ZAP // Dall-E 2
The Incompleteness Theorem may seem obvious, but not to mathematicians, who try to find order in the world. There are indeterminable things — and there are incalculable numbers. Now we know how this can influence physics.
Em 1931, Kurt Gödel formulated the Incompleteness Theorem, which states that It will never be possible to prove all mathematical truths: there will always be truths that escape the basic mathematical structure and that are impossible to prove.
For example, It will never be possible to clarify how many real numbers there are within the mathematical framework currently in use. And unsolvable problems are not limited to mathematics.
For example, in certain card and computer games (such as Magic: The Gathering), situations may arise in which it is impossible to determine which player will win, explains the
In physics, it is known that it is not always possible to predict whether a crystal system will conduct electricity or not. But now, a new one published in October explains that, In other fields of physics, this theorem also applies.
Toby Cubitt and his colleagues explained that it is impossible to determine the critical parameter at which a particle system undergoes a phase transition — a change similar to that which occurs when water freezes below a temperature of zero degrees Celsius. “Our result… illustrates how non-calculable numbers can manifest themselves in physical systems“, the article writes.
Unpredictability can therefore occur in realistic systems. In this work, Cubitt’s team analyzed a finite square lattice containing an arrangement of several particles that each interact with its nearest neighbor.
In the Cubitt model, the strength of the interaction between electrons depends on a parameter f. Cubitt and his team determined the value of f from which this gap occurs. And it corresponds to the so-called Chaitin constant Ω — a number that may sound familiar to math lovers, as it is among the few known examples of numbers that cannot be calculated.
There is no algorithm that, if run for an infinite time, will produce Ω. So if Ω cannot be calculated, so it is also not possible to specify when a phase transition occurs in the system studied by Cubitt and his colleagues.
Chaitin’s constant Ω corresponds to the probability that the theoretical model of a computer (a Turing machine) stopping for a given input:
In this equationp denotes all programs that stop after a finite execution time, and |p| describes the length of the program in bits.
To calculate Chaitin’s constant accurately, it would be necessary to know which programs are maintained and which are not maintained — which is not possible, according to the maintenance problem.
Although, in 2000 the mathematician Cristian Calude and his colleagues have to calculate the first digits of the Chaitin constant (0.0157499939956247687…) it will never be possible to find all the decimal places.
Cubitt’s team was thus able to prove mathematically that their physical model undergoes a phase transition to a value of f = Ω: changes from conductor to insulator. However, since Ω cannot be calculated exactly, the phase diagram of the physical system is also undefined.
“Our results illustrate that non-computable numbers can to emerge as phase transition points in physics-like models, even when all the underlying microscopic data are fully computable,” they write in the paper.
Nevertheless, the precision with which Chaitin’s constant can be specified makes it senough for real-world applications.
Still, the work of Cubitt and his colleagues once again illustrates the reach of Gödel’s idea—and may even question the idea that ever we will achieve a Theory of Everything.
