1. This seems to be a widely misunderstood concept (including by me), yet one commonly used wrongly. One of the common mistakes, for instance, is describing something as simply infinite without specifying exactly what quality of the thing is supposed to be infinite. So let me make some statements and ask some questions regarding infinity and I would like anyone properly knowledgeable to verify their validity, if not make some corrections.

It seems to me that structure can exist within infinity. One example of this would be that in a hypothetical universe that is infinite in time, space and matter, that matter could exist as one particle per light year or as an infinite gas cloud and that both scenarios would have infinite matter.

As I understand it, it is considered in mathematics that . But does this necessarily have to be the case in reality? If you have an infinite data set where is supposed to describe the probability of a certain property to be exhibited by each element of the data set, does that mean that among any number of elements in the infinite data set you would care to consider that all of them would exhibit this particular property? (hope that was coherent :? )

2.

3. Originally Posted by KALSTER
This seems to be a widely misunderstood concept (including by me), yet one commonly used wrongly. One of the common mistakes, for instance, is describing something as simply infinite without specifying exactly what quality of the thing is supposed to be infinite. So let me make some statements and ask some questions regarding infinity and I would like anyone properly knowledgeable to verify their validity, if not make some corrections.

It seems to me that structure can exist within infinity. One example of this would be that in a hypothetical universe that is infinite in time, space and matter, that matter could exist as one particle per light year or as an infinite gas cloud and that both scenarios would have infinite matter.

As I understand it, it is considered in mathematics that . But does this necessarily have to be the case in reality? If you have an infinite data set where is supposed to describe the probability of a certain property to be exhibited by each element of the data set, does that mean that among any number of elements in the infinite data set you would care to consider that all of them would exhibit this particular property? (hope that was coherent :? )

I like what you are bringing up.

Many years ago the scientists I really liked, the Universal Scientists. Said that their accuracy in all honesty based on their groups collective knowledge.
Their rather well known strive for pure truth, and their very humble nature. Put purity of elements at 99.99 percent accuracy. That was as sure as they could ever be about anything. They did not feel any other experiments on earth would yield anything higher then that level of purity.

They had already noted that industries use of radio active substances were contaminating the sources for standard elements. There were a few magnet factory explosions. Where they used radio active substances.

99.99 percent accuracy means that there is an expected and proven error of contamination in any element they purified of, 1/10,000 parts. Or one in ten thousand parts.

You do not have to believe me however I feel they were more accurate then, then we are today.
Not the bulk of the industrial products of the past, surly on the whole the purity of our supplies has improved.

But the level of sterility in their laboratories. And the life long dedication to purifying the elements. I do not believe is being matched today.

One thing Universal Scientists were very big about is that 0.0001 parts contamination. They found that by introducing or switching out one contaminate in a purified element. They could cause the entire element to take on new properties.

Although this was very real and very provable. It was another sticking point to just educating people about science. One more outrageous variable to pass along without any gravity of its importance. Amongst Universal Scientists it was a hot subject. But it may have been the death knell for Universal Science.

Although amongst themselves they felt rather assured that the many tests they did, proved the isolation of the elements. For science writers that just wanted yes and know answers to pass along to readers, it did not go well.

Consider an element known to react a certain way. Used in quantity, everyday across the world, suddenly man made. Sometimes the process they use changes the natural contaminate. Suddenly this element in certain situations starts to act differently in reactions. Even fatally wounding workers.

All because of the leverage of the contaminate in the element. They were underway with work to show the importance of the contaminate in elements. When Chadwick sidetracked that work.

Chadwick did a whole bunch of experiments using radio active elements that were not even isolated yet. The way he portrayed his experiments captured a lot of fans. And money started to go to him. And not the Universal Scientists that were really grinding away at the mysteries.

Believe me a Universal Scientist needed no Manhattan project to end the world.

Universal Scientists actually stated that the contaminate in the element appears to be more powerful then the element itself in some cases in some reactions. I thought that was fascinating.

Sincerely,

William McCormick

5. hahaha. I wonder what infinity is too. I think its common to think about it as a line going on and on forever. Is that really possible in reality though? Could that just be something we assume can happen? Do we even know if forever is real? Im under the impression that we think infinity is something going on and on and on and on. That doesnt sound infinit enough to me though. Could Infinity be more like everything fused together where all patterns that could ever be are ever present in one space. Kinda like a sphere of energy that is the source of everything when divided. Just an idea I had about infinity I really have no idea what im talking about.

6. Originally Posted by KALSTER
My point, was that infinitely small trace quantities of contaminants in purified elements could alter the 99.99 percent pure elements properties. To a point that the element itself as used by millions everyday, was no longer recognized as that element.

This was the monumental work done by Universal Scientists into the infinite power of such a small lever, the contaminate, and its abilty to effect the huge quantity of matter in wild often shocking ways.

That is why even an infinitely small amount of a radio active substance may cause severe adverse effects to equipment and supplies. I was only highlighting the power of infinity. Even when it is infinitely small and hidden.

