Math and physics have always been seen as disciplines that can provide answers to any question or problem. However, a groundbreaking discovery made by Kurt Gödel in 1931 shattered this belief. Gödel’s incompleteness theorem revealed that there will always be truths that cannot be proven within the framework of mathematics. This idea was shocking to mathematicians who believed that they could construct a system based on a few axioms and prove every truth within that system.
The implications of Gödel’s theorem extend beyond mathematics. It has been observed in various fields, including physics. Recently, physicist Toby Cubitt and his team at University College London made a significant discovery that further illustrates the incompleteness theorem in physics. They described a particle system undergoing a phase transition, similar to the freezing of water, but with a critical parameter that cannot be calculated, unlike traditional phase transitions.
Cubitt’s team studied a simple system—a finite square lattice with particles interacting with their nearest neighbors. The strength of interaction between the particles depends on a parameter called φ. As φ varies, the behavior of the system changes, leading to different energy states. The researchers found that there is a value of φ at which the system undergoes a phase transition from a conductor to an insulator. Surprisingly, this critical value corresponds to the Chaitin constant Ω—an incomputable number that cannot be calculated precisely.
The Chaitin constant was defined by mathematician Gregory Chaitin using the halting problem from computer science. This number represents the probability that a theoretical computer will halt for a given input. While some digits of Ω have been calculated, it is impossible to determine all decimal places due to the nature of noncomputable numbers.
Cubitt’s team’s findings demonstrate how uncomputable numbers can manifest as phase transition points in physics-like models. Despite the precision with which the Chaitin constant can be specified, the phase diagram of the physical system remains undefined. This discovery emphasizes the far-reaching implications of Gödel’s incompleteness theorem, even in the realm of physics.
The work by Cubitt and his colleagues highlights the ongoing relevance of Gödel’s insights, even after more than 90 years. It raises questions about how Gödel’s incompleteness theorems may impact broader physical problems, such as the search for a theory of everything. This research underscores the interconnectedness of mathematics and physics, revealing the inherent limitations of human knowledge and understanding.
This article was originally published in Spektrum der Wissenschaft and has been reproduced with permission.
