(March 5, 2018 at 10:18 am)polymath257 Wrote:

(March 5, 2018 at 9:18 am)RoadRunner79 Wrote: 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.

---

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.

Well we are just repeating ourselves anyway.... and you refuse to look at the logic.   I don't dispute that in a conceptual geometric model, that you have a theoretical point in time, for any infinitesimal point in distance.  However the question is, in something real, what does this "point" represent?   What are you saying is the actual infinite?  As well, when you actually follow the model you are presenting to show and actual infinite and do the math, at no point will you ever reach the end (which equates to the definition of infinite).    This is not an assumption, but a fact.  You can do that calculations by hand, you can make a computer model, neither will ever reach the end (which is why you use those methods to show an infinite to begin with).    You seem to be assuming an infinite, and assume that any "point" you can make up on paper, correlates to something in real motion, and then ignore the logical contradiction of your assumptions.  As an engineer, I would say you need to get out of the office (or classroom) a little more.

(edit to add)   Also as for bringing up modern physics theories and such, there is the notion of a Planck length.  Where it is theorized, that classical ideas concerning space time, break down, and quantum effects take over.   That at this point, any distinction between two points is indistinguishable.  You might also keep in mind, that things are always moving (vibrating back and forth)

Since we don't seem to be getting anywhere... I leave you to get ready for your vacation; I hope you have fun.

(March 5, 2018 at 10:56 am)RoadRunner79 Wrote:

(March 5, 2018 at 10:18 am)polymath257 Wrote: 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.

---

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.

Well we are just repeating ourselves anyway.... and you refuse to look at the logic.   I don't dispute that in a conceptual geometric model, that you have a theoretical point in time, for any infinitesimal point in distance.  However the question is, in something real, what does this "point" represent?   What are you saying is the actual infinite?  As well, when you actually follow the model you are presenting to show and actual infinite and do the math, at no point will you ever reach the end (which equates to the definition of infinite).    This is not an assumption, but a fact.  You can do that calculations by hand, you can make a computer model, neither will ever reach the end (which is why you use those methods to show an infinite to begin with).    You seem to be assuming an infinite, and assume that any "point" you can make up on paper, correlates to something in real motion, and then ignore the logical contradiction of your assumptions.  As an engineer, I would say you need to get out of the office (or classroom) a little more.

(edit to add)   Also as for bringing up modern physics theories and such, there is the notion of a Planck length.  Where it is theorized, that classical ideas concerning space time, break down, and quantum effects take over.   That at this point, any distinction between two points is indistinguishable.  You might also keep in mind, that things are always moving (vibrating back and forth)

Since we don't seem to be getting anywhere... I leave you to get ready for your vacation; I hope you have fun.

A point represents a location between 0 and 1. Wasn't that obvious?

And yes, there is an infinite number of such points.

And yes, we *do* reach the end. The *logic* that I presented shows that. Your *claim* is based on a faulty *assumption* that you cannot a complete an infinity. And no, that is NOT a fact. Remember, to be infinite *only* means that it cannot be put into correspondence with some counting number. it does NOT mean 'has no bound'. it does NOT mean 'goes on forever'. it does NOT mean 'has no end'. The example of the sequence we have been considering shows the differences.

(March 5, 2018 at 12:05 pm)polymath257 Wrote:

(March 5, 2018 at 10:56 am)RoadRunner79 Wrote: Well we are just repeating ourselves anyway.... and you refuse to look at the logic.   I don't dispute that in a conceptual geometric model, that you have a theoretical point in time, for any infinitesimal point in distance.  However the question is, in something real, what does this "point" represent?   What are you saying is the actual infinite?  As well, when you actually follow the model you are presenting to show and actual infinite and do the math, at no point will you ever reach the end (which equates to the definition of infinite).    This is not an assumption, but a fact.  You can do that calculations by hand, you can make a computer model, neither will ever reach the end (which is why you use those methods to show an infinite to begin with).    You seem to be assuming an infinite, and assume that any "point" you can make up on paper, correlates to something in real motion, and then ignore the logical contradiction of your assumptions.  As an engineer, I would say you need to get out of the office (or classroom) a little more.