Contaminates are often spread evenly throughout an element. It was believed that these contaminants, in elements could spread or open avenues through the element that caused it to have a different effect on ambient radiation, and in turn the ambient radiation would have a different effect on the object.
Just like a molecule often hides the emissions of the substances, the elements, it is made from. That is the power of ambient radiation and structure.

I do not want to bother your topic. I just thought you were interested in infinity from all angles. This is what interested me about infinity.

Sincerely,

William McCormick

7. If you don't want to contaminate my thread, stop talking about universal scientists. Stop disguising your nonsense as relavent by throwing one or two "infinite's" in there. I don't want to hear about universal scientists or their science and neither does anyone else. Get it?

8. Originally Posted by KALSTER
If you don't want to contaminate my thread, stop talking about universal scientists. Stop disguising your nonsense as relavent by throwing one or two "infinite's" in there. I don't want to hear about universal scientists or their science and neither does anyone else. Get it?
No and you are not God, so do not speak for everyone else.

I am totally being honest, my posts about infinity are my only interests or unknowns about infinity. You attack me.

Why do I have to have an opinion about reality that is just like yours, as long as I allow you, your opinion?

Of those contaminants that are just 1/10,000th of the whole. Only a small percentage of the contaminants, themselves being altered. Could cause the change in the purified element. Those amounts would be infinitely small.

That is really what I was looking to clarify, or get to. I was having trouble putting it into words. Now I have it.

Well, thanks for helping me get to it, despite the hard feelings.

Sincerely,

William McCormick

9. One way to look at the 0.9999... question is to consider this: what is 1 - 0.9999...? The answer, 0. Therefore, yes, 0.9999... = 1. Actually, there are many, many ways to show that this is true, at least for the real numbers. There are number systems out there where it isn't true anymore, but those systems aren't what we're used to.

BTW, the real world doesn't seem to contain any true infinites. The universe is composed of finitely many particles of a finite size with a finite amount of energy existing within a finite amount of space over a finite amount of time. :/

10. Originally Posted by MagiMaster
One way to look at the 0.9999... question is to consider this: what is 1 - 0.9999...? The answer, 0.
How do you work that one out?

11. Originally Posted by MagiMaster
One way to look at the 0.9999... question is to consider this: what is 1 - 0.9999...? The answer, 0. Therefore, yes, 0.9999... = 1. Actually, there are many, many ways to show that this is true, at least for the real numbers. There are number systems out there where it isn't true anymore, but those systems aren't what we're used to.

BTW, the real world doesn't seem to contain any true infinites. The universe is composed of finitely many particles of a finite size with a finite amount of energy existing within a finite amount of space over a finite amount of time. :/

Finite to God maybe. But not anyone I know.

Sincerely,

William McCormick

12. Originally Posted by MagiMaster
One way to look at the 0.9999... question is to consider this: what is 1 - 0.9999...? The answer, 0. Therefore, yes, 0.9999... = 1. Actually, there are many, many ways to show that this is true, at least for the real numbers. There are number systems out there where it isn't true anymore, but those systems aren't what we're used to.

BTW, the real world doesn't seem to contain any true infinites. The universe is composed of finitely many particles of a finite size with a finite amount of energy existing within a finite amount of space over a finite amount of time. :/
arent there an infinite amount of points in any given space?

13. If something, THING, can "apply" within "infinity", why not aply that also commonly used term "beyond zero"?

"Within infinity....beyond zero".

Anyone who wants to be OUTSIDE that "scale"......of things, obviously useful to inter-related sexual-exercises that produce nothing other than doubt..............

%

14. in mathematics infinity is any number which is always greater than any large number you can think of.
in physics infinity means space in which you keep on moving without reaching an "end"?

15. Originally Posted by JaneBennet
Originally Posted by MagiMaster
One way to look at the 0.9999... question is to consider this: what is 1 - 0.9999...? The answer, 0.
How do you work that one out?
Well, consider the partial series:
1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.999 = 0.001
.
.
.

What would the limit of this series be? After 1 step it's . After 2 steps it's . Etc. So 1 - 0.9999... would be which, in the real numbers, is 0. (Other number systems might define an infintesimal, which is a number smaller than any positive real, but still larger than 0. This is what confuses most people. In the real numbers, there is no infintesimal.)

16. Originally Posted by randynewman666
Originally Posted by MagiMaster
One way to look at the 0.9999... question is to consider this: what is 1 - 0.9999...? The answer, 0. Therefore, yes, 0.9999... = 1. Actually, there are many, many ways to show that this is true, at least for the real numbers. There are number systems out there where it isn't true anymore, but those systems aren't what we're used to.

BTW, the real world doesn't seem to contain any true infinites. The universe is composed of finitely many particles of a finite size with a finite amount of energy existing within a finite amount of space over a finite amount of time. :/
arent there an infinite amount of points in any given space?

Yes, you can keep dividing a linear space, a one inch straight line, until no human could ever in his lifetime conclude one way or the other, if the divisions are actually there.
A human would not be able to count them or measure them in one lifetime or measure them by any human standard. But mathematically and realistically they could exist even if there was no way to verify them.

In other words when electrons move, I highly doubt they jump from the smallest grid or increment we can measure, to the next grid or increment we can measure. I would say they move along smoothly in infinitely small increments.

