Actual Infinity in Reality?

[polymath257]

Let's be clear about everything here.

1. What is the process we are using in going from x=0 to x=1?  Answer: We move with a speed of .2 units per second. In this, the time variable, t, goes from 0 to 5. For convenience, we can also assume we continue going past t=5, say to t=6 and thereby reach x=1.2.

2. What does it mean to say we go through every point? Answer: For each x, there is a t such that we are at x at time t.

Now, is it true that we go through every x between 0 and 1, inclusive? Answer yes. If we are curious what time we are at x, then t=5x will work.

So, is there a time for each point of the sequence of halves? Answer: Yes. In fact, for *every* x, there is a t, so we go through every point. So, in particular, for each x value in that sequence there is a corresponding t value when we pass through it.

Now to answer your questions specifically.

1. Do we 'jump out' of the sequence at some point? Actually, we jump in and out of the sequence many times. We are only in the sequence at times t=2.5, 3.75, 4.375,... ALL the rest of the times between t=0 and t=5, we are out of the sequence. So, for example, when t=2, we are at x=.4 which is not in the sequence. But, we are *past* that sequence at t=5, in which case, x=1. At no point *in the sequence* are we equal to, or greater than 1. But we get out of the sequence, none the less.

2. Since you assumed X<1, there is no time when X=1. But there is a *time* when X=1, namely T=5. Again, each point of the sequence has a time in which we go through it *and* there is a time in which we go through x=1.

3. I think the error is the implicit assumption that we cannot go through an infinite sequence of points. In fact, each one of those points is gone through and we can figure out what time each is passed.

The definition of 'infinite' you use in this case doesn't apply. The sequence of halves that you focus on *does* have a limit: x=1. Every single one of them is smaller than 1, so that is a limit. There are two aspects here:

1. For every point in the sequence, there is a point after that point of the sequence.

2. There is something larger than every point of the sequence.

According to 1, the sequence is unbounded. According to 2, the sequence is bounded. Both are true, but there are two different versions of 'bounded' in use here. There is no contradiction. So which version of 'bounded' do you want to use? is this sequence infinite or not?

I would say it is infinite since there are an infinite sets of point in that sequence. But, for my definition, an infinite set can be bounded and even have an end. The problem is in your ambiguous definition, not in the math.

I appreciate all the effort you are putting in, but it seems like you are trying to over complicate things, and answer a number of other question that where not being posed.   It doesn't matter what time the train will arrive in Boston, and the dispute is not that you can reach the end.  

We have a line with a start and an end point.  We will assume that there are an infinite number of points between these two positions.
We progress through this line towards the end point, passing through each point in succession along the way.
From any given point, along that line we will always have more points between the current position and the end position.
All prior points must be reached, in order to reach the end position
If there is always another point, that is not the end, and which precedes the end, then end cannot be reached.
Therefore the end position is not reachable if there is an infinite number of points which must be traveled.  

The disagreement is not that you cannot reach the end position (I believe that motion is fairly well evidenced).  There is not a problem with the logic here.   The problem is that if you have to complete something that never ends, then you will never complete.  To say that it is infinite and that it ends, is contradictory.  Something cannot be both A and !A at the same time.

The two statements in bold are false. The failure of the first bold statement invalidates the second. The rest of the statements are all correct.

And, in fact, if you look at my description, you see why the bold statements are false.

Remember, to go through all those points means that there is a time corresponding to each point. And that is true.

So, yes, if you have not reached the end, there is always another point to go through (in fact, an infinite number of points to go through) and you do reach the end.

Your assumption that we cannot go through an infinite number of points is invalidated.

[RoadRunner79]

I appreciate all the effort you are putting in, but it seems like you are trying to over complicate things, and answer a number of other question that where not being posed.   It doesn't matter what time the train will arrive in Boston, and the dispute is not that you can reach the end.  

We have a line with a start and an end point.  We will assume that there are an infinite number of points between these two positions.
We progress through this line towards the end point, passing through each point in succession along the way.
From any given point, along that line we will always have more points between the current position and the end position.
All prior points must be reached, in order to reach the end position
If there is always another point, that is not the end, and which precedes the end, then end cannot be reached.
Therefore the end position is not reachable if there is an infinite number of points which must be traveled.  