(edit to add)   Also as for bringing up modern physics theories and such, there is the notion of a Planck length.  Where it is theorized, that classical ideas concerning space time, break down, and quantum effects take over.   That at this point, any distinction between two points is indistinguishable.  You might also keep in mind, that things are always moving (vibrating back and forth)

Since we don't seem to be getting anywhere... I leave you to get ready for your vacation; I hope you have fun.

A point represents a location between 0 and 1. Wasn't that obvious?

And yes, there is an infinite number of such points.

And yes, we *do* reach the end. The *logic* that I presented shows that. Your *claim* is based on a faulty *assumption* that you cannot a complete an infinity. And no, that is NOT a fact. Remember, to be infinite *only* means that it cannot be put into correspondence with some counting number. it does NOT mean 'has no bound'. it does NOT mean 'goes on forever'. it does NOT mean 'has no end'. The example of the sequence we have been considering shows the differences.

So are you saying that counting numbers have an end?   Or that only non-counting numbers do not end?  It seems to me, that you are attempting to make a distinction without a difference.

Also just to clarify,  would you agree that these locations, do not necessarily correspond to anything physical on the line, or involved in the motion.  They serve a purpose, for an academic model but are not an actual infinite number of things?

(March 5, 2018 at 12:05 pm)polymath257 Wrote:

(March 5, 2018 at 10:56 am)RoadRunner79 Wrote: Well we are just repeating ourselves anyway.... and you refuse to look at the logic.   I don't dispute that in a conceptual geometric model, that you have a theoretical point in time, for any infinitesimal point in distance.  However the question is, in something real, what does this "point" represent?   What are you saying is the actual infinite?  As well, when you actually follow the model you are presenting to show and actual infinite and do the math, at no point will you ever reach the end (which equates to the definition of infinite).    This is not an assumption, but a fact.  You can do that calculations by hand, you can make a computer model, neither will ever reach the end (which is why you use those methods to show an infinite to begin with).    You seem to be assuming an infinite, and assume that any "point" you can make up on paper, correlates to something in real motion, and then ignore the logical contradiction of your assumptions.  As an engineer, I would say you need to get out of the office (or classroom) a little more.

(edit to add)   Also as for bringing up modern physics theories and such, there is the notion of a Planck length.  Where it is theorized, that classical ideas concerning space time, break down, and quantum effects take over.   That at this point, any distinction between two points is indistinguishable.  You might also keep in mind, that things are always moving (vibrating back and forth)

Since we don't seem to be getting anywhere... I leave you to get ready for your vacation; I hope you have fun.

A point represents a location between 0 and 1. Wasn't that obvious?

And yes, there is an infinite number of such points.

And yes, we *do* reach the end. The *logic* that I presented shows that. Your *claim* is based on a faulty *assumption* that you cannot a complete an infinity. And no, that is NOT a fact. Remember, to be infinite *only* means that it cannot be put into correspondence with some counting number. it does NOT mean 'has no bound'. it does NOT mean 'goes on forever'. it does NOT mean 'has no end'. The example of the sequence we have been considering shows the differences.

Of course 'infinity' means every one of those things you say it does not mean! The very first sentence in the Wikipedia article:

Quote:Infinity (symbol: ∞) is a concept describing something without any bound or larger than any natural number. 

Need more?

in·fi·nite
ˈinfənət/
adjective
adjective: infinite

1. 1.
   limitless or endless in space, extent, or size; impossible to measure or calculate.
   synonyms:
   boundless, unbounded, unlimited, limitless, never-ending, interminable; More
   • very great in amount or degree.
     "he bathed the wound with infinite care"
     
   • MATHEMATICS
     greater than any assignable quantity or countable number.
     
   • MATHEMATICS
     (of a series) able to be continued indefinitely.
     

Where are you getting your definition?

OK, then what do *you* call it when something doesn't correspond in quantity to a counting number?

---

(March 5, 2018 at 12:14 pm)RoadRunner79 Wrote:

(March 5, 2018 at 12:05 pm)polymath257 Wrote: A point represents a location between 0 and 1. Wasn't that obvious?

And yes, there is an infinite number of such points.

