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Author Topic:   A definition of infinity?
fallacycop
Member (Idle past 5521 days)
Posts: 692
From: Fortaleza-CE Brazil
Joined: 02-18-2006


Message 16 of 41 (372308)
12-26-2006 1:52 PM
Reply to: Message 15 by Hyroglyphx
12-26-2006 12:41 PM


Would the outcome be different if we were to employ a different measuring device?
I'll quote myself
quote:
No matter what you do, you cannot at the same time know which slit the electron went through, and still get the interference pattern. It`s just not possible.

This message is a reply to:
 Message 15 by Hyroglyphx, posted 12-26-2006 12:41 PM Hyroglyphx has replied

Replies to this message:
 Message 18 by Hyroglyphx, posted 12-26-2006 2:19 PM fallacycop has replied

  
Brad McFall
Member (Idle past 5033 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 17 of 41 (372309)
12-26-2006 2:05 PM
Reply to: Message 9 by Chiroptera
12-25-2006 9:17 PM


Re: Pedantic point.
Wow chirp!!
Based on your other posts off EVC, I had no idea you knew and thought all this stuff. Cool!

This message is a reply to:
 Message 9 by Chiroptera, posted 12-25-2006 9:17 PM Chiroptera has not replied

  
Hyroglyphx
Inactive Member


Message 18 of 41 (372312)
12-26-2006 2:19 PM
Reply to: Message 16 by fallacycop
12-26-2006 1:52 PM


Outcomes
No matter what you do, you cannot at the same time know which slit the electron went through, and still get the interference pattern. It`s just not possible.
I think we all agree with that premise but wonder why that it is. Maybe its just not possible as of yet. I mean, who would have thought that matter behaved like a wave function or that by trying to observe the phenomenon would actually cause it to behave differently than when not directly observed?

"A man can no more diminish God's glory by refusing to worship Him than a lunatic can put out the sun by scribbling the word, 'darkness' on the walls of his cell." -C.S. Lewis

This message is a reply to:
 Message 16 by fallacycop, posted 12-26-2006 1:52 PM fallacycop has replied

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Hyroglyphx
Inactive Member


Message 19 of 41 (372315)
12-26-2006 2:28 PM
Reply to: Message 13 by dogrelata
12-26-2006 6:58 AM


I imagine a sphere one meter in diameter, with a line extending infinitely from its surface, I will have a semi-infinite line. If I then imagine the diameter of the sphere to be halved, I still have a semi-finite line. I guess my question is, what prevents me halving the diameter of the sphere indefinitely? In other words, if I were to follow the line on its journey towards an ever-shrinking sphere, must I inevitably reach a point where the line can extend no further, and therefore be considered semi-finite?
I don't know. But I do object to the term "semi-finite." I mean, something is either finite or infinite. What in-between would exist?

"A man can no more diminish God's glory by refusing to worship Him than a lunatic can put out the sun by scribbling the word, 'darkness' on the walls of his cell." -C.S. Lewis

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 Message 13 by dogrelata, posted 12-26-2006 6:58 AM dogrelata has replied

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fallacycop
Member (Idle past 5521 days)
Posts: 692
From: Fortaleza-CE Brazil
Joined: 02-18-2006


Message 20 of 41 (372316)
12-26-2006 2:44 PM
Reply to: Message 18 by Hyroglyphx
12-26-2006 2:19 PM


Re: Outcomes
I think we all agree with that premise but wonder why that it is.
It's a consequence of the uncertainty principle. In the first experiment, (Without the observer) the position of the electron is uncertain (We don't know which slit the electron went through). that allows for a less uncertain momentum which is the same as a more shaply defined wave-length (By the de Broglie's relation p=h/lambda). This shaply defined wave-length leads to the observed interference pattern.
In the second experiment the position of the electron becomes more determined and, by the uncertainty principle, its wave-length becomes blured (more uncertain) destroying the interference pattern.
Maybe its just not possible as of yet. I mean, who would have thought that matter behaved like a wave function or that by trying to observe the phenomenon would actually cause it to behave differently than when not directly observed?
I don't think so. the validity of quantum mechanics (QM for short) has been confirmed by a huge range of experiments to the point that it is not unreasonable to consider it incontrovertable.

