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March 1, 2018 at 7:45 am (This post was last modified: March 1, 2018 at 8:06 am by polymath257.)
(February 28, 2018 at 9:03 pm)SteveII Wrote:
(February 28, 2018 at 6:41 pm)polymath257 Wrote: Yes, of course you put them in sets: the collection of bricks for each side. You don't *have* to, but you can then discuss the bricks that will go to each side. That is a collection, i.e, a set.
Arithmetic is a specialized mathematical subject governed by a whole set of axioms. To justify those axioms requires the use of collections, i.e, sets.
So, when you add, what are you doing? You are taking two collections (or whatever sizes) and putting them together into a single collection and counting the number of things in the new set. That is how addition is *defined*.
The same goes for multiplication: How do you define multiplication? You repeat a set of one size, one copy for each element in a different set. Then we look at how many total elements there are. So to define multiplication, you need some sort of set theory.
But, more, to show those operations are well defined (that they give the same result no matter how you rearrange things) is crucial and also depends on set theory.
So, yes, if you are even attempting to *define* 10^10, you will need the concepts of set theory.
Not only are you correct when you say "you don't *have* to" put the rooms in sets, but why would you? The desk clerk is not pulling out paper and using {G1, G2, G3...} to move the guests around. He is making changes with real rooms and real guests and does not have to resort to creating abstract objects by grouping them together.
You do not need sets to preform any basic operations like addition, subtraction, multiplication or division. I'll give you a chance to back away from that assertion.
The only reason you desperately want to hang on to putting those rooms into sets is so you can apply unwarranted external constraints (set theory) to them and dismiss the contradiction and absurdities that would otherwise surface.
Yes, of course you need sets for addition (for example). Without them, how do you even define addition? Try to define the notion of addition without using collections. You cannot do it. The same goes for multiplication. Subtraction and division require *extensions* of those notions and the way to extend them is by careful consideration of the sets involved.
So, a challenge: how do you *define* addition? Use *only* those logical axioms that you have accepted prior to this. They are just not sufficient to the task. For that matter, even to define the concept of number requires the use of collections.
And *you* were the one wanting to get results like infinity+infinity=infinity from moving people around. YOU were the one claiming subtraction needs to be well defined for infinite quantities (with no reason).
(February 28, 2018 at 7:39 pm)RoadRunner79 Wrote:
(February 28, 2018 at 6:41 pm)polymath257 Wrote:
Yes, of course you put them in sets: the collection of bricks for each side. You don't *have* to, but you can then discuss the bricks that will go to each side. That is a collection, i.e, a set.
Arithmetic is a specialized mathematical subject governed by a whole set of axioms. To justify those axioms requires the use of collections, i.e, sets.
So, when you add, what are you doing? You are taking two collections (or whatever sizes) and putting them together into a single collection and counting the number of things in the new set. That is how addition is *defined*.
The same goes for multiplication: How do you define multiplication? You repeat a set of one size, one copy for each element in a different set. Then we look at how many total elements there are. So to define multiplication, you need some sort of set theory.
But, more, to show those operations are well defined (that they give the same result no matter how you rearrange things) is crucial and also depends on set theory.
So, yes, if you are even attempting to *define* 10^10, you will need the concepts of set theory.
And if each step of the process took the same amount of time, you would never finish. But if it takes geometrically less time, then you will. In fact, we *do* go through an infinite sequence. So the basic assumption that this is impossible is just false. Space and time are continua and not discrete.
Yes, Achilles actually does catch up with the Tortoise in a finite amount of time. If you look at Zeno's paradox, you realize his fundamental assumption is that you cannot go through an infinite number of points. That is what is incorrect. Not only is it possible, but it is required for motion.
In your question of what happened one step before, there is some ambiguity. Achilles does not take a *step* at each stage of this process. In fact, the tail end of the process happens in the interval of a fraction of one of his steps.
And there isn't a 'step before' in this process. It is an infinite, completed process. As a function of time, the graph of the stages taken is not continuous, but the motion itself is. That just shows the stages aren't a good description.
I'm sorry, I'm going to have to flunk you for not following directions (this is not common core). You need to follow what Zeno had described. And you do not reach the end of 1 (which is necessary) to be called infinite. Infinitely small times still do not help you. It's not a matter of time. But I do find that you saying that more time, would not allow you to finish, that you need infinity less time.... That is funny.
For the last part, perhaps I worded it incorrectly. What is your last point, before you reach your destination (1 or whatever the number is)? And you can declare it infinite as much as you like. You are assuming your conclusion in your premise (as Steve pointed out before).
