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Infinity and beyond, transfinite numbers Aleph

late nineteenth century the original mathematical Georg Cantor proposed a beautiful theory of finite or transfinite numbers, whereby the total number of fractions, integers and natural numbers are the same number you called Aleph transfinite sub-zero.

At first sight it seems reasonable, as one might think that the number of integers is greater than the number of natives, and that every natural number is an integer while some integers (negative) are not natural numbers. Similarly one might think, too, that the number of fractions is greater than the whole, but one thing is what it seems and one that is.


The key lies in the strange properties infinite numbers and relationships can be established between them. For finite objects of two different sets if we can establish a "one-to-one, between them, we can deduce that have the same number of elements. For a finite number of natural numbers the same thing happens, but what is obvious to cease to be finite numbers to infinite.

can establish a one-to-one relationship between the natural numbers and integers as follows: 0 ( whole) -> 0 (natural ) -1 (integer ) - > 1 (natural ) +1 ( whole) -> 2 (natural ) and so we continue indefinitely with the following table:

Each integer and every natural number appears once and only once in the table. This correspondence between each pair of numbers-natural whole is what sets in Cantor's theory that the number of elements in the integer column is equal to the number of elements in natural column. Therefore, the number of integers is the same as natural. Similarly, although somewhat more complicated, it can prove that the set of fractions (rational) has the same number of elements as the set of integers. The number is infinite, but no matter, it is the same number.

The great mathematician David Hilbert was invented metaphor Hotel Infinity to intuitively explain the paradoxes we face the existence of an infinity of infinities:

"There was a hotel that had infinite rooms. One day there comes a new guest to stay there, but the concierge said he had bad luck, they were all full. The guest indignantly called the manager, and asks how it was possible in a hotel with infinite rooms. The manager agrees, but says he can not do anything, then the host responds quickly: "I know what you can do, which is in the room 1 it sends to the room 2, room 2 to 3 and so on, then Room 1 will be free for me. The manager

found ma Raville this solution and it did. "

" Some days later another guest arrives and asks to stay, to what responds that the hotel was full, but not to worry, they knew how to fix it. Then the host says there was a problem, he was not alone, but with a group of friends ... and that it was an infinite group. The manager again shocked I did not know what to do, but the host, skillful also told not to worry, to send the room 1 to 2, from 2 to 4, the 3 to 6 and so on. Thus all odd-numbered rooms would be free to its infinite friends. "

sets that can be placed on one-to-one with the natural numbers is called countable, so that infinite sets are countable sub aleph -zero elements.

Surprisingly, although the extension of the system from natural numbers to integers and rational, not really increased the number of objects that work !.

After this you might think that all infinite sets are countable, but not, not only one type of infinity, as the situation is very different to go to the real numbers. Cantor proved by the argument "crosscut" actually more real numbers than rational. The real number is the number transfinite C, continuously, another name for the set of real numbers.

might think about giving that number called aleph sub-one, for example. But that name represents the next transfinite number greater than sub-zero aleph to decide whether or sub C = Aleph-one is a famous unsolved problem, the so-called continuum hypothesis.

As a curiosity, since we are talking about infinite, the term googol (googol English) is a huge number 10 ¹⁰⁰ was coined in 1938 by Milton Sirotta, a 9-year-old nephew of American mathematician Edward Kasner . Kasner announced the concept in his book Mathematics and the imagination. Isaac Asimov once said about it: "We will have to suffer eternally a number invented by the baby."

The googol is of particular importance in mathematics and has no practical applications. Kastner he created to illustrate the difference between an unimaginably large number and infinity, and is sometimes used in this way in the teaching of mathematics. The google search engine was so named because of the number . The original founders were going to call Googol, but ended up with Google because of a spelling error Larry Page, one of the founders of Google.