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dogrelata
Member (Idle past 5312 days)
Posts: 201
From: Scotland
Joined: 08-04-2006


Message 21 of 41 (372397)
12-27-2006 7:32 AM
Reply to: Message 19 by Hyroglyphx
12-26-2006 2:28 PM


nemesis juggernaut writes:
I don't know. But I do object to the term "semi-finite." I mean, something is either finite or infinite. What in-between would exist?
Obviously I’m no expert in this area, which is one of the reasons I started the post, but I hope I’m starting to gain a better understanding than I previously had.
It begins to appear that the way we view things goes some way to defining degrees of infinity. If we take, for example, the set of positive integers, it could be defined as infinitely large. If we then pick a positive integer at random, and proceed to increment it by one, we can do so forever, as the number of positive integers is presumably infinitely large. However, if we choose to deduct by one each time instead of increment, we will eventually arrive at the number one, a point beyond which we cannot go as there are no positive integers less than one.
So it appears to me if we view the set of positive integers solely as that, a set, it may be defined as infinite. However, if we start to think of the set in terms of a sequence, it may be defined as semi-infinite, as there is a point beyond which it is impossible to go in one ”direction’.
There’s another interesting point regarding positive integers and infinity. It concerns what happens when you pair each positive integer with its square, i.e. 1 with 1, 2 with 4, 3 with 9, etc. The original set will be a complete set of positive integers, but the set of squares is not, yet both sets are exactly the same size! I believe this apparent paradox has been resolved, but I cannot remember by whom, and am not aware as to what the explanation is. Maybe somebody on here can help us.

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Phat
Member
Posts: 18262
From: Denver,Colorado USA
Joined: 12-30-2003
Member Rating: 1.1


Message 22 of 41 (372405)
12-27-2006 8:30 AM
Reply to: Message 1 by dogrelata
12-24-2006 7:34 AM


Forever and ever and ever
One thing I never figured out is how human minds (which are finite by definition) were able to quantify an infinite definition.

This message is a reply to:
 Message 1 by dogrelata, posted 12-24-2006 7:34 AM dogrelata has replied

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 Message 27 by JavaMan, posted 12-28-2006 7:39 AM Phat has not replied

  
dogrelata
Member (Idle past 5312 days)
Posts: 201
From: Scotland
Joined: 08-04-2006


Message 23 of 41 (372409)
12-27-2006 9:05 AM
Reply to: Message 22 by Phat
12-27-2006 8:30 AM


Re: Forever and ever and ever
Phat writes:
One thing I never figured out is how human minds (which are finite by definition) were able to quantify an infinite definition.
Is an ”infinite definition’ the same as a definition of the infinite though?

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Neutralmind
Member (Idle past 6124 days)
Posts: 183
From: Finland
Joined: 06-08-2006


Message 24 of 41 (372411)
12-27-2006 9:14 AM
Reply to: Message 14 by fallacycop
12-26-2006 8:00 AM


fallacycop
The problem isn`t the observer per se, but the means for observing. in order to observe the electron, you would have to shine some light on it. The act of shining that light changes the behaviour of the electron and distroys the interference pattern.
How about if we observed it by different means, something that doesn't involve light?

