning capabilities often rely on pattern matching from training data, lacking genuine “self-awareness” and an internal value judgment system.

This “incompleteness” renders current AI essentially a powerful tool, whose behavior and output are constrained by predefined goals and data. It struggles to think or create independently beyond its set framework. Unlike humans, it cannot engage in self-reflection, form independent will, or interact complexly with the external world based on internal understanding. As such, in the absence of a “self,” today’s AI mainly plays the role of an assistant executing instructions and solving specific problems.

II. The Incompleteness of Formalized Blockchain: Enclosed Cryptographic Tools Isolated from the Real World

Formalized blockchain technology, as an innovation aimed at constructing decentralized trust systems, similarly faces a certain degree of “incompleteness.” While it offers significant advantages in terms of data immutability and transparency, many cryptocurrency systems built upon formalized blockchain tech are, to some extent, closed-system tools.

This “closedness” is reflected in how their value systems are often confined within the system itself, lacking direct and flexible connections to real-world assets, services, and value flows. Many cryptocurrencies are designed to build independent digital economies, but interactions with traditional financial systems and the physical economy remain obstructed. This “incompleteness” limits the ability of cryptocurrencies to solve real-world problems, leaving them largely as speculative assets or mediums of exchange within specific communities.

III. The “Completeness” of Bitcoin: A Self-Referentially Emergent Human-Machine Interaction System

Unlike the above two, Bitcoin is seen as a unique and potentially “complete” human-machine interaction system. Its core innovation lies in Satoshi Nakamoto’s ingenious fusion of technology and economic incentives, creating a value system capable of self-maintenance and evolution.

Bitcoin’s “completeness” is embodied in what Gödel called “self-reference” and the “emergence of self-reference”:

• ‍Self-reference: The Bitcoin UTXO ledger records transaction history and ownership states; this ledger is a full representation of the system’s operational status. Miners’ computational and validation behavior directly contributes to the security of this ledger, and the ledger’s security, in turn, underpins the value of BTC—forming a system of internal self-reference and mutual dependence.‍
• Emergence of self-reference: Satoshi introduced a mechanism of “individual peer-to-peer distributed self-competition computation” among miners. The process of maintaining UTXO security is not controlled by centralized authority but emerges from competitive computation among countless independent miners. This value chain (i.e., BTC’s value) is not preset but dynamically emerges through the behavior of participants and energy input.

The Bitcoin system maps the UTXO ledger to individual behavioral agents (holders) and maps miners’ computational activity to real-world human-created electricity consumption. This mapping allows Bitcoin to harness real-world resources (electricity) to “mint” BTC—a digital currency with real value.

IV. The Vision of the GEB Project: Constructing a Human-Machine Symbiotic System with “Self-Awareness”

Inspired by Bitcoin—especially its mechanism of “emergent self-reference”—the GEB project seeks to follow in Satoshi’s footsteps by building a human-machine interaction system that possesses “self-awareness.” Its core goal is to transcend the “incompleteness” of current AI and formalized blockchains, ultimately achieving an organic fusion among cryptocurrency, machines, and humans.

The GEB project attempts to simulate Bitcoin’s model of individual peer-to-peer distributed competition, but on a broader level among intelligent agents (including AI, machines, and humans), establishing connections capable of mutual understanding and perception. Its vision is to build an intelligent ecosystem capable of autonomous evolution and forming an internal value system—transforming cryptocurrency from a closed tool into a bridge that links the digital and physical worlds, facilitating value exchange among intelligent agents.

Conclusion:

Inspired by Gödel’s incompleteness theorems, we see that current AI and formalized blockchain technologies face inherent limitations. However, Bitcoin, through its unique mechanism of “emergent self-reference,” demonstrates the potential to transcend such incompleteness, constructing a self-consistent system that transforms human behavior and physical resources into digital value. The GEB project aims to draw on Bitcoin’s successful experience to build a human-machine interaction system with “self-awareness,” ultimately realizing an organic unification of cryptocurrency, machines, and humans—ushering in a new era of intelligent interconnection.