And yes, we *do* reach the end. The *logic* that I presented shows that. Your *claim* is based on a faulty *assumption* that you cannot a complete an infinity. And no, that is NOT a fact. Remember, to be infinite *only* means that it cannot be put into correspondence with some counting number. it does NOT mean 'has no bound'. it does NOT mean 'goes on forever'. it does NOT mean 'has no end'. The example of the sequence we have been considering shows the differences.

So are you saying that counting numbers have an end?   Or that only non-counting numbers do not end?  It seems to me, that you are attempting to make a distinction without a difference.

Also just to clarify,  would you agree that these locations, do not necessarily correspond to anything physical on the line, or involved in the motion.  They serve a purpose, for an academic model but are not an actual infinite number of things?

The counting numbers do not 'end' if ordered in the usual way. But, for example, the *negative* counting numbers *do* have an 'end' when ordered int he usual way. The question of 'having an end' depends on the *order*, not on the quantity.

No, those locations correspond to points on the physical line. And yes, they are an *actual* infinity of points.

(March 5, 2018 at 12:52 pm)polymath257 Wrote: OK, then what do *you* call it when something doesn't correspond in quantity to a counting number?

---

(March 5, 2018 at 12:14 pm)RoadRunner79 Wrote: So are you saying that counting numbers have an end?   Or that only non-counting numbers do not end?  It seems to me, that you are attempting to make a distinction without a difference.

Also just to clarify,  would you agree that these locations, do not necessarily correspond to anything physical on the line, or involved in the motion.  They serve a purpose, for an academic model but are not an actual infinite number of things?

1.) The counting numbers do not 'end' if ordered in the usual way. But, for example, the *negative* counting numbers *do* have an 'end' when ordered int he usual way. The question of 'having an end' depends on the *order*, not on the quantity.

2.) No, those locations correspond to points on the physical line. And yes, they are an *actual* infinity of points.

1.)  Ok... if all positive integers end in one direction, and all negative integers end in one direction.  Would you agree, that they are infinite in at least one direction.  This would be a definition of  a ray.   Now are you telling me, that when definitions say that the ray proceeds infinitely in one direction, that this is telling me something about the order, not that the line goes on without end?

2.)  So the points correspond to locations, and the locations correspond to points.  My question is what does a point or a location or a  point like under a microscope?

(March 5, 2018 at 1:08 pm)RoadRunner79 Wrote:

(March 5, 2018 at 12:52 pm)polymath257 Wrote: OK, then what do *you* call it when something doesn't correspond in quantity to a counting number?

---

1.) The counting numbers do not 'end' if ordered in the usual way. But, for example, the *negative* counting numbers *do* have an 'end' when ordered int he usual way. The question of 'having an end' depends on the *order*, not on the quantity.

2.) No, those locations correspond to points on the physical line. And yes, they are an *actual* infinity of points.

1.)  Ok... if all positive integers end in one direction, and all negative integers end in one direction.  Would you agree, that they are infinite in at least one direction.  This would be a definition of  a ray.   Now are you telling me, that when definitions say that the ray proceeds infinitely in one direction, that this is telling me something about the order, not that the line goes on without end?

2.)  So the points correspond to locations, and the locations correspond to points.  My question is what does a point or a location or a  point  like under a microscope?

1) Both are infinite. Yes, having an 'end' is a matter of how things are ordered, NOT a matter of the quantity.

The interval from 0 to 1 has an infinite number of real numbers, but has an end at both ends.

2. ???? What does time look like under a microscope? Silly question.

(March 5, 2018 at 1:12 pm)polymath257 Wrote:

(March 5, 2018 at 1:08 pm)RoadRunner79 Wrote: 1.)  Ok... if all positive integers end in one direction, and all negative integers end in one direction.  Would you agree, that they are infinite in at least one direction.  This would be a definition of  a ray.   Now are you telling me, that when definitions say that the ray proceeds infinitely in one direction, that this is telling me something about the order, not that the line goes on without end?

2.)  So the points correspond to locations, and the locations correspond to points.  My question is what does a point or a location or a  point  like under a microscope?

1) Both are infinite. Yes, having an 'end' is a matter of how things are ordered, NOT a matter of the quantity.