The disagreement is not that you cannot reach the end position (I believe that motion is fairly well evidenced).  There is not a problem with the logic here.   The problem is that if you have to complete something that never ends, then you will never complete.  To say that it is infinite and that it ends, is contradictory.  Something cannot be both A and !A at the same time.

Ok, perhaps if you let go of your unscientific time "theory", you might see the answer clearly. I think the A-theory of time is your main hurdle right there, whether or not polymath's answer works just as well with the A-theory of time.

Was time a factor in what I said at all?

Now if you are arguing against motion from a different perspective or something, then make your case. Otherwise I don't' see how this fits it.

[polymath257]

Ok, perhaps if you let go of your unscientific time "theory", you might see the answer clearly. I think the A-theory of time is your main hurdle right there, whether or not polymath's answer works just as well with the A-theory of time.

Was time a factor in what I said at all?

Now if you are arguing against motion from a different perspective or something, then make your case.   Otherwise I don't' see how this fits it.

If you are discussing motion, then time is a factor.

[GrandizerII]

Ok, perhaps if you let go of your unscientific time "theory", you might see the answer clearly. I think the A-theory of time is your main hurdle right there, whether or not polymath's answer works just as well with the A-theory of time.

Was time a factor in what I said at all?

Now if you are arguing against motion from a different perspective or something, then make your case. Otherwise I don't' see how this fits it.

Of course time is a factor.

Heres the article which I linked to a few pages back, and which addresses Zenos motion paradoxes from a physics standpoint.

http://www.mathpages.com/rr/s3-07/3-07.htm

Read the second last paragraph at least, if you dont want to read the whole thing. Though the whole thing is an interesting read.

[Amarok]

Was time a factor in what I said at all?

Now if you are arguing against motion from a different perspective or something, then make your case.   Otherwise I don't' see how this fits it.

Of course time is a factor.

Heres the article which I linked to a few pages back, and which addresses Zenos motion paradoxes from a physics standpoint.

http://www.mathpages.com/rr/s3-07/3-07.htm

Read the second last paragraph at least, if you dont want to read the whole thing. Though the whole thing is an interesting read.

He won't read it and will just repeat the same thing again .

[RoadRunner79]

I appreciate all the effort you are putting in, but it seems like you are trying to over complicate things, and answer a number of other question that where not being posed.   It doesn't matter what time the train will arrive in Boston, and the dispute is not that you can reach the end.  

We have a line with a start and an end point.  We will assume that there are an infinite number of points between these two positions.
We progress through this line towards the end point, passing through each point in succession along the way.
From any given point, along that line we will always have more points between the current position and the end position.
All prior points must be reached, in order to reach the end position
If there is always another point, that is not the end, and which precedes the end, then end cannot be reached.
Therefore the end position is not reachable if there is an infinite number of points which must be traveled.  

The disagreement is not that you cannot reach the end position (I believe that motion is fairly well evidenced).  There is not a problem with the logic here.   The problem is that if you have to complete something that never ends, then you will never complete.  To say that it is infinite and that it ends, is contradictory.  Something cannot be both A and !A at the same time.

The two statements in bold are false. The failure of the first bold statement invalidates the second. The rest of the statements are all correct.

And, in fact, if you look at my description, you see why the bold statements are false.

Remember, to go through all those points means that there is a time corresponding to each point. And that is true.

So, yes, if you have not reached the end, there is always another point to go through (in fact, an infinite number of points to go through) and you do reach the end.

Your assumption that we cannot go through an infinite number of points is invalidated.

Those statements follow logically from those that proceeded it.
Here they are again

We have a line with a start and an end point.  We will assume that there are an infinite number of points between these two positions.
We progress through this line towards the end point, passing through each point in succession along the way.
From any given point, along that line we will always have more points between the current position and the end position.
All prior points must be reached, in order to reach the end position
[b]If there is always another point, that is not the end, and which precedes the end, then end cannot be reached.
Therefore the end position is not reachable if there is an infinite number of points which must be traveled. 

