Actual infinity.
Inspired by this post.
For the most part, mathematics operates with the concept of actual infinity, it is all analytical geometry, almost all mathematical analysis, the algebra of fields of characteristic zero and much more.
We are so accustomed to the icon ∞ , that we do not even notice how we use it, not only in mathematics, but also in real life. It is pleasant to consider time as continuous, but the concept of continuity is set through the limit, which in turn operates with infinity.
The concept of infinity is one of the most simple. Do not believe?
Tell me, what is easier for you to perceive - ∞ or the number 1307489315878623114365278078516974 ?
In fact, the real infinity begins very close. For example, take a number 10 10 10 , about which we can safely say that this is a real plus infinity. It is real, and not introduced at the level of abstraction.
For example, consider a lot of books that are less than 10,000 pages long, try to estimate their number, it is easy to see that we will describe almost all existing books and most of those that will be released in the future.
Suppose that on a page of 10,000 letters, each letter can be represented in just 100,000 in a number of different ways (in general, humanity fits into unicode so far - 98,884 characters ( Unicode 5.0 )).
The total content of a book consisting of exactly 10 4 pages can be represented by (10 5 ) 10 4 * 10 4 - the number of letters to the degree the number of letters per page can be multiplied by the number of pages, that is 10 5 * 10 8 .
The number of all books with the number of pages ≤ 10,000 will not exceed 10 5 * 10 5 * 10 8 = 10 5 + 5 * 10 8 <10 10 9 .
It should be borne in mind that two books are considered different, even if they differ by only one symbol.
Thus, by such a rather small number, the entire world literature was described for a sufficiently long period of time ahead. Moreover, a rough estimate from above is extremely rough.
So what is more endless? Is the number 10 10 10 or ∞ ?
For the most part, mathematics operates with the concept of actual infinity, it is all analytical geometry, almost all mathematical analysis, the algebra of fields of characteristic zero and much more.
We are so accustomed to the icon ∞ , that we do not even notice how we use it, not only in mathematics, but also in real life. It is pleasant to consider time as continuous, but the concept of continuity is set through the limit, which in turn operates with infinity.
The concept of infinity is one of the most simple. Do not believe?
Tell me, what is easier for you to perceive - ∞ or the number 1307489315878623114365278078516974 ?
In fact, the real infinity begins very close. For example, take a number 10 10 10 , about which we can safely say that this is a real plus infinity. It is real, and not introduced at the level of abstraction.
For example, consider a lot of books that are less than 10,000 pages long, try to estimate their number, it is easy to see that we will describe almost all existing books and most of those that will be released in the future.
Suppose that on a page of 10,000 letters, each letter can be represented in just 100,000 in a number of different ways (in general, humanity fits into unicode so far - 98,884 characters ( Unicode 5.0 )).
The total content of a book consisting of exactly 10 4 pages can be represented by (10 5 ) 10 4 * 10 4 - the number of letters to the degree the number of letters per page can be multiplied by the number of pages, that is 10 5 * 10 8 .
The number of all books with the number of pages ≤ 10,000 will not exceed 10 5 * 10 5 * 10 8 = 10 5 + 5 * 10 8 <10 10 9 .
It should be borne in mind that two books are considered different, even if they differ by only one symbol.
Thus, by such a rather small number, the entire world literature was described for a sufficiently long period of time ahead. Moreover, a rough estimate from above is extremely rough.
So what is more endless? Is the number 10 10 10 or ∞ ?
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Source: https://habr.com/ru/post/24224/
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