Well, we can turn Zeno's arguments around and show that it *is* possible to go through an infinite collection of things. For example, to go forward in time one second requires we go through the first half second, the next quarter second, the next eighth of a second, etc. Since we *do*, in fact, manage to get past the one second mark, we do, in fact, manage to go through that infinity. It *is* completed once we hit the second mark on our clocks.
And the point for the spatial distances is that each of those infinitely many distances is paired with one of those infinitely many times. Once those infinitely many times have passed (which they will!), we have also gone through the infinitely many distances.
Perhaps what is confusing you is that we can add an infinite number of positive quantities and get a finite answer. But that is trivial to see:
This all shows two things: 1. Finishing an infinite process is possible if the time for each step decreases geometrically. 2. It is the fact that space and time are *both* infinite in the same way that allows for motion.
(February 28, 2018 at 6:11 pm)RoadRunner79 Wrote: However this doesn't address the dichotomy paradox by Zeno. It's not about motion, or answering how we get from position 0 to position 1.
The whole point of Zeno's paradoxes is that they are arguments against motion, continuous or discrete.
Do you really think that reason that Zeno's paradoxes are still being discussed today, is because a large number of people question motion? Personally, if that is all you are after, then I like the guy who responded by getting up, walked around the room, and sat back down without saying a thing. You are missing or ignoring the larger issue; and why it is being brought up at all.
Quote:I love, by the way, how Steve has not responded to my last response or two and is just repeating the same old bullshit that has long been debunked. So much for intellectual honesty.
Wow.... you had the restraint to wait a response or two, before questioning his intellectual integrity. You should check with that admins; that may be a new AF.org record in the principle of Charity. You should get a cookie or something!
Perhaps it was just lost in the mix of things.
It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. - Alexander Vilenkin If I am shown my error, I will be the first to throw my books into the fire. - Martin Luther
(February 28, 2018 at 10:03 pm)Grandizer Wrote: The whole point of Zeno's paradoxes is that they are arguments against motion, continuous or discrete.
Do you really think that reason that Zeno's paradoxes are still being discussed today, is because a large number of people question motion? Personally, if that is all you are after, then I like the guy who responded by getting up, walked around the room, and sat back down without saying a thing. You are missing or ignoring the larger issue; and why it is being brought up at all.
Oh, stop it. Maybe Christian apologists like you like to turn it into a larger issue (because you need it to), but Zeno's paradoxes have always been about motion.
(February 28, 2018 at 9:03 pm)SteveII Wrote: Not only are you correct when you say "you don't *have* to" put the rooms in sets, but why would you? The desk clerk is not pulling out paper and using {G1, G2, G3...} to move the guests around. He is making changes with real rooms and real guests and does not have to resort to creating abstract objects by grouping them together.
You do not need sets to preform any basic operations like addition, subtraction, multiplication or division. I'll give you a chance to back away from that assertion.
The only reason you desperately want to hang on to putting those rooms into sets is so you can apply unwarranted external constraints (set theory) to them and dismiss the contradiction and absurdities that would otherwise surface.
Yes, of course you need sets for addition (for example). Without them, how do you even define addition? Try to define the notion of addition without using collections. You cannot do it. The same goes for multiplication. Subtraction and division require *extensions* of those notions and the way to extend them is by careful consideration of the sets involved.
So, a challenge: how do you *define* addition? Use *only* those logical axioms that you have accepted prior to this. They are just not sufficient to the task. For that matter, even to define the concept of number requires the use of collections.
And *you* were the one wanting to get results like infinity+infinity=infinity from moving people around. YOU were the one claiming subtraction needs to be well defined for infinite quantities (with no reason).
(February 28, 2018 at 7:39 pm)RoadRunner79 Wrote: I'm sorry, I'm going to have to flunk you for not following directions (this is not common core). You need to follow what Zeno had described. And you do not reach the end of 1 (which is necessary) to be called infinite. Infinitely small times still do not help you. It's not a matter of time. But I do find that you saying that more time, would not allow you to finish, that you need infinity less time.... That is funny.
For the last part, perhaps I worded it incorrectly. What is your last point, before you reach your destination (1 or whatever the number is)? And you can declare it infinite as much as you like. You are assuming your conclusion in your premise (as Steve pointed out before).
Well, we can turn Zeno's arguments around and show that it *is* possible to go through an infinite collection of things. For example, to go forward in time one second requires we go through the first half second, the next quarter second, the next eighth of a second, etc. Since we *do*, in fact, manage to get past the one second mark, we do, in fact, manage to go through that infinity. It *is* completed once we hit the second mark on our clocks.
And the point for the spatial distances is that each of those infinitely many distances is paired with one of those infinitely many times. Once those infinitely many times have passed (which they will!), we have also gone through the infinitely many distances.
