Graham's number and glance to infinity

Peering into infinity can be different. You can imagine the ever-increasing astronomical numbers and compare them with physical phenomena. You can peer at a selected point of the Mandelbrot fractal, smoothly increasing the scale by ^10,198 times (you can do more, but for the sake of speed, visibility suffers). A fractal, no matter how small it is taken, remains self-similar and retains its fractional structure.

And you can imagine the Graham number as it is represented by the author of the article “The Graham Number on Fingers” . Graham's number is so great that even if you imagine some monstrously large astronomical number, and then raise it to an equally monstrous degree, and then repeat all this monstrous number of times - then you will not even budge on the scale of that path, which leads to the Graham number. In order to count up to the Graham number, one will have to learn to count in a completely different way than we are accustomed to - imagining that the path to infinity lies through writing zeros to the astronomical numbers known to us. In this counting system, the finger bending on the hand will correspond not to adding one or a million to the number, not adding zero or hundreds of zeros at once, but a step from addition to multiplication, from multiplication to exponentiation and further into unimaginable distances.
')
Immediately I warn you that all these exercises are not inferior - do not get carried away, take care of your mental health. However, it is sometimes useful to peer into infinity in order to understand where you are and what you, as a person, can oppose.

For me, at one time an infinity look, similar to the Graham number “on the fingers”, was given by the Ackermann function (which is cited as an example of a complex recursive function in the theory of algorithms). It is closely related to the Knut switch used in the article about the Graham number.

The idea is very simple. Take the increment operation by 1 as the zero step, increment. Those. X + 1. As the first step, take the increment repeated Y times. We get X + Y, i.e. addition operation. As a second step, take the addition of X with itself, repeated Y times. We get X · Y, i.e. multiplication operation. In the third step, we obtain the exponentiation operation, X ^Y. In the fourth, we get a “turret” of degrees X ^{X X} long Y. On the fifth, a “turret of turrets” (what the author of an article about the Graham number on his fingers called “bezbashny”). Well, and so on.

If we take a natural (i.e. non-negative integer) number and apply an operation of an order equal to that number to it, then we will get approximately the Ackermann function (in fact, it is more difficult to determine from three or two arguments, but not the essence) .

Ackermann’s function is growing very fast, it is growing inexpressibly fast, it is growing faster than anything you can imagine. Already in the fifth step, it goes beyond the boundaries of the universe. But in order to count up to the Graham number in the foreseeable number of steps, even that is not enough. We need to take the function of Ackermann "second order". Those. Ackermann function of Ackermann function of Ackermann function - and so Y times. It turns out a kind of "turret" of the functions of Akkerman. This is the “tower” with a height of 64 floors just up to the number of Graham and counts.

It seems that the realization of the ineffable magnitude of this number can crush a person. But do not rush to conclusions. The author of this article, trying to assess the approaches to this number, compares its elements with the number of particles in the Universe, compares the height of the “turrets” with the distance between the planets. But all this seeming inexpressibleness comes down to the number "one and a half". Okay, let "two and a half."

I will explain. It is necessary to count “infinity” (in quotes - for any number is, of course) not by how many sand grains it contains, but by how many times the quantity goes into quality, how many non-trivial ideas are in it. We calculate how many nontrivial ideas in the number of Graham. Ackermann's function with its order of arithmetic operation as a function argument is an idea of ​​times. Applying the Ackermann function to itself does not even pull on a full-fledged idea, so to half (and you can imagine a third-order Akkerman function to get even more numbers - but the more clearly the degeneracy of the idea). Let us add, in fact, a description of the problem, within which the Graham number appeared (painting in a random combination of two colors of diagonals of multidimensional hypercubes), in order to have an idea where to stop in our account - and we get two and a half ideas.

It seems, on the one hand, almost immeasurable infinity - and on the other hand, triviality. Put two mirrors in front of each other, stand between them - and you will see an infinite number of increasingly dull reflections. There are an infinite number of reflections, but they have only one original - only you yourself are reflected.

If in some phenomenon you notice that from some point only worsening (at best, the same) copies of what has happened before begin to repeat, then this is bad infinity, false. Movement on its scale is only an appearance of life, but in fact it is a trap for your consciousness.

For example, you get acquainted with some work - a book, a movie, a video game - and you notice that from a certain moment the work begins to repeat itself. Perhaps most of all, video games are doing this — endless quests “kill so many such monsters”, “bring this and that”, the exponentially rising cost of ever more elaborate weapons and armor to fight against more and more resilient enemies more play money. If the repetition ceased to reveal the original idea and became an end in itself, then leave this work - it fell into a bad infinity and only drains you from the true path.

Or there was a good original - and they made him a sequel, a prequel or an offshoot of the plot. What to fill? It is well known how to take everything the same as in the original, but in large quantities and otherwise combined. There was one idea, it became a half. These sequels can now make an infinite number, earning money for those who liked the original. And again before us is a bad infinity.

In general, take any genre - and most of it will be repetition, degraded copies of the founder of the genre. If you feel that you are suffocating in the dominance of these similar reflections - swim against the current, look for the source of the reflections. Only in this way can you find the true path in the labyrinth of evil infinity.