But the fact that we cannot see them. Means we assume that they don't jump from one place to another. They might move like Mexican jumping beans for all we know.

Sincerely,

William McCormick

17. *Ia m either here to be good or bad.
None of you have worked for me yet.

I may as well be bad.

18. Originally Posted by William McCormick
Originally Posted by randynewman666
Originally Posted by MagiMaster
One way to look at the 0.9999... question is to consider this: what is 1 - 0.9999...? The answer, 0. Therefore, yes, 0.9999... = 1. Actually, there are many, many ways to show that this is true, at least for the real numbers. There are number systems out there where it isn't true anymore, but those systems aren't what we're used to.

BTW, the real world doesn't seem to contain any true infinites. The universe is composed of finitely many particles of a finite size with a finite amount of energy existing within a finite amount of space over a finite amount of time. :/
arent there an infinite amount of points in any given space?

Yes, you can keep dividing a linear space, a one inch straight line, until no human could ever in his lifetime conclude one way or the other, if the divisions are actually there.
A human would not be able to count them or measure them in one lifetime or measure them by any human standard. But mathematically and realistically they could exist even if there was no way to verify them.

In other words when electrons move, I highly doubt they jump from the smallest grid or increment we can measure, to the next grid or increment we can measure. I would say they move along smoothly in infinitely small increments.

But the fact that we cannot see them. Means we assume that they don't jump from one place to another. They might move like Mexican jumping beans for all we know.

Sincerely,

William McCormick

The idea of being able to divide a space in reality infinitly has always boggled my mind. When electrons move I wouldnt think they jump from one finite place to another but how can they be described as moving smoothly along infinitely small increments? It would still be moving in increments. We try to describe infinity in math, which doesnt seem right to me. I think Infinity exists in nature but I think we just dont comprehend it yet.

19. There may be some limits on how small things can be, etc. Check out Planck length and Planck time. Of course, this falls into the "no one quite knows" category.

20. Originally Posted by randynewman666
Originally Posted by William McCormick
Originally Posted by randynewman666
Originally Posted by MagiMaster
One way to look at the 0.9999... question is to consider this: what is 1 - 0.9999...? The answer, 0. Therefore, yes, 0.9999... = 1. Actually, there are many, many ways to show that this is true, at least for the real numbers. There are number systems out there where it isn't true anymore, but those systems aren't what we're used to.

BTW, the real world doesn't seem to contain any true infinites. The universe is composed of finitely many particles of a finite size with a finite amount of energy existing within a finite amount of space over a finite amount of time. :/
arent there an infinite amount of points in any given space?

Yes, you can keep dividing a linear space, a one inch straight line, until no human could ever in his lifetime conclude one way or the other, if the divisions are actually there.
A human would not be able to count them or measure them in one lifetime or measure them by any human standard. But mathematically and realistically they could exist even if there was no way to verify them.

In other words when electrons move, I highly doubt they jump from the smallest grid or increment we can measure, to the next grid or increment we can measure. I would say they move along smoothly in infinitely small increments.

But the fact that we cannot see them. Means we assume that they don't jump from one place to another. They might move like Mexican jumping beans for all we know.

Sincerely,

William McCormick

The idea of being able to divide a space in reality infinitly has always boggled my mind. When electrons move I wouldnt think they jump from one finite place to another but how can they be described as moving smoothly along infinitely small increments? It would still be moving in increments. We try to describe infinity in math, which doesnt seem right to me. I think Infinity exists in nature but I think we just dont comprehend it yet.

I would think that electrons slow and positively accelerate as they move through matter. But their overall movement is probably a very smooth motion. I would say infinitely smooth.

I know where moving objects are caught on high speed film, or movie film, they show a jittering jerky movement. We often relate that to the actuality. The truth is that it is just the restricted speed of the recording device, that creates the jerky movements.

I would imagine if you tried to film an electron you would get that same jerky motion. However the reason would be the particles carrying the picture of the single electrons movement would come to the recording device intermittently and in different areas of the media.
Causing a guaranteed jerky picture. And loss of focus, as you wait for enough particles to hit the entire recording surface, and create a change in that surface, representative of the actuality.

To me this is all an amazing speculation though. Because I know they were here in the early 1900's and they all agreed that an electron will never be seen. And they were mighty crafty back then.

If you tried to film an electron, you would have to wait till the electrons that bring you a picture get to the recording device, and recreate the photo, the photon within the recording device. By that time the electron would be gone. You would never get a second glimpse of the same electron.
Your field of view is going to be to small compared to the distance from the electron being filmed and the recording device. The time it takes the electrons bringing the information is going to be longer then the field of view is going to be able to capture the electrons presence. So you are guaranteed to never get a second glimpse. Even if you could get one glimpse.

I was taught that infinite just meant that in one lifetime one human could not count or positively verify the occurrence, whether it be the number of particles, the length of the object, the location of the object, or whatever cannot be confirmed by one individual.

Part of the race in the Olympics where they hand off the baton from one runner to the next, is part of this proof of infinity. Or when you whisper a predetermined statement from one student to the next in a room.
All to often this exchange of information and responsibility causes error. And in science there can be no error. So one individual would have to work on the counting or tallying for a truly scientific experiment.