If all points prior to the destination must be passed through sequentially, before reaching the destination; and there is an infinite number of points prior to destination so that the next point is always not the destination.  Then the end position will not be reachable.

It cannot be without end, and end at the same time... this is the contradiction.   Now perhaps you assume that your model is infinite, and you can calculate any infinitely small point and corresponding time.  However, this is why I asked numerous times previously, what exactly your points represent, in a real world motion.  If your model is only conceptual, then I suppose you can have whatever you want.   You don't actually traverse an infinite number of points in your model, you just assume that you do.  And if you actually follow the same math that shows that it is infinite, it also shows you that it cannot be completed.    Your model appears to be logically incoherent.

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Was time a factor in what I said at all?

Now if you are arguing against motion from a different perspective or something, then make your case.   Otherwise I don't' see how this fits it.

If you are discussing motion, then time is a factor.

Sure, so are accelerations and deceleration, and if your dealing with actual motion, then it's likely not going to be perfectly consistent all the way through.

However what I was saying didn't involve time, and I don't see where adding time in, effects it.  So while it is assumed that there is motion, that one is traversing the path of the line in some manner; the amount of time, or the even constant motion is not a factor.

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Was time a factor in what I said at all?

Now if you are arguing against motion from a different perspective or something, then make your case.   Otherwise I don't' see how this fits it.

Of course time is a factor.

Heres the article which I linked to a few pages back, and which addresses Zenos motion paradoxes from a physics standpoint.

http://www.mathpages.com/rr/s3-07/3-07.htm

Read the second last paragraph at least, if you dont want to read the whole thing. Though the whole thing is an interesting read.

What do you think in that addresses the logical contradiction being presented in Zeno's paradox?

[polymath257]

The two statements in bold are false. The failure of the first bold statement invalidates the second. The rest of the statements are all correct.

And, in fact, if you look at my description, you see why the bold statements are false.

Remember, to go through all those points means that there is a time corresponding to each point. And that is true.

So, yes, if you have not reached the end, there is always another point to go through (in fact, an infinite number of points to go through) and you do reach the end.

Your assumption that we cannot go through an infinite number of points is invalidated.

Those statements follow logically from those that proceeded it.
Here they are again

If there is always another point, that is not the end, and which precedes the end, then end cannot be reached.
Therefore the end position is not reachable if there is an infinite number of points which must be traveled. 
[/qutote][/b]


If all points prior to the destination must be passed through sequentially, before reaching the destination; and there is an infinite number of points prior to destination so that the next point is always not the destination.  Then the end position will not be reachable.

It cannot be without end, and end at the same time... this is the contradiction.   Now perhaps you assume that your model is infinite, and you can calculate any infinitely small point and corresponding time.  However, this is why I asked numerous times previously, what exactly your points represent, in a real world motion.  If your model is only conceptual, then I suppose you can have whatever you want.   You don't actually traverse an infinite number of points in your model, you just assume that you do.  And if you actually follow the same math that shows that it is infinite, it also shows you that it cannot be completed.    Your model appears to be logically incoherent.

And, again, it is two different notions of 'having an end'. If the difference is kept in mind, the issue resolves itself.

1. In the sequence: whenever you are in the sequence, there is another term of the sequence later on. In this sense, the sequence does not end. But this is irrelevant to whether you go through every point.

2. In the interval between 0 and 1: In this sense, the sequence has an 'end'. More technically, it has an upper bound. And 1 is an upper bound. This is the sense that is important in asking if we go through every point. Since each point corresponds to a time, we do, in fact, finish that sequence.

The point here is that the sequence has an upper bound, but that upper bound is not in the sequence. There is no contradiction here, just an opportunity to learn.

My points represent locations between the two points. Again, what does it mean to 'traverse an infinite number of points'? It *means* that for each point, there is a time when we are at that point. And that is the case here. So, yes, we do traverse an infinite number of points. You intuition that this is impossible is just wrong.

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[quote='polymath257' pid='1711158' dateline='1520172357']

The two statements in bold are false. The failure of the first bold statement invalidates the second. The rest of the statements are all correct.

And, in fact, if you look at my description, you see why the bold statements are false.

Remember, to go through all those points means that there is a time corresponding to each point. And that is true.

