Circumstantial Evidence for "God" (mathematical-logical-scientific view)

[Malkom]

Many have probably heard of Gödel's Incompleteness Theorem and Turing's Halting Problem. Here's an expanded version. So, let's get started, let's get intrigued. Don't rush, don't spoil anything, and don't jump to conclusions, even if it triggers you. Enjoy the journey of the mind.

I'll just say right away that this material was created in collaboration with "Rubicon of Knowledge," and I also used an English translator, so there may be some confusion. Please keep that in mind and allow for that. And since I'm involved in science, I do it. Therefore, I find this material quite interesting, but don't take it at face value.

«Imagine a number that mathematicians fear, not because it's infinite, not because it's unpronounceable, but because its very knowledge could destroy mathematics. This number isn't hidden in fantasy or conjecture. It's real, existing on the number line between zero and one, where probability and harmony typically reside. And yet, it brings chaos. If someone were to know all its digits, all of mathematics would collapse, theorems would become meaningless, proofs unnecessary, and the truth too easy. This number contains all the answers, but that's precisely why no one should know it. It sounds like a legend, but it's not a myth; it's pure, proven mathematics. Encoded within this number are the fates of all algorithms, all formulas, all questions that could ever be asked. It knows which programs will stop, and which will spin in an endless loop forever. It knows which equations have solutions and which don't, and hidden somewhere in its infinite digits are the answers to all the mysteries that have tormented humanity since the time of Euclid. This number is called the Gregory-Chaitin (Catalan) constant, or more simply, "Omega." Imagine there is a number that can be written as the decimal fraction 0. ... and then... an infinite, completely random sequence of digits. But each digit isn't random in the usual sense; it's not simply the result of a coin toss; it's the result of an infinite number of calculations. If you knew enough of the first digits of this sequence, you could answer any mathematical question. Any. From the Riemann hypothesis to the fate of infinite series. Everything that can be formalized as a program is already woven into this fraction. This number doesn't just store answers; it encodes the very principles of cognition; it's like the shadow of logic itself, the imprint of the limits of what can be calculated. There's no mysticism in his definition, but rather a cold, mathematical poetry: Omega is the probability that a randomly written program will eventually terminate. It seems like a simple idea. But beyond it lies the fundamental limit of knowledge. And if you feel a slight unease, you're right, because where there's a limit, there's always the temptation to cross it. The history of Omega began long before Gregory Chaitin conceived it. This Argentine-American mathematician worked in the 1960s, trying to understand the boundary between knowledge and ignorance. But before him, others laid the foundations for the world of algorithms. Almost a century ago, a young Englishman named Alan Turing asked a question that changed the course of science. In 1936, he presented the idea of a universal machine, a theoretical device capable of performing any calculation if given the right instructions. Essentially, it was the prototype of the modern computer. But Turing posed a deeper question: could a program be created that could analyze other programs and accurately determine whether they would ever terminate or run forever? The question sounds technical. But it conceals a philosophical horror: the question of the predictability of thought, the boundary between reason and chaos. Turing proved that such a universal program does not exist; no algorithm can determine whether any other program will terminate. This is known as the Halting Problem.

Imagine you're writing a program, you run it, and wait—a minute, an hour, a day, a month passes. You don't know if it's frozen forever or just hasn't finished its calculations yet. It would seem possible to create a freeze detector, a program that would simply tell you the truth, but Turing showed that this is impossible in principle. He did this with stunning elegance—he assumed such a program existed and called it H (from "Halt"). H takes any program as input and decides whether it will halt or not. Everything is logical until Turing suggested playing with logic itself. Turing creates a new program, P. It takes a description of the program as input and checks what H predicts about it. If H decides that the program will halt, P does the opposite—it goes into an infinite loop. If H predicts that the program won't halt, P immediately terminates. Now try running P on itself; what will H predict? If P predicts that P will halt, the program hangs; if it predicts that it won't halt, the program halts immediately. In both cases, the prediction turns out to be false. This is a perfect logical loop, a program that breaks the very idea of prediction. Thus, Turing proved that there is no universal algorithm capable of solving all problems; there are programs whose behavior is impossible to predict in advance, not because of complexity, but because of the very nature of computation. And at that moment, mathematics first saw a crack in its foundations, and from this crack, 30 years later, Omega was born. Turing left behind a question that made mathematicians uneasy. If we cannot know the fate of each individual program, perhaps we can learn something on average, for example, what percentage of all possible programs ever halt. At first glance, this seems like statistics, but answering this question requires an understanding of the entire structure of the computational world. In the 1960s, a young mathematician, Gregory Chaitin, tried to look at this problem differently. He proposed imagining all possible programs, short and long, simple and chaotic, written in all possible combinations of zeros and ones. Imagine flipping a coin. Heads is 1, tails is 0. You get a random sequence of bits—that's your program. You run it. It might print "hello, world" and stop, or it might loop and never finish. Then Chaitin asked a question that at first glance sounds harmless: what is the probability that a randomly generated program will ever stop? The answer was the Omega number. Chaitin's constant is the probability that a random program will stop. This number is between zero and one, like any probability, but it has a strange property. It's absolutely precisely defined, although it can't be calculated. Each digit of Omega carries information about the fate of an infinite number of programs. If you knew the first, say, one hundred thousand digits of Omega, you could predict the behavior of all programs up to one hundred thousand bits long. Every program that could be written would either stop or not. And the digits of Omega already contain this answer. To get a sense of scale, imagine that for any mathematical problem, you can write a program that will search for a solution. If a solution exists, the program will find it and terminate; if not, it will loop forever. Knowing enough Omega digits, you can tell for sure whether the program will find the answer, and thus know the solution to the problem. Everything comes down to a single sequence of digits. Omega is like a library of all mathematical truths, only the page in this book is forever locked. But long before Chaitin, there was a man who realized that logic itself has a hidden limit: his name was Kurt Gödel. In 1931, he published two theorems that shook the foundations of mathematics. The first asserted that in any sufficiently complex system, there will always be statements that are true but cannot be proven within that system. The second went even further, arguing that a system cannot prove its own consistency—in other words, mathematics cannot prove that mathematics is infallible. These ideas sounded like philosophy, but they were the most rigorous mathematical proofs. Gödel showed that in any closed logical structure there is a statement that is outside its language, as if chess could not prove that there is no way to make an infinite move.