That is why I am an amateur scientist. In this world with the current poor labelling or mislabeling of important things, I am surely only an amateur scientist, until I get it straightened out.

If you look at the individual runner they will almost never drop a baton as they are running alone. You will almost always see the error caused while exchanging information or the baton.

Sincerely,

William McCormick

21. Originally Posted by William McCormick
Originally Posted by randynewman666
Originally Posted by William McCormick
Originally Posted by randynewman666
Originally Posted by MagiMaster
One way to look at the 0.9999... question is to consider this: what is 1 - 0.9999...? The answer, 0. Therefore, yes, 0.9999... = 1. Actually, there are many, many ways to show that this is true, at least for the real numbers. There are number systems out there where it isn't true anymore, but those systems aren't what we're used to.

BTW, the real world doesn't seem to contain any true infinites. The universe is composed of finitely many particles of a finite size with a finite amount of energy existing within a finite amount of space over a finite amount of time. :/
arent there an infinite amount of points in any given space?

Yes, you can keep dividing a linear space, a one inch straight line, until no human could ever in his lifetime conclude one way or the other, if the divisions are actually there.
A human would not be able to count them or measure them in one lifetime or measure them by any human standard. But mathematically and realistically they could exist even if there was no way to verify them.

In other words when electrons move, I highly doubt they jump from the smallest grid or increment we can measure, to the next grid or increment we can measure. I would say they move along smoothly in infinitely small increments.

But the fact that we cannot see them. Means we assume that they don't jump from one place to another. They might move like Mexican jumping beans for all we know.

Sincerely,

William McCormick

The idea of being able to divide a space in reality infinitly has always boggled my mind. When electrons move I wouldnt think they jump from one finite place to another but how can they be described as moving smoothly along infinitely small increments? It would still be moving in increments. We try to describe infinity in math, which doesnt seem right to me. I think Infinity exists in nature but I think we just dont comprehend it yet.

I would think that electrons slow and positively accelerate as they move through matter. But their overall movement is probably a very smooth motion. I would say infinitely smooth.

I know where moving objects are caught on high speed film, or movie film, they show a jittering jerky movement. We often relate that to the actuality. The truth is that it is just the restricted speed of the recording device, that creates the jerky movements.

I would imagine if you tried to film an electron you would get that same jerky motion. However the reason would be the particles carrying the picture of the single electrons movement would come to the recording device intermittently and in different areas of the media.
Causing a guaranteed jerky picture. And loss of focus, as you wait for enough particles to hit the entire recording surface, and create a change in that surface, representative of the actuality.

To me this is all an amazing speculation though. Because I know they were here in the early 1900's and they all agreed that an electron will never be seen. And they were mighty crafty back then.

If you tried to film an electron, you would have to wait till the electrons that bring you a picture get to the recording device, and recreate the photo, the photon within the recording device. By that time the electron would be gone. You would never get a second glimpse of the same electron.
Your field of view is going to be to small compared to the distance from the electron being filmed and the recording device. The time it takes the electrons bringing the information is going to be longer then the field of view is going to be able to capture the electrons presence. So you are guaranteed to never get a second glimpse. Even if you could get one glimpse.

I was taught that infinite just meant that in one lifetime one human could not count or positively verify the occurrence, whether it be the number of particles, the length of the object, the location of the object, or whatever cannot be confirmed by one individual.

Part of the race in the Olympics where they hand off the baton from one runner to the next, is part of this proof of infinity. Or when you whisper a predetermined statement from one student to the next in a room.
All to often this exchange of information and responsibility causes error. And in science there can be no error. So one individual would have to work on the counting or tallying for a truly scientific experiment.

That is why I am an amateur scientist. In this world with the current poor labelling or mislabeling of important things, I am surely only an amateur scientist, until I get it straightened out.

If you look at the individual runner they will almost never drop a baton as they are running alone. You will almost always see the error caused while exchanging information or the baton.

Sincerely,

William McCormick
So you agree that infinity is usually thought of as something going on and on and 1 person cannot prove if that actually exists. If something is going on and on that means it hasnt reached certain points, wouldnt infinity be all points? If an electron doesnt move from 1 finite point to another It would move infinitly smooth. I dont think anyone can imagine that, something moving infinitly smooth. I also dont think we can explain it in math terms because math would just try to make infinity finite somehow. This just makes be belive humans dont know the true nature of infinity yet.

22. Infinity in physics, in context of relativity, suggests to me infinity with no fixed reference e.g. zero. Moreover, without reference, if it points in one direction it must point the other way as well because a "start" is a fixed reference.

23. Originally Posted by randynewman666
So you agree that infinity is usually thought of as something going on and on and 1 person cannot prove if that actually exists. If something is going on and on that means it hasnt reached certain points, wouldnt infinity be all points? If an electron doesnt move from 1 finite point to another It would move infinitly smooth. I dont think anyone can imagine that, something moving infinitly smooth. I also dont think we can explain it in math terms because math would just try to make infinity finite somehow. This just makes be belive humans dont know the true nature of infinity yet.