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Son Goku
Inactive Member


Message 25 of 41 (372435)
12-27-2006 12:23 PM


Infinity and QM
There are questions about two things here so I'll answer them in order.
So it appears to me if we view the set of positive integers solely as that, a set, it may be defined as infinite. However, if we start to think of the set in terms of a sequence, it may be defined as semi-infinite, as there is a point beyond which it is impossible to go in one ”direction’.
Formally that quality is expressed by saying that the positive integers are bounded. They are still infinite in the sense that they do not have a finite collection of elements.
You've spotted the difference yourself though. As a set (as you said) the positive integers are infinite, but when you think of it another way they are only "semi-infinite"(bounded).
This other way of thinking is Topology, where being bounded is an important property. If something is bounded from both ends and it is closed (if you know what closed means) it is called compact. Being compact is probably one of the most important topological properties.
There’s another interesting point regarding positive integers and infinity. It concerns what happens when you pair each positive integer with its square, i.e. 1 with 1, 2 with 4, 3 with 9, etc. The original set will be a complete set of positive integers, but the set of squares is not, yet both sets are exactly the same size! I believe this apparent paradox has been resolved, but I cannot remember by whom, and am not aware as to what the explanation is. Maybe somebody on here can help us.
It was resolved by Cantor at the turn of the century. Any infinite set of objects that can be put into correspondence with the positive integers is called countably infinite or aleph-null. (Believe it or not, the set of all fractions is countable)
Any set which cannot be put in correspondence with the positive integers is called uncountable. An example would be the real numbers.




Now with regard to the double slit experiment, let's call the slits hole A and hole B,. When you attempt to detect which hole the particle went through, you must set up experimental equipment which can measure what hole the particle went through. (This might sound like an obvious tautology but it's a very important point that I'll explain later.)
This will mean the equipment can return two results.
These results being A or B. Corresponding to each of these two measurements there is a quantum mechanical state/wavefunction. I'll call these wavefunctions |A> and |B>, being in these states means the particle is localized (or concentrated) at hole A or B.
Measurement of either A or B by the equipment will mean the particle is now in state |A> or state |B>.
(To sum up, if that sounded muddled, if I measure B (i.e. I detect the particle at B), that means the particle-wave is now concentrated at B and not spread out all over the place.)
So I let the experiment run and my equipment detects the particle at hole A, which localizes the particle to A (puts it in the state |A>). After the measurement the particle spreads out like a wave from A and hits the detector screen. However I get no interference pattern. Why?
Take a look at this picture, where interference is occurring:
For interference to occur the particle-wave had to come from the source, hit both holes and propagate outwards from each hole and then have the waves from both holes meet and interfere with each other.
The reason I got no interference pattern was because I localized my particle to A, which meant there was no wave coming from hole B and therefore nothing for my particle-wave from A to interfere with.
Now it doesn't matter what method of detection I use, because all methods will have {A,B} as their set of possible results and therefore localize the particle to A or B and prevent interference. Therefore the result holds for any experimental equipment which can measure what hole the particle went through regardless of the method it uses.

Replies to this message:
 Message 26 by dogrelata, posted 12-28-2006 6:19 AM Son Goku has replied

  
dogrelata
Member (Idle past 5312 days)
Posts: 201
From: Scotland
Joined: 08-04-2006


Message 26 of 41 (372510)
12-28-2006 6:19 AM
Reply to: Message 25 by Son Goku
12-27-2006 12:23 PM


Re: Infinity and QM
Son Goku writes:
Now it doesn't matter what method of detection I use, because all methods will have {A,B} as their set of possible results and therefore localize the particle to A or B and prevent interference. Therefore the result holds for any experimental equipment which can measure what hole the particle went through regardless of the method it uses.
That sound you just heard was my head exploding!
I can get my head around the Schrodinger’s cat thought experiment. As far as I understand it, there is no way of knowing whether the cat is alive or dead until the box is opened. Although I have seen this characterized in some places, as being the cat is simultaneously alive and dead, my standard worldview kicks in and dismisses this as purely hypothetical. A different analogy might be, a football match has taken place, but I am unaware of what the result is, so until I learn of the result, the outcome remains a matter of probability-based conjecture.
I think the problem for those who hold the ”standard’ worldview as regards the double slit experiment is more complex. On the face of it, the interference pattern observed when no attempt is made to identify which slit the particle passed through, can be seen as the outcome of the event. Using the football analogy again, when I’m told the result, it is A wins 45% of the time, B 25%, the draw 30%. But if I want to know what actually happened, I need to have observed the game take place.
However, I guess what you’re saying is that the interference pattern observed is not the outcome of the event, but the spectrum of possible outcomes, had we taken the trouble to observe it. Or am I still confused?