The world of formulas turned out to be not a flawless machine, but a system with an internal shadow, a zone where knowledge ceases to be attainable. When Chaitin created Omega, he unexpectedly combined the ideas of Turing and Gödel. The halting problem showed that the fate of any program cannot be predicted, and Gödel's theorem showed that all truths cannot be proven. Omega became a bridge between these limitations. If we could calculate all its digits, we would know which programs halt and which don't, we would know all mathematical truths, which theorems are provable and which are false. We could discover whether mathematics itself is consistent, because one can write a program that searches for proofs of a contradiction, and if one exists, the program will halt; if not, it will run forever. Omega already knows how this search will end; if someone could read its digits to the end, they would discover whether mathematics itself is self-destructive. This knowledge would destroy the entire idea of proof, because if you can simply look at a number and find the answer, mathematics itself ceases to be a process of reasoning; it becomes a database where everything has already been decided. Fortunately, Omega is protected from us by the very nature of computation. Chaitin proved that it is uncomputable, which doesn't just mean it's too complex; it means it's impossible to create a program that could calculate its digits, because to do so would require solving the halting problem, which, as Turing showed, is fundamentally impossible. You can figure out the first few digits by experimenting with short programs that clearly halt, but the further you go, the more frequently you encounter programs whose fates are indeterminate: some may terminate in a second, others in billions of years, and still others never. And you won't know which of the three categories the next one belongs to until you've lived forever. Omega exists. It has a precise value, but it's forever hidden, not because someone hid it, but because logic itself has forbidden the path to it. This number exists, but it's unattainable, like a horizon that recedes as you take a step forward. The digits of Omega appear random; if you write them out in a row, you'll find no pattern, no formula, no symmetry, no repeating fragments—and this isn't just an illusion. Chaitin proved that they are truly random in the strict mathematical sense. This means that no algorithm can reproduce an Omega sequence shorter than the sequence itself. In other words, Omega is incompressible information, each digit a bit of pure chaos, hiding the answer to an infinite number of logical questions. This is paradoxical. After all, in mathematics, we're accustomed to seeking patterns, simplifications, and symmetry, but here we have perfect order, which appears as complete disorder. Omega is chaos that obeys the strictest possible laws. And somewhere between these opposites resides a strange beauty, as if the universe itself decided to joke about our desire to explain everything.

With Omega, we saw for the first time the limits not just of calculations, but of knowledge itself. It demonstrates that there are truths that objectively exist, but are fundamentally unattainable. They are independent of our minds or tools; they cannot be discovered at all, no matter the technology or the intellect. This isn't temporary ignorance, not "we don't know yet"—it's proven, something we will never know. It's not a death sentence, but a fundamental law woven into the fabric of logic. Mathematics has always seemed like an island of absolute order, but now we know that even on this island, there's a shore beyond which an ocean of the unknowable begins. We can see its waves, hear their roar, but we can't enter them. And perhaps this is precisely what makes mathematics human: it knows more than we do, yet still holds secrets. If you think about it, Omega is a protective mechanism of reality, like a fuse in an electrical circuit, preventing knowledge from exceeding a safe level. If we could calculate all its numbers, we would see all truths and all contradictions; mathematics would cease to be a system capable of standing on its own feet. Our mind strives to know everything, but not all knowledge can withstand it. "Omega" seems to be built into the very structure of existence, reminding us that there is a limit. If we could see everything at once, the very meaning of searching would disappear; knowledge without limits collapses the system, just as an infinite current burns out a wire. And perhaps this is precisely why "Omega" is uncalculable. The world itself has ensured that we do not touch what can destroy us not physically, but logically, from within. "Omega" is a reminder of humility, that even the most perfect structures contain mystery. The human mind is accustomed to seeing science as a ladder to absolute knowledge. But the higher we climb, the more clearly we understand there is no summit. And there is no defeat in this. This isn't an admission of powerlessness, but an awareness of scale. Our laws, formulas, and proofs aren't walls, but windows through which we peer into infinity. And if somewhere beyond the horizon lies a number that knows everything but doesn't reveal itself, then the world is wiser than we thought. Perhaps "Omega" truly does kill theory, but only theory that believes in the complete dominance of reason. It leaves room for mystery, for respect for what cannot be explained, and perhaps this is precisely what makes knowledge safe.»

But Reality isn't mathematics. Mathematics is part of Reality, logic is a part of mathematics. Moreover, in mathematics there are even moments when you find yourself on the verge of epistemology (but that's a separate discussion). . If we apply Niels Bohr's principle of complementarity, he once said that science is a way of understanding the world, but so is art, simply another aspect of it.

God is "Infinite, powerful Imagination."