Yes, that was the technical scientific term for infinity. It was a human based call.

If one human could hear a frequency or sense a frequency beyond 18,000 cycles a second. Then that would not be frequency, infinitely out of human hearing range.
However it should be noted that only some humans can sense that. And it should be checked to see if the fellow is sensing the sound, and not some kind of electrical or radio like emission.

But usually 18,000 cycles is out of human hearing range. You could sit there for all eternity and never hear the sound at those hertz or cycles. 18,000 hertz is usually infinitely out of the human hearing range.

Zero is an infinity, a nonexisting number of items. You will be looking for zero car keys for all eternity. Zero car keys are infinitely not there. You may also be looking for one set of car keys for all eternity. The might be infinitely lost. Ha-ha.

But those are two different based infinities.

Infinity was always thought of in terms of, "being able to prove it in one lifetime". If it takes longer then that. That is an infinitely long time.

A human is infinitely impossible to replace. A single human mind is infinitely unique. One scientists research is infinitely unique to his understanding. No one else could just say the exact same things about his work. No one else has the infinitely unique vantage points he has.

That is why I always hit up the forums for a glimpse into other individuals vantage points and observations unique to them. Infinitely theirs.

Never in one lifetime, is an eternity, an infinity. For one lifetime, is an eternity an infinity. When you die your gone and no one can replace you. You are infinitely gone. We lose something infinitely valuable.

The knowledge of God, or all knowing, is infinity to a single man.

Just thought I would share a few of my other view points on infinity.

Sincerely,

William McCormick

24. Originally Posted by MagiMaster
Well, consider the partial series:
1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.999 = 0.001
.
.
.

What would the limit of this series be? After 1 step it's . After 2 steps it's . Etc. So 1 - 0.9999... would be which, in the real numbers, is 0. (Other number systems might define an infintesimal, which is a number smaller than any positive real, but still larger than 0. This is what confuses most people. In the real numbers, there is no infintesimal.)
But even if you're working in a hyper-real number system that allows infinitesimals, it's not clear (to me, anyway) that 0.999... and 1 aren't still exactly equal. It's been a long time since I studied any real analysis (much less nonstandard analysis), but as I recall hyper-real infinitesimals are the reciprocals of "unlimited" numbers that are larger than any number that could be expressed as the limit of an infinite series. So if you can express the difference between 0.999... and 1 as something like , I'm not sure that it qualifies as a valid hyper-real infinitesimal.

Also, remember that the real numbers are a subset of the hyper-real numbers. Since 0.999... and 1 are both real numbers and exactly equal under the real number system, I don't think they would suddenly stop being equal in you add in the hyper-real numbers to the mix. I'm pretty sure that if E is your hyperreal infinitesimal, 1-E is smaller than 0.999... and 1-E is still exactly equal to 0.999...-E.

Of course I could be totally wrong about this, and if anyone out there has taken a class in nonstandard analysis lately I would love to hear what they have to say on this issue. But my point is that even if you work with hyper-real numbers, it still isn't clear to me that 0.999... and 1 are different.

25. I thought that, with regard to

1 - 0.999999...

the safe result is that the Limit approaches 0, not that it is actually = 0?

Or am I limiting the concept of a limit meaninglessly?

Tp my mind, therefore, when considering infinity (to respond to Kalster's original question), if there is a 0.99999999... chance of a member of a set having a characteristic, then it is not the same as a chance of 1, and there may even be infinite members of the set that do not share that characeristic.

But I'd love a proper mathematician to explain to me why I'm wrong.

26. Basically, 0.9999... is the limit. A limit implies a series, which in this case is {0.9, 0.99, 0.999}, etc. So yes, the limit of {1-0.9, 1-0.99, 1-0.999} is 0, which means that 1-0.9999... is 0.

27. Originally Posted by MagiMaster
Basically, 0.9999... is the limit. A limit implies a series, which in this case is {0.9, 0.99, 0.999}, etc. So yes, the limit of {1-0.9, 1-0.99, 1-0.999} is 0, which means that 1-0.9999... is 0.
Am I wrong though in saying that limit are only a way for us to deal with infinity? It involves choosing an infinitesimal that has to be added after a certain number of iterations of the series, no? The smaller the infinitesimal the more accurate the result that is attained. In reality you could reduce the size of the infinitesimal infinitely, ending up right back where you started. Or do I misunderstand something? :?

28. Originally Posted by sunshinewarrior
I thought that, with regard to

1 - 0.999999...

the safe result is that the Limit approaches 0, not that it is actually = 0?

Or am I limiting the concept of a limit meaninglessly?

Tp my mind, therefore, when considering infinity (to respond to Kalster's original question), if there is a 0.99999999... chance of a member of a set having a characteristic, then it is not the same as a chance of 1, and there may even be infinite members of the set that do not share that characeristic.

But I'd love a proper mathematician to explain to me why I'm wrong.
1 - 0.999... doesn't have a limit, because limits only apply to a series of calculations that stretch off to infinity. 1 - 0.999... is simply zero. You could express 0.999... as the limit of the series 0.9+0.09+0.009..., because the limit of that series is 1. Or 0.999..., however you prefer to write it. 1 and 0.999... are two different ways of writing the same number, just like 0.5 and 1/2 are different ways of writing the same number.

