Thursday, November 25, 2010

Dvd Third Party Decryption

Weird (but true)

0.999 .... (With infinite nines) is an integer.

Sunday, November 21, 2010

Nico Stremaing Megavidio

The Omegon and all that ... (Part 17 what end?)

( A part 16 - part 18 A )

The beginning and end

We said in the previous chapter, in the early 1870's, Georg Cantor came to Halle. Cantor began working under the direction of Eduard Heine who proposed the following problem: Is it only the Fourier series decomposition of a periodic function, even if it is at all time, an infinite number of singular points? (Heine had proved that the answer is positive when the number of singular points is finite.)

few months later, around 1872, Cantor got a first answer: the decomposition is unique provided that the singular points of the function are distributed on the real line in a certain way. But Cantor is not found, in principle, a clear and precise way to state what conditions must comply with this distribution.

As a theorem not only be shown, but this demonstration must be written so that it is understandable [the shows are written and read in human beings], Cantor was devoted to the problem, essentially linguistic, find a way clearly explaining what the assumptions to be met by the set of singular points for the decomposition in Fourier series was unique. And it was in the course of solving this problem which Cantor created a concept that would later career in mathematics: the concept of accumulation point .

is not necessary here a precise definition of this concept (this link is a very brief explanation - in this post extending ). Suffice to say that, given a set P of real numbers, we can define (the terminology and notation are Cantor) a set P ', which is called the derived set of P , which contains all accumulation points P. This set P 'can be equal to P, or P may contain, or may be the empty set, etc. For example, if P = [0.1], then P 'is set to be the same [0.1].

But we can also calculate the derivative of the derivative, P '. And the derivative of the derivative of derivative P''', etc. In the case of P = [0.1] all these successive derivatives remain the [0.1], but in other cases different sets are obtained. For example, if P = {0, 1, 1 / 2, 1 / 3, 1 / 4 ,...} then P '= {0} and P is the empty set.

Cantor noted that in some cases these successive derivations ending in the empty set (as in the second example), while in other cases this never happened (as in the case of [0.1]). Cantor sets of first type called the first and second type to second. and stated his theorem as follows: if the singular points of a periodic function are a set of first type write sing its Fourier series is unique. Since finite sets out to be the first type, the result of Cantor included as a special case that of Heine.

But Cantor was not consistent with this result and kept thinking. He noted that this process of successive derivatives could define a "derivative infinite" means the limit of P (n) (nth derivative of P) with n tending to infinity. Sien embargo, so far, still was not making use "dangerous" of infinity, still in the familiar terrain of infinite potential, infinite limit. However, he was on the verge of great discovery, which occurred when, at one point, he found an example of a set P such that P (infinity) = {0} ... and therefore (P (infinity)) '= P (infinity + 1) = {0}' = empty.

What was this "infinity + 1"? He was not the infinite potential, the limit because the limit for both "infinite" as "infinity + 1" are the same. But in this case was not the case, as P (infinity) and P (infinity + 1) were different outfits. Then it was a different concept of infinity.

must have been very traumatic for Cantor the face to face with him for so many centuries banned and feared infinite potential. So much so that it took ten years to accept that what I had on hand was nothing more, nothing less, than a way of counting beyond infinity. And when I finally accepted in 1883, published an article entitled Fudamentos for a General Theory of Sets , where Cantor includes his much-quoted phrase: "I was taken by the logic of my research to break with traditions that had taught me to respect "[some translations say" worship "]. In this work, also changed the usual symbol of infinity (the "horizontal eight) by the Greek letter omega, in order to emphasize that" his "infinite was not the limit. In that work, also dubbed "ordinal" these new "infinite numbers" and began the story I have narrated.

And here the story ends, or perhaps begins, because the paradoxes of Cantor's theory led to the Crisis of Foundations, the program of Hilbert, Gödel's Theorem, to Turing machines, jobs Frege, the intuitionism of Brouwer, ... Remains in the desire of the reader deep into the maze, a labyrinth of which we have shown only the entry and that, fortunately, has no outlet.

The End? - It seems that no ...

(16 A Part - A part 18)