Actual Infinity in Reality?

[RoadRunner79]

I'm not saying that you have to stop at the point. Passing  through is fine.  

Your demonstration did not offer a rebuttal, but further re-enforces my point.  This is exactly what I was saying. 

It's your own point that you are saying there is a contradiction with, even though you keep saying that there is no contradictions.  This is why myself and Steve are saying that you will never reach the destination of 1. 

This is using the same reasoning that you used to say that there is an infinite number of points, and showing that by that reasoning you cannot reach the destination.


Now if you would want to get into what you are calling an assumption that there is a point prior to our destination (A) where it is (!A); we can work through that.  However any way you get around this, I believe is going to cause you to abandon your model which is giving you an infinite number of "points".

The contradiction is to the assumption that there is a last number before 1. That was a proof that there is not.

This is called a proof by contradiction. You assume something, get a contradiction, and thereby show it is false. In this case, the assumption was that there is a last number before 1. There isn't. But there is no reason to think there would be.

If you are saying that you never reach the destination (A), then I would agree.  It was my understanding that you where saying that A was reached, as opposed to Zeno's conclusion that motion was impossible or you have your contradiction.

Here is my reasons for there needing to be an last point (note reasons, not an assumption). 

So we take a line between two points (0 and our destination) 
We'll define A as any point >= the destination, making any point less than the destination !A

We start at point 0 (!A)  and progress through the line of points (either finite or infinite) where the current point is always greater than the previous point.
If you reach the destination (A) then the last point you passed where !A is your final point.

I believe that your equation where you progress through where X<Y<A  is analogous to Zeno's paradox.   Where Y is the average between X and A;  and you keep repeating this sequence where then Y becomes X and you solve for Y again.

This involves the first principles of the Law of identity, the law of non-contradiction, and the law of excluded middle applied as we move along our line.  It assumes that we are progressing along the line, and that we do reach the destination.   We start off !A and progress through the point along the line until we reach A.

Now what myself and Steve have been saying, and that a number of people seem to not want to do the math on, is that given this approach; you will never reach A.  And I would agree, given that you never reach A, there will not be a last point prior to A (it's a nonsensical question in that case).  The contradiction would be in saying that both A and !A are true.  And what makes your claim of infinity also prevents you from ever ending at point A.

I think that we can take for granted, we are presented this in the middle.  Can I place another number in there... Sure.   However the problem arises when you start doing so on the ends, and realize that you either cannot finish or cannot start your journey with such thinking, and lead Zeno to conclude that motion is impossible.  Zeno according to the tale was a strict logician and even though it was demonstrated that motion was in fact possible, demanded that he be shown logically where he was wrong.

I hope that we are in agreement, that motion is possible. To me and others, Zeno's problem is not in that math, nor in his logic.  His problem was with his starting assumption of a continuum or having to cross an infinite.

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If it is said, that a line contains a continuum of points (however you choose to define them).  Despite the fact, that this supposed infinity ends at 1 which is contradictory to saying that it is infinite in number in itself. (note:  I'll use one as a destination in this writing, although it may be another length)   What is the point immediately prior to 1?  There is necessarily an instance, where you transition from "not 1" to "1" while traveling along this line.  

Here we should introduce the concept of the infinitesimal. Let's call it dx. dx is simply the limit, as x goes to zero, of x, or, in notation: lim(x->0) x.

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The value directly before 1 would be: 1 - lim(x->0) x, or 1 - dx.

This is the only way I know how to represent this "number".
The infinitesimal is often used in derivative and integral algebra, where the derivative of a function of x is represented as df/dx, usually meaning derivative of f with respect to x, but the df/dx nomenclature is actually more powerful than just a simple representation of the derivative, as it allows some infinitesimal algebra when you want to integrate a derivative:
∫df/dx dx = f  (+ a constant, but let's not go there)

Care for a simple example?

f(x) = x^2 = x*x

df/dx = f'(x) = lim(t->0) [f(x+t) - f(x)] / t
As you can see, this will always lead to an indeterminate 0/0, which is annoying, but it can be lifted, in most cases.
For our case:
df/dx = f'(x) = lim(t->0) [ (x+t)^2 - x^2 ] / t
= lim(t->0) [x^2 + t^2 + 2xt - x^2 ] / t
= lim(t->0) (t^2 + 2xt)/t
= lim(t->0) (t +  2x)
= 0 + 2x = 2x

So, the derivative of x^2 is 2x.
You can do it fox x^n and the result will be nx^(n-1)

Integration is the inverse operation and is always done, to the best of my knowledge, as a reduction of the function to be integrated into known derivatives.
Knowing that the derivative of x^2 is 2x, we can easily say that the integral of 2x is x^2.
Integration can also be though of as a successive sum of the function.

∫ 2x dx = Σ 2x dx, but remember that dx is an infinitesimal, so a normal sum isn't feasible.... however, computer numerical implementations do this all the time.

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How would you describe the limit as X goes to zero?

I suspect that when you include this, we are not that far apart in our thinking about the topic. That this allows you to end the infinite regression (or progression) and reach the destination.