29. Okay, I read up a little on non-standard analysis and hyper-real numbers, and it appears that infinitesimals do NOT allow you to avoid 0.999... exactly equaling 1. The hyper-real numbers are an extension of the real numbers - in other words, all real numbers are also hyper-real but not all hyper-real numbers are real. 0.999... and 1 are both real numbers, and their values don't change simply because you allow for the existence of hyper-real numbers, so they are still equal.

Also, there is a theorem which proves that any result that is true in standard analysis (meaning working with the real numbers) will also be true in nonstandard analysis (when you're also working with infinitesimals). If 0.999...=1 is true in the real number system (and it is), it's still going to be true in a hyper-real system.

30. Originally Posted by Scifor Refugee
Okay, I read up a little on non-standard analysis and hyper-real numbers, and it appears that infinitesimals do NOT allow you to avoid 0.999... exactly equaling 1. The hyper-real numbers are an extension of the real numbers - in other words, all real numbers are also hyper-real but not all hyper-real numbers are real. 0.999... and 1 are both real numbers, and their values don't change simply because you allow for the existence of hyper-real numbers, so they are still equal.

Also, there is a theorem which proves that any result that is true in standard analysis (meaning working with the real numbers) will also be true in nonstandard analysis (when you're also working with infinitesimals). If 0.999...=1 is true in the real number system (and it is), it's still going to be true in a hyper-real system.
Lets say that below is a representation of 1/100ths of an inch. Each one of those marks means 1/100th of an inch. In the first row there are 99 marks. It means 99/100ths. You could represent this with 0.99 in base ten decimal, it means the same thing. Or you could use 99/100ths all the same thing. All totally accurate.
/////////////////////////////////////////////////////////////////////////////////////////////////

In this second row there are 100 1/100ths of an inch or a whole inch. You could represent this with 1.0 in base ten. Or you could say 100/100ths
//////////////////////////////////////////////////////////////////////////////////////////////////

But they are not the same thing respectively. One is exactly, one one hundredth larger then the other.

We used to use reminders in school and they are highly accurate. What ever the reminder is that is what the remainder is exactly.

So if you use long division on 23/3 to express it in base ten decimal. Often it looks like this represented in base ten decimal as 7.6666666666666666666666666666666666667 they round up the six at the end.

23/3 is a perfect representation of a real ratio in base ten. It represents something real, exactly as it is.

If you use the remainder method you end up with three goes into 23 seven times, with a remainder of 2. That means, that three goes into 23 seven times, and 2/3rds of the three again.
But you can divide things exactly using the reminder, most people just do not understand base ten decimal or fraction. Or the other bases. Especially in relation to real things.

I will be honest I approximately counted out those hash marks but there could be more or less. But it is the principle that I am getting across.

Sincerely,

William McCormick

31. Originally Posted by theQuestIsNotOver
*Ia m either here to be good or bad.
None of you have worked for me yet.

I may as well be bad.
You know what Santa says. You Better watch out, you better not cry. You better not pout, I am telling you why. Santa Claus is coming to town.

Sincerely,

William McCormick

32. Originally Posted by KALSTER
As I understand it, it is considered in mathematics that 0.99999^. = 1. But does this necessarily have to be the case in reality? If you have an infinite data set where 0.99999^. is supposed to describe the probability of a certain property to be exhibited by each element of the data set you would care to consider that all of them would exhibit this particular property? (hope that was coherent :? )
I believe so. I have an actual source of actual proof of the concept infinite in this case, to which I hope helps.

There is a video game called 'Breath Of Fire III'. In this game at the end of the first chapter in the game, you fight 2 undefeatable enemies known as Balio and Sunder. Many people have tried to beat these enemies, but it is vital that they defeat you in order for the story of the game to progress. And if you have not levelled enough (and I mean a lot), you will get slaughtered.

In the game code, they do not have life (HP), in others words they can never be defeated.

I managed to get to a point where Balio and Sunder no longer attacked me, where they as a result could not win as long as I continued to fight. And I could now inflict damage on them forever without dying.

This caused a stalemate. Now by Einsteins concept of nothing travelling faster than light (where is where I get this explanation from). Something with mass can go faster than light if it has infinite energy to use, or acts for an infinite amount of time.

Using this notion, I thus conclude that Balio and Sunder were defeatable, if I fought for eternity (an infinitie time). Now then, it doesn't actually mean that I would beat them, but the concept that I can beat them is there.

Here is a video of:

The way the fight is supposed to go.

And the way that I made it go (infinite).

I know this explanation of the phenomena is vauge and a bit wierd, but I believe it is a down to Earth and is an understandable notion of infinite. I hope this has helped.

33. Using the theories expressed - if the universe is infinitely big, or lasts for an infinitely long time, surely every event will happen (perhaps simultaneously).

Using that, I can prove that we'll discover intelligent life, provided humanity lasts long enough. It's there. Perhaps instead of parallel universes, we have parallel galaxies yet to be discovered.