Perhaps what is confusing you is that we can add an infinite number of positive quantities and get a finite answer. But that is trivial to see:
This all shows two things: 1. Finishing an infinite process is possible if the time for each step decreases geometrically. 2. It is the fact that space and time are *both* infinite in the same way that allows for motion.
So you just add "...." and then you finished? That you are finishing a process that you are claiming is endless is a contradiction. And the math in your dichotomy addition will never add up to what you want it to. That is why it is an infinite chain (until you stop anyway).
Also a distance infers that you have two points or a segment. You cannot add an infinite number of distances, and end in a result period (again by the nature of claiming that it is infinite). The only exception perhaps is perhaps if you are adding a 0 distance an infinite number of times.
I also think that you are confusing "showing" with "assuming". In my job, we sometimes run into engineers, that while they may be book smart, are said, to have not had enough time in the field. The don't understand that the abstraction is not equal to actuality. Your concept may have a perfect sphere, but in actuality, we get as close to a perfect sphere as we can (or hopefully, at least what we need). In reality, we don't deal with points of zero size, if they have zero size, then they are not really anything. It does depend on what your abstraction represents. Can you actually cut a thing into a perfect 1/3, can you have a perfect circle, can we know that they are? How many of these abstracts refer to something in them that is zero size? What I think that you are showing, is that these things are not infinite at all, but do come to completion.
It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. - Alexander Vilenkin If I am shown my error, I will be the first to throw my books into the fire. - Martin Luther
(March 1, 2018 at 7:45 am)polymath257 Wrote: Yes, of course you need sets for addition (for example). Without them, how do you even define addition? Try to define the notion of addition without using collections. You cannot do it. The same goes for multiplication. Subtraction and division require *extensions* of those notions and the way to extend them is by careful consideration of the sets involved.
So, a challenge: how do you *define* addition? Use *only* those logical axioms that you have accepted prior to this. They are just not sufficient to the task. For that matter, even to define the concept of number requires the use of collections.
And *you* were the one wanting to get results like infinity+infinity=infinity from moving people around. YOU were the one claiming subtraction needs to be well defined for infinite quantities (with no reason).
Well, we can turn Zeno's arguments around and show that it *is* possible to go through an infinite collection of things. For example, to go forward in time one second requires we go through the first half second, the next quarter second, the next eighth of a second, etc. Since we *do*, in fact, manage to get past the one second mark, we do, in fact, manage to go through that infinity. It *is* completed once we hit the second mark on our clocks.
And the point for the spatial distances is that each of those infinitely many distances is paired with one of those infinitely many times. Once those infinitely many times have passed (which they will!), we have also gone through the infinitely many distances.
Perhaps what is confusing you is that we can add an infinite number of positive quantities and get a finite answer. But that is trivial to see:
This all shows two things: 1. Finishing an infinite process is possible if the time for each step decreases geometrically. 2. It is the fact that space and time are *both* infinite in the same way that allows for motion.
So you just add "...." and then you finished? That you are finishing a process that you are claiming is endless is a contradiction. And the math in your dichotomy addition will never add up to what you want it to. That is why it is an infinite chain (until you stop anyway).
Also a distance infers that you have two points or a segment. You cannot add an infinite number of distances, and end in a result period (again by the nature of claiming that it is infinite). The only exception perhaps is perhaps if you are adding a 0 distance an infinite number of times.
I also think that you are confusing "showing" with "assuming". In my job, we sometimes run into engineers, that while they may be book smart, are said, to have not had enough time in the field. The don't understand that the abstraction is not equal to actuality. Your concept may have a perfect sphere, but in actuality, we get as close to a perfect sphere as we can (or hopefully, at least what we need). In reality, we don't deal with points of zero size, if they have zero size, then they are not really anything. It does depend on what your abstraction represents. Can you actually cut a thing into a perfect 1/3, can you have a perfect circle, can we know that they are? How many of these abstracts refer to something in them that is zero size? What I think that you are showing, is that these things are not infinite at all, but do come to completion.
Again, the definition of 'infinite' that you are using isn't the one that others use. That infinite sum does, in fact, add to be 1/3. The *limit* is exactly 1/3. We can, in fact, evaluate the answer without going through the whole process.
Yes, you can, in fact, add an infinite number of distances and obtain a finite distance. That is precisely what limits do. 1/2 + 1/4 + 1/8 +...=1. Exactly.
Yes, the infinite aspects do come to completion in a finite time. That only shows your definition isn't working.
(March 1, 2018 at 8:30 am)RoadRunner79 Wrote: Do you really think that reason that Zeno's paradoxes are still being discussed today, is because a large number of people question motion? Personally, if that is all you are after, then I like the guy who responded by getting up, walked around the room, and sat back down without saying a thing. You are missing or ignoring the larger issue; and why it is being brought up at all.
