The infinite and further away, the numbers transfinitos Aleph

At the end of the XIXth century the original mathematician Georg Cantor proposed a beautiful theory on the finite numbers or transfinitos, according to which the entire number of fractions, entire numbers and natural numbers are the same number transfinito to the one that Aleph sub-zero called.
At first sight it does not seem slightly reasonable, since it might think that the number of points is major than the number of natives, since any natural number is a point while some points (the negatives) are not natural numbers. Of similar form it might think, also, that the number of fractions is major than that of points, but a thing is what seems and other one what is.
The key is in the strange properties of the infinite numbers and the relations that can be established between them. For finite objects of two different sets if we can establish a "mail one uno - or", between both, it is possible to deduce that they have the same number of elements. For a finite number of natural numbers it happens the same, but what is clear for finite numbers it stops being for infinites.
One one can establish a mail uno - or between the natural numbers and the entire numbers of the following form: 0 (point)-> 0 (native);-1 (I inform)-> 1 (native); +1 (point)-> 2 (native) and this way we continue indefinitely with the following table:

Every point and every natural number appear one and only once in the table. This mail between every pair of numbers point - native is what it establishes in the Singer's theory that the number of elements of the column of points is equal to the number of elements in the natives' column. Consequently, the number of points is the same that that of natives. Of similar form, although slightly more complicated, it is possible to prove that the set of (rational) fractions has the same number of elements as the set of points. The number is infinite, but it does not matter, it is the same number.
The big mathematician David Hilbert invented the metaphor of the Infinite Hotel to explain of intuitive form the paradoxes to which we are faced by the existence of infinity of infinites:
"There was a hotel that had infinite rooms. A new guest comes one day to stay there, but the caretaker says to him that it had no luck, that all floods were. The guest, infuriated calls the manager, and asks him how it was possible in a hotel with infinite rooms. The manager gives him the account, but he says that it cannot do anything, then the guest answers quickly: ' already what it is possible to do; there orders in that it is in the room 1 to the room 2, to that of the room 2 to 3 and this way successively, then the room 1 will remain free for me. The gerenteencontró ma ravillosa this solution and this way it did it".
"Some days later another guest comes and he asks of staying, to what they answer him that the hotel was full, but him not to worry, that they knew how to solve it. Then this guest says that there was a problem, that he was not alone, but with a group of friends … and that was an infinite group. The manager, again dismayed did not know what to do, but the guest, also very skillful says to him that he should not worry, that it should order that of the room 1 2, that of 2 4, that of 3 6 and this way successively. Thus all the rooms with odd numbers would remain free for his friendly infinites."
The sets that can be put in mail one uno - or with the natural numbers they are called numerables, so that the infinite sets numerables have aleph sub-zero element.
Surprisingly, although the system is extended from the natural numbers to the points and to the rational ones, we do not increase really the number of objects with which we work!.
Later all this we might think that all the infinite sets are numerables, but it is not like that, not only there is a type of infinite, so the situation is very different on having gone on to the real numbers. Singer demonstrated by means of the argument of the "diagonal court" that really there are more numbers real that rational. The number of real is the number transfinito C, constantly, another name that receives the system of the real numbers.
We might think of giving him to this number the name of aleph the sub-one, for example. But this name represents the following number transfinito major than aleph sub-zero and deciding if really C = Aleph constitutes the sub-one a famous not decisive problem, the called hypothesis of the continuous one.
Like curiosity, since we are speaking about infinites, the term gugol (in English googol) is an enormous number 10100 there was minted in 1938 by Milton Sirotta, a 9-year-old child, nephew of the American mathematician Edward Kasner. Kasner announced the concept in his book The mathematics and the imagination. Isaac Asimov said in an occasion on this matter: "We will have to endure eternally a number invented by a baby".
The gúgol is not of particular importance in the mathematics and neither it has practical uses. Kastner created it to illustrate the difference between a number inimaginablemente big and the infinite, and sometimes it is used this way in the mathematics education. The engine of search of google was called this way due to this number. The original founders were going to call it Googol, but they ended with Google due to an error of spelling of Larry Page, one of the founders of Google.