# Graham number and look at infinity

You can peer into infinity in different ways. One can imagine the ever-increasing astronomical numbers and compare them with physical phenomena. You can peer at the selected point of the Mandelbrot fractal, gradually increasing the scale by 10

^{198}times (it can be more, but visibility suffers for the sake of speed). The fractal, however small a portion of it is taken, remains self-similar and retains a fractional structure.

And you can imagine the Graham number as it is represented by the author of the article "Graham Number on the Fingers". The Graham number is so great that even if you imagine some kind of 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 Graham’s number. To count to the Graham number, you have to learn to count in a completely different way than we are used to - imagining that the path to infinity lies through adding zeros to the astronomical numbers we know. In this counting system, bending a finger on a 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 raising to a power and further into unimaginable distances.

I immediately warn you that all these exercises are not unsupportive - do not get carried away, take care of your mental health. However, sometimes it’s useful to peer into infinity to understand where you are and what you, as a person, can oppose to it.

For me, at one time, the view of infinity, similar to the Graham number described on fingers, was given by the Ackerman function (which is given as an example of a complex recursive function in the theory of algorithms). It is closely related to Knuth's arrow notation used in the article about Graham's number.

The idea is very simple. Take the increment operation by 1, the increment, as the zero step. 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 to obtain the degree of erection operation, X

^{Y}. On the fourth, we get a “turret” of degrees X

^{X X of}length Y. On the fifth, we get a “turret” of turrets (what the author of the article called Graham’s number on his fingers called “turretless”). Well and so on.

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

Ackerman's function is growing very fast, it is growing unspeakably fast, it is growing faster than anything you can imagine. Already at the fifth step, she goes beyond the boundaries of the universe. But to count to Graham's number for the foreseeable number of steps, even it is not enough. We need to take the Akkerman function of the “second order”. Those. Ackerman function of Ackerman function of Ackerman function - and so Y times. It will be a kind of “turret” of Akkerman's functions. Here is a "tower" with a height of 64 floors, just up to Graham’s number and counts.

It seems that awareness of the inexpressible value of this number can crush a person. But do not rush to conclusions. The author of the mentioned article, trying to evaluate the approaches to this number, compares its elements with the number of particles in the Universe, compares the height of the "towers" with the distance between the planets. But all this seemingly inexpressibility is reduced to the number "one and a half". Okay, let’s be two and a half.

I will explain. It is necessary to consider “infinity” (in quotation marks — for any number is nevertheless finite) not by how many grains of sand it contains, but by how many times the quantity goes into quality, how many non-trivial ideas are in it. Let's count how many non-trivial ideas are in Graham’s number. Ackerman's function with its order of arithmetic operations as an argument to the function is an idea once. The application of the Ackerman function to itself - even for a full-fledged idea, does not pull in half, (and you can imagine the Ackerman function of the third order in order to get an even greater number - but all the more distinct the degeneracy of the idea). Let us also add, in fact, a description of the problem within which the Graham number appeared (painting in a random combination of two colors of the diagonals of multidimensional hypercubes) in order to have an idea of where to stay in our account - and we will get two and a half ideas.

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

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

For example, you get acquainted with some work - a book, a film, a video game - and notice that at some point the work begins to repeat itself. Perhaps most of all, video games are guilty of this - endless quests “kill so many such and such monsters”, “bring this and that”, the exponentially increasing cost of ever more elaborate weapons and armor to fight ever more tenacious enemies that give everything 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 evil infinity and only drags 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 known by what - to take all the same as in the original, but in large quantities and otherwise combined. There was one idea, it became one and a half. These sequels can now be done an infinite number, making money for those who fell in love with the original. And again, before us is evil infinity.

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