Sorry this doesn't quite fit into the flow of things but..

Saying 0.9999^./1 as a probability is a theoretical probability. The reality is that it could happen the first time it occurs.

34. Originally Posted by KALSTER
Originally Posted by MagiMaster
Basically, 0.9999... is the limit. A limit implies a series, which in this case is {0.9, 0.99, 0.999}, etc. So yes, the limit of {1-0.9, 1-0.99, 1-0.999} is 0, which means that 1-0.9999... is 0.
Am I wrong though in saying that limit are only a way for us to deal with infinity? It involves choosing an infinitesimal that has to be added after a certain number of iterations of the series, no? The smaller the infinitesimal the more accurate the result that is attained. In reality you could reduce the size of the infinitesimal infinitely, ending up right back where you started. Or do I misunderstand something? :?
If you really want an understanding of the notion of "infinity" you need to take a look at the notion of cardinal and ordinal numbers. It turns out that there are in fact many different sizes of "infinity".

A set is infinite if it can be placed in one-to-one correspondence with a proper subset.

The smallest infinity is the size of the natural numbers. This turns out to be the same as the size of the integers and even of the rational numbers. Each of these sets can be put in one-to-one correspondence with one another.

The real numbers are a large size of infinity. It can be proved that it is impossible to put the real numbers in one-to-one correspondence with the natural numbers.

A good reference for this material is the little book Naive Set Theory by Paul Halmos.

35. Originally Posted by KALSTER
This seems to be a widely misunderstood concept (including by me), yet one commonly used wrongly. One of the common mistakes, for instance, is describing something as simply infinite without specifying exactly what quality of the thing is supposed to be infinite. So let me make some statements and ask some questions regarding infinity and I would like anyone properly knowledgeable to verify their validity, if not make some corrections.

It seems to me that structure can exist within infinity. One example of this would be that in a hypothetical universe that is infinite in time, space and matter, that matter could exist as one particle per light year or as an infinite gas cloud and that both scenarios would have infinite matter.

As I understand it, it is considered in mathematics that . But does this necessarily have to be the case in reality? If you have an infinite data set where is supposed to describe the probability of a certain property to be exhibited by each element of the data set, does that mean that among any number of elements in the infinite data set you would care to consider that all of them would exhibit this particular property? (hope that was coherent :? )
In going back to the initial question, the question was relevant to what we are suppose to exist "within", what "infinity" we are suppose to exist within when using the term "infinity". Basically, what is infinity of space and time. Infinity can also be used in conceptual mathematics, but ultimately in using the idea of "infinity" mathematically that is pointing to the greater structure of space-time we live within and experiment within, "infinity" is the carrot in front of the donkey. It is impossible to "prove", namely infinite time and infinite space, and so the best we can be with the idea is "pure theory", pure mathematics. It is the same with the idea of "zero", namely zero space and zero time. Purely theoretical. If you want to be purely theoretical with mathematics regarding space and time, use "zero" and "infinity". And, in understanding the basics of physics, you would know that the superstructure, the infinity, relates to the sub structure, the zero, that the laws of physics applies to both almost as "one". Nothing strange about that.

Now, if you want to theorise space-time using the mathematical variables of zero and infinity for space and time, you will NEED to step OUTSIDE of the experimental-square. The theory you would come up with would not be like theories we have derived through research and experiment. But, the theory would be able to represent a unique "way" of explaining space-time, not entirely incorrect. It would be pure theory, but it would represent what exactly infinity and zero would be to space and time, namely "a little out there". The theory though would be useful in making suggestions about how space-time works. If though any such theory could form a space-time link between zero and infinity, it may be on the money. The thing is though is that no one would know it (it being on the money), because it would be an "outside the experimental square" theory, one not derived from experiment but one derived from using "zero" and "infinity", being purely theoretical, non-experimental, and then proposing the ideas/forces/energies/matter that link space-time between zero and infinity for space and time. Any such theory may be a more "efficient" way of explaining space-time. And if it is a more "efficient" way of explaining space-time, it may also lead to more efficent ways of using space-time (mass-energy).

36. Originally Posted by Lawrence
Using the theories expressed - if the universe is infinitely big, or lasts for an infinitely long time, surely every event will happen (perhaps simultaneously).
Just because an infinite amount of things happen doesn't mean that every conceivable thing must happen. There are infinitely many numbers between 3 and 4, but none of them are greater than 5.

37. That's right.

Why get upset.

We are being teased into conversation!

38. Originally Posted by Scifor Refugee
Originally Posted by Lawrence
Using the theories expressed - if the universe is infinitely big, or lasts for an infinitely long time, surely every event will happen (perhaps simultaneously).
Just because an infinite amount of things happen doesn't mean that every conceivable thing must happen. There are infinitely many numbers between 3 and 4, but none of them are greater than 5.
They do happen, all the increments between three and four. We just assume them. They are there, we just can never count them all.