The interval from 0 to 1 has an infinite number of real numbers, but has an end at both ends.

2. ???? What does time look like under a microscope? Silly question.

So what order equates them the description of being infinite?  And infinity is used often (I would say normally) in regards to quantity.  Which would seemingly be the case in your "points" / "Locations".  Or are you saying now that there is not an endless number of points (that they can be completed)?

(March 5, 2018 at 1:17 pm)RoadRunner79 Wrote:

(March 5, 2018 at 1:12 pm)polymath257 Wrote: 1) Both are infinite. Yes, having an 'end' is a matter of how things are ordered, NOT a matter of the quantity.

The interval from 0 to 1 has an infinite number of real numbers, but has an end at both ends.

2. ???? What does time look like under a microscope? Silly question.

So what order equates them the description of being infinite?  And infinity is used often (I would say normally) in regards to quantity.  Which would seemingly be the case in your "points" / "Locations".  Or are you saying now that there is not an endless number of points (that they can be completed)?

Yes, of course I am saying they can be completed. That is what it means to be an actual infinity, after all.

The problem is that you have two very different notions of having an 'end'. One uses a list of the elements. The other is based on the order properties. if you want to list the elements of an infinite set one by one, you won't ever end that process. But that isn't required in the Zeno paradoxes. ALL that is required there is that every position has a time associated with it. THAT'S ALL.


So, if you use 'infinity' to describe quantity, then 'not having an end' is NOT the description you can use. That doesn't describe a quantity: it describes a process.

(March 5, 2018 at 12:42 pm)SteveII Wrote:

(March 5, 2018 at 12:05 pm)polymath257 Wrote: A point represents a location between 0 and 1. Wasn't that obvious?

And yes, there is an infinite number of such points.

And yes, we *do* reach the end. The *logic* that I presented shows that. Your *claim* is based on a faulty *assumption* that you cannot a complete an infinity. And no, that is NOT a fact. Remember, to be infinite *only* means that it cannot be put into correspondence with some counting number. it does NOT mean 'has no bound'. it does NOT mean 'goes on forever'. it does NOT mean 'has no end'. The example of the sequence we have been considering shows the differences.

Of course 'infinity' means every one of those things you say it does not mean! The very first sentence in the Wikipedia article:

Quote:Infinity (symbol: ∞) is a concept describing something without any bound or larger than any natural number. 

Need more?

in·fi·nite
ˈinfənət/
adjective
adjective: infinite

1. 1.
   limitless or endless in space, extent, or size; impossible to measure or calculate.
   synonyms:
   boundless, unbounded, unlimited, limitless, never-ending, interminable; More
   • very great in amount or degree.
     "he bathed the wound with infinite care"
     
   • MATHEMATICS
     greater than any assignable quantity or countable number.
     
   • MATHEMATICS
     (of a series) able to be continued indefinitely.
     

Where are you getting your definition?

I think it's the same place as words such as "end", "contradiction", "assume", "never", "last" and the like.

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(March 5, 2018 at 1:23 pm)polymath257 Wrote:

(March 5, 2018 at 1:17 pm)RoadRunner79 Wrote: So what order equates them the description of being infinite?  And infinity is used often (I would say normally) in regards to quantity.  Which would seemingly be the case in your "points" / "Locations".  Or are you saying now that there is not an endless number of points (that they can be completed)?

Yes, of course I am saying they can be completed. That is what it means to be an actual infinity, after all.

The problem is that you have two very different notions of having an 'end'. One uses a list of the elements. The other is based on the order properties. if you want to list the elements of an infinite set one by one, you won't ever end that process. But that isn't required in the Zeno paradoxes. ALL that is required there is that every position has a time associated with it. THAT'S ALL.


So, if you use 'infinity' to describe quantity, then 'not having an end' is NOT the description you can use. That doesn't describe a quantity: it describes a process.

So what is it about the order that makes a thing infinite?  Why is it, that you cannot list the elements, but you seem to think that you can go through them all sequentially?  Are not both a process?  When cutting in half in the dichotomy paradox, would that then demonstrate that the process; not the number of points is infinite?