In Part 17 - A 19
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Addendum on collection points
In this post I want to expand the explanation of the role played in the development of ordinal theory, the concept of "point of accumulation." It was mentioned in previous chapters, but now I'm going to delve into concepts mentioned above a little pass.
We said a few chapters ago that, in the early 1870's, Georg Cantor began working at the University of Halle. There Eduard Heine, the director, he posed the following problem: we have a periodic function f (x) we have developed in Fourier series if in every period the number of singular points of f (x) (ie points of discontinuity of f (x) or points where the series is divergent) is infinite then can we ensure that writing in Fourier series of f (x) is the only possible (or, conversely, may be another different set to converge to the same function)?
remember that Heine had already resolved the question affirmatively for a finite number of singular point, Cantor now faces the "infinite case."
We also said that a few months after the issue raised, Cantor had an initial response, you can ensure that writing is only as long as the singular points are distributed on the line in a certain way. But Cantor, in the first instance, failed to find a clear and direct way to describe what were the conditions to be met by that distribution. After a while managed to get that description is clear and simple, and it created the concept of "point of accumulation."
What is an accumulation point? I give the definition given Cantor, he referred specifically to the case of real numbers (later the concept was much broader contexts). Previously need to remember which means that a sequence of numbers converges to a limit L.
A succession of (1), a (2), a (3 ),... converges to L if, any distance fixed epsilon, exists a natural number n (which depends on epsilon) such that if m> n then the distance between a (m) and L is less than epsilon . Translated into Castilian more vague but perhaps less arid regions: a (1), a (2), a (3 ),... converges to L if, taking n large enough, all terms of the sequence from a (n) approach L as much as you want.
The classic example is the sequence a (n) = 1 / n, whose terms are 1, 1 / 2, 1 / 3, 1 / 4, ... and converges to 0.
Definition: If P is a subset of real numbers, we say that b is accumulation point of P if there exists a sequence a (n) is not constant and consists entirely of elements of P such that a (n) converges to b.
For example, if P = (0.1), then 0 is a point of accumulation of P. For example, a sequence consisting of elements of P and converges to 0 is a (n) = 1 / (n + 1) (always take n = 1, 2, 3, 4 ,...). In fact, it is easy to see that the set of all accumulation points of P is [0.1].
Exercise for the reader: Show, from the above definition that if P is finite then there is no accumulation points.
Definition: will call P 'the resulting of P, the set of all accumulation points of P.
Therefore, for P = (0.1), we have P '= [0.1].
Let another example. Now take P = {0, 1, 1 / 2, 1 / 3, 1 / 4, 1 / 5 ,...}. That is, P consists of the 0 and all numbers of the form 1 / n with n = 1, 2, 3, 4, ... It is clear that 0 is accumulation point of P. What about the other numbers?
Here, 1 is no accumulation point of P. If it were, there should be other elements of P as close to 1 as desired (these elements would be the terms of the sequence a (n) of which he speaks the definition.) But this does not happen, since no other points of P at the hands of 1 / 2 away from 1. That is, over the range (1 - 1 / 2, 1 + 1 / 2) no P elements very different from 1. Therefore, the 1 is an isolated point of P and is not about accumulation.
The same applies to the 1 / 2, since there are no points of P at a distance less than 1 / 6 of it. And it is with the 1 / 3, the 1 / 4, etc. All points of the form 1 / n are isolated points of P. On the other hand, it is easy to see that the points not belonging to P are not points of accumulation. Therefore, for P = {0, 1, 1 / 2, 1 / 3, 1 / 4, 1 / 5 ,...} worth that P '= {0}.
Of course, we also define the derivative of the derivative and P is (P ')' = P ". And the derivative of this: (P) '= P''', etc. To those who call derived second, third derivative , etc.
Observe that if P = (0.1) then P '= [ 0.1], P = [0.1], P'''= [0.1], etc.
On the other hand, if P = {0, 1, 1 / 2, 1 / 3, 1 / 4, 1 / 5 ,...}, then P '= {0} and P is the empty set (the derivative of {0}, and for every finite set is a vacuum). We have so the second derivative of P is the empty set.
Is it possible to find a set P such that its derivative third is the gap (But none of the above)? The answer is yes, but the study in the next chapter ...
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