So, yes, if you have not reached the end, there is always another point to go through (in fact, an infinite number of points to go through) and you do reach the end.

Your assumption that we cannot go through an infinite number of points is invalidated.

Those statements follow logically from those that proceeded it.
Here they are again

No, you added another assumption: that you cannot go through an infinite sequence of points. That assumption is wrong, and I showed how it is possible.

Here are two situations:

1. At time t=1, we go through the first point. At time t=2, we go through the second point. At time t=3, we go through the third point, we continue and go through the n^th point at t=n.

If *this* is your scenario, then we will never get through all of the points.

2. At t=2.5, we go through the first point (located at x=.5), at t=3.75, we go through the second point (located at x=.75), at t=4.375, we go through the third point (located at x=.875). We continue, and go through the point located at x at time t=5*x.

In *this* scenario, we get through all of the points and we get to x=1 when t=5.

As far as I can tell, you are imagining the first scenario. But the second scenario is how it actually happens.

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If you are discussing motion, then time is a factor.

Sure, so are accelerations and deceleration, and if your dealing with actual motion, then it's likely not going to be perfectly consistent all the way through.

However what I was saying didn't involve time, and I don't see where adding time in, effects it.  So while it is assumed that there is motion, that one is traversing the path of the line in some manner; the amount of time, or the even constant motion is not a factor.

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Time is a factor because to 'go through a point' means that there is a time when you are at that point. So the timing is everything. The constant speed simplifies things so we can say *exactly* when we go through any particular point.
Once again, for each point in that sequence, there is a time when we are at that point. AND there is a time when we are past every term in that sequence.
The sequence is bounded above, but that bound is not in the sequence. That seems to be your main issue. But that isn't a contradiction.

[RoadRunner79]

[quote='polymath257' pid='1711158' dateline='1520172357']

The two statements in bold are false. The failure of the first bold statement invalidates the second. The rest of the statements are all correct.

And, in fact, if you look at my description, you see why the bold statements are false.

Remember, to go through all those points means that there is a time corresponding to each point. And that is true.

So, yes, if you have not reached the end, there is always another point to go through (in fact, an infinite number of points to go through) and you do reach the end.

Your assumption that we cannot go through an infinite number of points is invalidated.

Those statements follow logically from those that proceeded it.
Here they are again

If there is always another point, that is not the end, and which precedes the end, then end cannot be reached.
Therefore the end position is not reachable if there is an infinite number of points which must be traveled. 
[/qutote][/b]


If all points prior to the destination must be passed through sequentially, before reaching the destination; and there is an infinite number of points prior to destination so that the next point is always not the destination.  Then the end position will not be reachable.

It cannot be without end, and end at the same time... this is the contradiction.   Now perhaps you assume that your model is infinite, and you can calculate any infinitely small point and corresponding time.  However, this is why I asked numerous times previously, what exactly your points represent, in a real world motion.  If your model is only conceptual, then I suppose you can have whatever you want.   You don't actually traverse an infinite number of points in your model, you just assume that you do.  And if you actually follow the same math that shows that it is infinite, it also shows you that it cannot be completed.    Your model appears to be logically incoherent.

And, again, it is two different notions of 'having an end'. If the difference is kept in mind, the issue resolves itself.

1. In the sequence: whenever you are in the sequence, there is another term of the sequence later on. In this sense, the sequence does not end. But this is irrelevant to whether you go through every point.

2. In the interval between 0 and 1: In this sense, the sequence has an 'end'. More technically, it has an upper bound. And 1 is an upper bound. This is the sense that is important in asking if we go through every point. Since each point corresponds to a time, we do, in fact, finish that sequence.

The point here is that the sequence has an upper bound, but that upper bound is not in the sequence. There is no contradiction here, just an opportunity to learn.

My points represent locations between the two points. Again, what does it mean to 'traverse an infinite number of points'? It *means* that for each point, there is a time when we are at that point. And that is the case here. So, yes, we do traverse an infinite number of points. You intuition that this is impossible is just wrong.

So, how does that effect the logic I presented?
I don't see where saying that there is an upper bound of 1 is of any consequence, if the problem lies in reaching 1.   Again, this doesn't effect the logical issue;  it obfuscates it.