Oh, stop it. Maybe Christian apologists like you like to turn it into a larger issue (because you need it to), but Zeno's paradoxes have always been about motion.
Actually they are about Zeno's arguments.... which you seem to be avoiding by changing them into something else.
I may not be the best at math, but I can see that the infinite series being talked about will never end, and never be equal to the point that is being attempted to be reached. Since an infinity by definition never ends... I don't think that throwing more infinities at it is the answer.
It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. - Alexander Vilenkin If I am shown my error, I will be the first to throw my books into the fire. - Martin Luther
(March 1, 2018 at 8:48 am)Grandizer Wrote: Oh, stop it. Maybe Christian apologists like you like to turn it into a larger issue (because you need it to), but Zeno's paradoxes have always been about motion.
Actually they are about Zeno's arguments.... which you seem to be avoiding by changing them into something else.
You're not only not good at math, you're also bad at the history of philosophy. I can't help you there if you remain stubborn about your falsehoods.
Quote:I may not be the best at math, but I can see that the infinite series being talked about will never end, and never be equal to the point that is being attempted to be reached. Since an infinity by definition never ends... I don't think that throwing more infinities at it is the answer.
See my latest post. That there is an actual infinity.
March 1, 2018 at 10:03 am (This post was last modified: March 1, 2018 at 10:06 am by RoadRunner79.)
(March 1, 2018 at 9:37 am)polymath257 Wrote:
(March 1, 2018 at 9:31 am)RoadRunner79 Wrote: So you just add "...." and then you finished? That you are finishing a process that you are claiming is endless is a contradiction. And the math in your dichotomy addition will never add up to what you want it to. That is why it is an infinite chain (until you stop anyway).
Also a distance infers that you have two points or a segment. You cannot add an infinite number of distances, and end in a result period (again by the nature of claiming that it is infinite). The only exception perhaps is perhaps if you are adding a 0 distance an infinite number of times.
I also think that you are confusing "showing" with "assuming". In my job, we sometimes run into engineers, that while they may be book smart, are said, to have not had enough time in the field. The don't understand that the abstraction is not equal to actuality. Your concept may have a perfect sphere, but in actuality, we get as close to a perfect sphere as we can (or hopefully, at least what we need). In reality, we don't deal with points of zero size, if they have zero size, then they are not really anything. It does depend on what your abstraction represents. Can you actually cut a thing into a perfect 1/3, can you have a perfect circle, can we know that they are? How many of these abstracts refer to something in them that is zero size? What I think that you are showing, is that these things are not infinite at all, but do come to completion.
Again, the definition of 'infinite' that you are using isn't the one that others use. That infinite sum does, in fact, add to be 1/3. The *limit* is exactly 1/3. We can, in fact, evaluate the answer without going through the whole process.
Yes, you can, in fact, add an infinite number of distances and obtain a finite distance. That is precisely what limits do. 1/2 + 1/4 + 1/8 +...=1. Exactly.
Yes, the infinite aspects do come to completion in a finite time. That only shows your definition isn't working.
There was another post, where I was going to mention this, but this seems like a good one as well.
There have been a number of times where you do the typical atheistic thing where you dismiss logic and philosophy (seemingly to avoid it). You question the definitions of infinity, even the use of the terms actual and potential infinities. And demand that everything to be changed to dealing with infinities in math.
If math has something to add to the discussion, then that is good. However the OP was not about mathematical infinite sets that exists only in the abstraction of the mind. It is about actual infinity, and infinity in regards to philosophy. That you keep wanting to change things, says to me, that you are not talking about the same ideas.... I also think this is why you do not see the obvious contradictions.
1/2 + 1/4 + 1/8 +...<1 It is never equal to 1. If you where to reach 1, then you would stop your infinite chain, and it would not be infinite.
(March 1, 2018 at 9:52 am)Grandizer Wrote:
(March 1, 2018 at 9:46 am)RoadRunner79 Wrote: Actually they are about Zeno's arguments.... which you seem to be avoiding by changing them into something else.
You're not only not good at math, you're also bad at the history of philosophy. I can't help you there if you remain stubborn about your falsehoods.
Quote:I may not be the best at math, but I can see that the infinite series being talked about will never end, and never be equal to the point that is being attempted to be reached. Since an infinity by definition never ends... I don't think that throwing more infinities at it is the answer.
See my latest post. That there is an actual infinity.
Do you mean that 1 = 0.99999......
I think that says it all right there.
It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. - Alexander Vilenkin If I am shown my error, I will be the first to throw my books into the fire. - Martin Luther
