Gregory Chaitin, a mathematics researcher at IBM's T. J. Watson Research Center in Yorktown Heights, New York, has shown that mathematicians can't actually prove very much at all. Doing maths, he says, is just a process of discovery like every other branch of science: it's an experimental field where mathematicians stumble upon facts in the same way that zoologists might come across a new species of primate.
The reason for Chaitin's provocative statements is that he has found that the core of mathematics is riddled with holes. Chaitin has shown that there are an infinite number of mathematical facts but, for the most part, they are unrelated to each other and impossible to tie together with unifying theorems. If mathematicians find any connections between these facts, they do so by luck. "Most of mathematics is true for no particular reason," Chaitin says. "Maths is true by accident."
Decades later, in the 1960s, Chaitin took up where Turing left off. Fascinated by Turing's work, he began to investigate the halting problem. He considered all the possible programs that Turing's hypothetical computer could run, and then looked for the probability that a program, chosen at random from among all the possible programs, will halt. The work took him nearly 20 years, but he eventually showed that this "halting probability" turns Turing's question of whether a program halts into a real number, somewhere between 0 and 1.
Chaitin named this number Omega. And he showed that, just as there are no computable instructions for determining in advance whether a computer will halt, there are also no instructions for determining the digits of Omega. Omega is uncomputable.
Gregory Chaitin's Omega is a number that represents the probability or complexity of a mathematical object or function. It is also known as the "halting probability" and is used to measure the randomness or unpredictability of a mathematical system.
Gregory Chaitin developed Omega as part of his work in algorithmic information theory. He was inspired by the work of mathematician and computer scientist, Alan Turing, on the concept of algorithmic randomness.
Omega has significant implications for mathematics, specifically in the areas of algorithmic information theory, computational complexity, and the foundations of mathematics. It has been used to prove the existence of undecidable problems and has opened up new avenues for research in the field of randomness and complexity.
Omega is calculated by using a computer program to generate a series of random bits and then analyzing the sequence for patterns or structure. The more random the sequence, the higher the value of Omega. However, Omega is an uncomputable number, meaning it cannot be calculated exactly and is only known to a certain degree of precision.
One potential future application of Omega is in the field of cryptography, as it can be used to generate truly random numbers. It also has implications for artificial intelligence and the study of complex systems. Additionally, understanding Omega can help us better understand the limits of computation and the nature of randomness in the universe.