The Transfinite Exploration of Computation: From Leibniz to Bitcoin

Human exploration of “computation” is not merely a matter of mechanical operation, but a profound and prolonged transfinite iteration. Its historical narrative can be traced back to Leibniz’s grand vision and has evolved over generations of scientific thought, advancing toward the goal of mathematical axiomatization, ultimately crystallizing in modern technologies such as the Turing Machine and Bitcoin.

I. Prologue: Leibniz’s Axiomatic Dream and the Dawn of Computation

The story begins in the 17th century with Gottfried Wilhelm Leibniz. He first proposed a revolutionary axiomatic hypothesis: everything can be determined through computation.

Leibniz believed that through a universal language (characteristica universalis) and a calculus of reasoning (calculus ratiocinator), all human disputes—even philosophical ones—could be resolved through pure computation, just like mathematical problems. This laid the foundation for mathematical axiomatization and sowed the seeds for the modern concept of “computation.” His vision foreshadowed a world in which logic and reasoning could be formalized and mechanized.

II. First Iteration: Cantor and Hilbert — The Challenge of the Infinite and the Pursuit of Axiomatization

Leibniz’s vision matured over nearly 300 years. At the turn of the 19th and 20th centuries, the mathematical world witnessed an important iteration:

Georg Cantor was an “outlier” in this iteration. He broke free from the traditional mathematical constraints of the finite by introducing the concepts of transfinite numbers and transfinite ordinals. Cantor’s work on the infinite challenged Leibniz’s “everything can be computed” vision by revealing the layers and complexities of the infinite. At the same time, it spurred mathematicians to explore how to formalize and axiomatize these new mathematical objects. As explored in von Neumann’s early papers, Cantor’s work opened up the possibility of defining the infinite through axioms.
Following Cantor came David Hilbert, considered Cantor’s “vanguard.” Hilbert sharply perceived the looming crisis of mathematics being marginalized, and thus strongly advocated for the axiomatization of the entire mathematical system. His goal was to establish a complete, consistent, and decidable mathematical framework—thus inheriting and deepening Leibniz’s axiomatic dream, attempting to control even the concept of infinity through rigorous axioms.

III. Second Iteration: von Neumann’s Axiomatization of Set Theory and the Vision of Computational Machines

Building on Cantor and Hilbert’s foundation, John von Neumann entered the stage and pushed this exploration to new heights.

Von Neumann’s academic career began with the axiomatization of set theory. His third paper, An Axiomatization of Set Theory, directly responded to Hilbert’s call for mathematical axiomatization and contributed significantly to modern set theory. Through rigorous axioms, he sought to tame Cantor’s infinite within a controllable logical framework.
However, von Neumann’s vision extended far beyond this. He applied his understanding of axiomatic systems to the nature of computation. His life’s work was described as “starting with axiomatic set theory and ending with inventing computers modeled after the human brain”—this was no coincidence. His deep insights into formal logic and algorithms made him one of the founders of modern computer architecture. He translated abstract logical operations into executable instructions and data processing, paving the way for the emergence of the general-purpose computing machine.

IV. Third Iteration: Gödel’s Insight into Incompleteness and the Advent of the Turing Machine Era

On the road toward mathematical axiomatization and formal computation, Kurt Gödel introduced the most profound and shocking insights.

Gödel’s 1931 Incompleteness Theorems struck like a bolt of lightning. They revealed that any sufficiently powerful formal system containing Peano arithmetic is inherently incomplete. There will always be true propositions in such systems that cannot be proven within the system itself. Moreover, if such a system is consistent, its consistency cannot be proven internally. This directly challenged Hilbert’s grand plan for full axiomatization, showing that any formal system involving computation has inherent limitations and cannot achieve full completeness.
Nevertheless, Gödel held great admiration for Leibniz. He believed that Leibniz had, centuries earlier, already glimpsed the deep principles of theories such as von Neumann’s game theory. Gödel dedicated his life to studying Leibniz’s unpublished manuscripts, hoping to find evidence to support his own advanced conjectures. This reflects his deep understanding of and longing for Leibniz’s axiomatic hypothesis that “everything is computable.”
While Gödel’s Incompleteness Theorems highlighted the limitations of formal systems, they also indirectly refined the definition of “computation.” Against this backdrop, Alan Turing proposed the Turing Machine in 1936. This abstract computational model, in its minimalist form, precisely defined the limits of computability. It demonstrated that any algorithm could be executed by this simple model, providing a concrete operational definition for the formal systems that Gödel had examined.
The von Neumann architecture for computers is a physical embodiment of the Turing Machine. Modern computers operate based on the principles of the Turing Machine, using stored programs and data to achieve universal computation. In this way, humanity’s vision of computation—born from the axiomatic hypothesis—was finally “realized,” though Gödel had already reminded us that this realization would always have boundaries.

V. Fourth Iteration: Oracle Turing Machines, Transfinite Craftsmanship, and the Birth of Bitcoin

With the rapid development of computer technology, humanity’s understanding of “computation” did not stop—it continued iterating toward transfinite dimensions, ultimately giving rise to the groundbreaking invention of Bitcoin.

The underlying logic of Bitcoin can be seen as a further extension of the Turing Machine concept, particularly in relation to Oracle Turing Machines and Transfinite Craftsmanship. The Oracle Turing Machine is a hypothetical Turing Machine that can access an “oracle” to instantly solve problems that a traditional Turing Machine cannot solve in finite time. While Bitcoin itself is not an Oracle Turing Machine, it constructs a decentralized “trust oracle” through cryptography and distributed ledger technology, enabling global value transfer and consensus without centralized authority.
Transfinite Craftsmanship can be understood as the subtle art of weaving and ordering information and value—an idea hinted at in Turing’s doctoral thesis Systems of Logic Based on Ordinals. The blockchain structure of Bitcoin is, at its core, an immutable, time-sequenced ledger of transactions, where each block occupies a unique “ordinal” position. Through ingenious use of hash functions, Proof of Work, and the chained block structure, Bitcoin achieves decentralized consensus and tamper-resistance, creating an unprecedented “trust machine.”
The birth of Bitcoin represents another milestone in combining mathematical axiomatization, computational theory, and practical application. It extends “computation” beyond data processing to the creation and management of digital scarcity and value, realizing computable trust on a global distributed network. This construction of trust is another “transfinite” extension of Leibniz’s original axiomatic dream—bringing societal consensus and value systems into a realm of computability and verifiability.

From Leibniz’s grand vision on paper, to Cantor’s exploration of the infinite, to Hilbert’s call for axiomatization, Gödel’s profound insights, von Neumann’s machine realization, Turing’s theoretical foundation, and finally Bitcoin’s practical innovation—humanity’s transfinite exploration of computation is an ongoing evolutionary process.

Each generation of scientists, building on the thoughts of their predecessors, has step by step advanced Leibniz’s vision through the lens of mathematical axiomatization, continually expanding its practical boundaries. Even today, this process continues.

Gödel’s Incompleteness Theorems constantly remind us that even in the most rigorous mathematical and computational systems, there remain mysteries beyond our complete grasp—thus fueling humanity’s eternal quest to transcend its cognitive limits.