I notice that you keep using the words "assume" and "intuition"  whenever you talk about the case I'm making.  This despite the fact, that you ignore where I have shown that it logically follows to be the case.  On the other hand, you seem to just mostly insist that it is infinite, even though the method you used to demonstrate it as infinite also leads to a contradiction, when you also say it can be completed by sequentially following the points (also note, that this follows definitionally as well).  

It doesn't surprise me, that you have a group that gives this kudo's.   It doesn't surprise me, if talking about upper level math may wow some, that they don't really think about the issue.  There is one of two, who is giving you kudo's, who not that long ago, was dismissing me, as just an apologetic (when I was agreeing with them).   I'm an engineer, whenever possible, I think it is best to keep things simple, before looking at a more complicated solution.  I'm also cautious, when I ask a question, and the salesman starts going on with a bunch of technical sounding babble, that talks about any number of things, but doesn't address my question.

[polymath257]

Those statements follow logically from those that proceeded it.
Here they are again


And, again, it is two different notions of 'having an end'. If the difference is kept in mind, the issue resolves itself.

1. In the sequence: whenever you are in the sequence, there is another term of the sequence later on. In this sense, the sequence does not end. But this is irrelevant to whether you go through every point.

2. In the interval between 0 and 1: In this sense, the sequence has an 'end'. More technically, it has an upper bound. And 1 is an upper bound. This is the sense that is important in asking if we go through every point. Since each point corresponds to a time, we do, in fact, finish that sequence.

The point here is that the sequence has an upper bound, but that upper bound is not in the sequence. There is no contradiction here, just an opportunity to learn.

My points represent locations between the two points. Again, what does it mean to 'traverse an infinite number of points'? It *means* that for each point, there is a time when we are at that point. And that is the case here. So, yes, we do traverse an infinite number of points. You intuition that this is impossible is just wrong.

So, how does that effect the logic I presented?
I don't see where saying that there is an upper bound of 1 is of any consequence, if the problem lies in reaching 1.   Again, this doesn't effect the logical issue;  it obfuscates it.

I notice that you keep using the words "assume" and "intuition"  whenever you talk about the case I'm making.  This despite the fact, that you ignore where I have shown that it logically follows to be the case.  On the other hand, you seem to just mostly insist that it is infinite, even though the method you used to demonstrate it as infinite also leads to a contradiction, when you also say it can be completed by sequentially following the points (also note, that this follows definitionally as well).  

It doesn't surprise me, that you have a group that gives this kudo's.   It doesn't surprise me, if talking about upper level math may wow some, that they don't really think about the issue.  There is one of two, who is giving you kudo's, who not that long ago, was dismissing me, as just an apologetic (when I was agreeing with them).   I'm an engineer, whenever possible, I think it is best to keep things simple, before looking at a more complicated solution.  I'm also cautious, when I ask a question, and the salesman starts going on with a bunch of technical sounding babble, that talks about any number of things, but doesn't address my question.

You reach x=1 when t=5. Simple enough. You reach each and every one of those points also. There is no contradiction here.

Yes, for each one of those points, there are infinitely many you still have to go through to get to 1.

Yes, you do go through all those points.

Yes, you also go through 1.

No, you did NOT show it follows logically that you cannot get through every point. You are making assumptions that it is impossible to go through an infinite number of points. THAT is your mistake. And no, you did NOT show a contradiction: we have an infinite set that is also bounded. Your problem is a mix of different notions of 'bounded' or having an 'end'. The fact that there are two distinct concepts here is part of your confusion.

I'm sorry, but it is *you* that isn't thinking about the issue. In the scenario we have been discussing, you *do* go through every point of the sequence **and* you reach 1. There is no contradiction there *unless* you assume that it is impossible to go through an infinite number of points.

Your question has been addressed multiple times. Evidently you have refused to learn enough to understand the answer.

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Just a heads up: I'm going to be going on vacation starting on Friday and won't be able to post for about 10 days. Don't expect any answers from 3/9 through 3/19.

[The Grand Nudger]

Every step we take traverses an infinitely divisible distance, infinite points..and yet bounded on both sides.