Re: Calculus XOR Probability

Tony Orlow wrote:
Virgil said:
In article <MPG.1ed290581a4f392198acd4@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
For the last time, no. If the limit of the staircase is anything different from the diagonal, which it is, then there is no contradiction.
There is no mathematically valid model in which the limit of the sequence of staircase functions is anything but the diagonal function.

If TO wished to claim otherwise, then he must create and present to us the entire system in which he claims his allegations hold, as they do not hold in any current system.

Okay. Here goes.

Rather than a set of points, let us define both the staircase and the diagonal as sequences of segments defined as a pair of reals which represent the x and y coordinate differences between subsequent points. Let us compare the two thus in a segment-wise manner, maintaining the same number of segments in each, and see if the segments which describe the staircase approach those that describe the diagonal. Where n=1, we have two segments to the staircase, {0,1} and {1,0}, for a total change of {1,1}. Dividing the diagonal into two segments we have {1/2,1/2} and {1/2,1/2}, also for a total change of {1,1}. Now, as n increases we have {1,1}=sum(x=1->n: {1/n,0}+{0,1/n}) for the diagonal, and sum
{1,1}=(x=1->n: {1/2n,1/2n}+{1/2n,1/2n}) for the diagonal. While the locations of the points in each segment become arbitrarily close, the vectors defining the segments of which the lines are made never become close, but are always at a 45 degree angle to their corresponding segments in the other line.

When you look at the distance traveled, you sum all the x components of the vectors in each line and sum all the y components, and you get {1,1} in both cases, and the distance is sqrt(2).

When you look at the lengths of each, you sum the length of each vector in the line. For the staircase we have sum(x=1->n: 1/n+1/n)=2. For the diagonal we have sum(x=1->n: 1/sqrt(2)+1/sqrt(2))=sqrt(2). Because of the difference in vector direction, even at the infinitesimal scale, the staircase is longer than the diagonal.

Is that an "entire" enough "system" for you? :D

Well, do you agree that /if/ the limit of the staircase is /not/ "anything different from" the diagonal, then there /is/ a contradiction?
Of course. If there were not distinguishing characteristics between the diagonal and the staircase in the limit, or some other explanation for the discrepancy, then I would have to admit that you may have a real counterexample to refute the validity of infinite induction.
And in standard mathematics there are no such distinguishing characteristics. I have, in fact, given a specific and concrete example of the staircases as parametric functions of the diagonal distance whose limit is the diagonal itself.

No, you made a leap in saying that the points become the same set in the limit, just because they are arbitrarily close.

First of all, it's not that "the points become the same set in the limit". It's
that the limits of the two sets of points are identical (the same set of
points). Nothing "becomes anything in the limit".

Second, the statement Virgil is making is not a leap; it's a consequence of
the definition of limit. Unless you have a different definition.

Matt

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Relevant Pages

  • Re: Calculus XOR Probability
    ... According to TO's infinite induction principle it would necessarily be ... The length of the staircase in the limit IS 2, ... The segments have directions, which alternate between vertical and horizontal, ... We're not defining the curve by the endpoints any more. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... Rather than a set of points, let us define both the staircase and the diagonal ... as sequences of segments .... ... of the sequence of curves, which clearly distinguishes between the two. ... We're not defining the curve by the endpoints any more. ...
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  • Re: Calculus XOR Probability
    ... sequence of curves "staircases". ... So, the diagonal line as a whole is the limit of the staircase as a whole, as ... the size of them shrunk accordingly, in the infinite limit of this process, the ... points persay, but line segments. ...
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  • Re: Calculus XOR Probability
    ... contradiction. ... sequence of staircase functions is anything but the diagonal function. ... Rather than a set of points, let us define both the staircase and the diagonal ... as sequences of segments defined as a pair of reals which represent the x and y ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... different from the diagonal, which it is, then there is no contradiction. ... sequence of staircase functions is anything but the diagonal function. ... Rather than a set of points, let us define both the staircase and the diagonal as sequences of segments defined as a pair of reals which represent the x and y coordinate differences between subsequent points. ...
    (sci.math)