This message is a reply to:
 Message 25 by Son Goku, posted 12-27-2006 12:23 PM Son Goku has replied

Replies to this message:
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JavaMan
Member (Idle past 2319 days)
Posts: 475
From: York, England
Joined: 08-05-2005


Message 27 of 41 (372514)
12-28-2006 7:39 AM
Reply to: Message 22 by Phat
12-27-2006 8:30 AM


Re: Forever and ever and ever
One thing I never figured out is how human minds (which are finite by definition) were able to quantify an infinite definition.
I'm not sure what 'quantify[ing] an infinite definition' means, but the origin of our notion of infinity is fairly commonplace, I think.
We observe that we can take any number and add one to it, and add one to that number, and so on without end. That gives us our notion of an infinite set of things.
And we observe that space extends in all directions as far as we can see. If we imagine ourselves at the edge of space, pushed up against the boundary, as it were, we can imagine that we could stretch out just a little further... And that gives us our notion of infinite extension.
And, just as with numbers, we can imagine adding another second to the time, and another second, and another without end, giving us our notion of infinite duration.
Is there anything more to it than that, do you think?

'I can't even fit all my wife's clothes into a suitcase for travelling. So you want me to believe we're going to put all of the planets and stars and everything into a sandwich bag?' - q3psycho on the Big Bang

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Replies to this message:
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Son Goku
Inactive Member


Message 28 of 41 (372518)
12-28-2006 9:15 AM
Reply to: Message 26 by dogrelata
12-28-2006 6:19 AM


Re: Infinity and QM
I'll get back to your post tomorrow, stuck for time at the moment.
However I will give you something to chew on, the difference between regular everyday probability and quantum probability.
The example I'll choose is a ball/particle (ball for the classical case, particle for the quantum) being kicked from A to B along three different tunnels.
In classical probability there is a 33.3% chance of the ball travelling down either tunnel, resulting in a 100% chance in total for the ball to be found at B. Nothing weird there.
In fact it'll usually be n% chance for one hole and a m% and l% for the other two holes.
With m%+n%+l%=100%
At the quantum mechanical scale, probabilities come in after I've obtained complex number associated with each path. I won't go into the details but basically sometimes the rules of the mathematics mean I have to subtract probabilities.
An example would be:
I get 33.3% for path 1, 33.3% for path 2 and 66.6% for path three. However the maths says I have to subtract the last probability from the other two.
So I get a 33.3% + 33.3% - 66.6% = 0% chance of the particle arriving at point B. A result you never get at the classical scale. The more paths you have to a location never reduces your chance of getting there.
(To those who know the details of the mathematics, I know I'm skipping a lot, but I think this captures one of the major differences between standard probability and QM)

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2ice_baked_taters
Member (Idle past 5851 days)
Posts: 566
From: Boulder Junction WI.
Joined: 02-16-2006


Message 29 of 41 (372703)
12-29-2006 5:40 AM


To me the question of infinity is incorectly approached when considering a point of refference. Once a point of reffernce is accepted a destination can be reached. Boundaries can be defined. The question becomes infinity by what refference.

  
JavaMan
Member (Idle past 2319 days)
Posts: 475
From: York, England
Joined: 08-05-2005


Message 30 of 41 (373692)
01-02-2007 12:24 PM
Reply to: Message 28 by Son Goku
12-28-2006 9:15 AM


Re: Infinity and QM
An example would be:
I get 33.3% for path 1, 33.3% for path 2 and 66.6% for path three. However the maths says I have to subtract the last probability from the other two.
So I get a 33.3% + 33.3% - 66.6% = 0% chance of the particle arriving at point B. A result you never get at the classical scale. The more paths you have to a location never reduces your chance of getting there.
Could you explain this a little further? Classical probability is just a statistical thing: if path X has a probability of 0.2, all this means is that in 100 cases of the ball travelling from A to B, then the ball is likely to follow path X in 20 of them. What does the probability of 0.2 mean in the quantum case?

'I can't even fit all my wife's clothes into a suitcase for travelling. So you want me to believe we're going to put all of the planets and stars and everything into a sandwich bag?' - q3psycho on the Big Bang

This message is a reply to:
 Message 28 by Son Goku, posted 12-28-2006 9:15 AM Son Goku has not replied

  
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