Sincerely,

William McCormick

39. Originally Posted by DrRocket
The real numbers are a large size of infinity. It can be proved that it is impossible to put the real numbers in one-to-one correspondence with the natural numbers.
And for a light-hearted attempt to show this, or rather the line of reasoning that leads to this conclusion, look at this post

40. Originally Posted by theQuestIsNotOver
Originally Posted by KALSTER
This seems to be a widely misunderstood concept (including by me), yet one commonly used wrongly. One of the common mistakes, for instance, is describing something as simply infinite without specifying exactly what quality of the thing is supposed to be infinite. So let me make some statements and ask some questions regarding infinity and I would like anyone properly knowledgeable to verify their validity, if not make some corrections.

It seems to me that structure can exist within infinity. One example of this would be that in a hypothetical universe that is infinite in time, space and matter, that matter could exist as one particle per light year or as an infinite gas cloud and that both scenarios would have infinite matter.

As I understand it, it is considered in mathematics that . But does this necessarily have to be the case in reality? If you have an infinite data set where is supposed to describe the probability of a certain property to be exhibited by each element of the data set, does that mean that among any number of elements in the infinite data set you would care to consider that all of them would exhibit this particular property? (hope that was coherent :? )
In going back to the initial question, the question was relevant to what we are suppose to exist "within", what "infinity" we are suppose to exist within when using the term "infinity". Basically, what is infinity of space and time. Infinity can also be used in conceptual mathematics, but ultimately in using the idea of "infinity" mathematically that is pointing to the greater structure of space-time we live within and experiment within, "infinity" is the carrot in front of the donkey. It is impossible to "prove", namely infinite time and infinite space, and so the best we can be with the idea is "pure theory", pure mathematics. It is the same with the idea of "zero", namely zero space and zero time. Purely theoretical. If you want to be purely theoretical with mathematics regarding space and time, use "zero" and "infinity". And, in understanding the basics of physics, you would know that the superstructure, the infinity, relates to the sub structure, the zero, that the laws of physics applies to both almost as "one". Nothing strange about that.

Now, if you want to theorise space-time using the mathematical variables of zero and infinity for space and time, you will NEED to step OUTSIDE of the experimental-square. The theory you would come up with would not be like theories we have derived through research and experiment. But, the theory would be able to represent a unique "way" of explaining space-time, not entirely incorrect. It would be pure theory, but it would represent what exactly infinity and zero would be to space and time, namely "a little out there". The theory though would be useful in making suggestions about how space-time works. If though any such theory could form a space-time link between zero and infinity, it may be on the money. The thing is though is that no one would know it (it being on the money), because it would be an "outside the experimental square" theory, one not derived from experiment but one derived from using "zero" and "infinity", being purely theoretical, non-experimental, and then proposing the ideas/forces/energies/matter that link space-time between zero and infinity for space and time. Any such theory may be a more "efficient" way of explaining space-time. And if it is a more "efficient" way of explaining space-time, it may also lead to more efficent ways of using space-time (mass-energy).
You can only use math on things you can measure and demonstrate. If you cannot demonstrate something then you cannot use math on it. Why? Because that is a double unknown variable.

You cannot prove one unknown variable with another unknown variable. That would be just pure speculation. It is not algebra at that point. It is not 2+A=4 And we fill in the A with two. And the equation works out and can be checked. "A" is out of our scope of understanding at this point.

We already know we cannot know what "A" is totally and exactly, that is all we are sure of, about all of "A".
We can though call it a variable in universal understanding. Demonstrate and calculate with what we do have, right in front of us. Right in front of us we can say that "A" is two or we have two "A"s. But we don't know how much "A" there is in the whole universe.

I get concerned because most of the multi subatomic particle scientists do not want to know what is right in front of them. I do not mind them dividing by zero. The answer used to be infinity. Which causes an error to a computer if the computer does not realize it.

But when these modern scientists come up with a real number after dividing by zero I get worried. In other words we have zero data or infinite data and they come up with three.

Sincerely,

William McCormick

41. Hello everyone.

Personally, I don't believe in the existence of Infinity or any properties that can assume an Infinite value. I believe the concept of Infinity is classical notion. Infinity cannot exist in a world where Quantum effects are prevalent. The reason is that Quantum mechanics shows us that everything in the Universe is granular in nature.
Let me explain :
First, I want to take you through a situation where infinity may arise, and from this, I will try to draw some important conclusions. Take the Time dilation formula for example:

t = to/(1-v^2/c^2)^0.5

as v which is the relative speed of the object, approaches a value of c, t approaches a value of Infinity or really to/0 .
Now suppose v>c , then we would get to/ni, where ni is an imaginary value. The i ( which is square root of -1) represents a direction reversal in time, but what is interesting to note is that the value to/0 is infinity for the set of values where v<c, but for the set of values where v>c, to/0 assumes the role of the value zero from which the imaginary values of time begin. The important point here is that Infinity and zero are essentially the same values only they represent different directions when moving through the number line.
So now that we have established that Infinity and zero are essentially the same things, we can now show using some simple Quantum mechanics that Infinity is impossible in nature. The Uncertainty principle disallows anything to have a value of zero, there are always zero-point fluctuations. Therefore, using our zero and Infinity equivalence, we can say the same about Infinity. The uncertainty principle disallows Infinity in nature.

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