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Thread: Zeno's paradox help

  1. #1 Zeno's paradox help 
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    How many people understand Zeno's paradox of the tortoise? and can explain it?

    -thnx


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    The "paradox" arises because many people incorrectly assume that the sum of an infinite number of discrete values must yield an infinite result. The paradox is usually stated along the lines of:

    "Suppose a tortoise and a rabbit (many other things are often substituted for the rabbit here) are racing. If the rabbit gives the tortoise a slight head start, it should be impossible for the rabbit to ever catch the tortoise - even though the rabbit is faster. The is because before the rabbit could catch the tortoise, it would first have to travel half-way to the tortoise. Then it would have to travel half-way to the tortoise again. And again...an infinite number of times, since there is an infinite number of half-distances between any two points."

    Many people will assume that since you have to travel through an infinite number of half-distances, it must take an infinite amount of time. Zeno's paradox was invented before calculus, which shows that you can sum an infinite set of numbers and get a non-infinite result. So even though you have to pass through an infinite number of discrete distances, it only takes a finite time to do it.


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    Quote Originally Posted by Scifor Refugee
    The "paradox" arises because many people incorrectly assume that the sum of an infinite number of discrete values must yield an infinite result. The paradox is usually stated along the lines of:

    "Suppose a tortoise and a rabbit (many other things are often substituted for the rabbit here) are racing. If the rabbit gives the tortoise a slight head start, it should be impossible for the rabbit to ever catch the tortoise - even though the rabbit is faster. The is because before the rabbit could catch the tortoise, it would first have to travel half-way to the tortoise. Then it would have to travel half-way to the tortoise again. And again...an infinite number of times, since there is an infinite number of half-distances between any two points."

    Many people will assume that since you have to travel through an infinite number of half-distances, it must take an infinite amount of time. Zeno's paradox was invented before calculus, which shows that you can sum an infinite set of numbers and get a non-infinite result. So even though you have to pass through an infinite number of discrete distances, it only takes a finite time to do it.
    As each step is halved, so is the time taken to travel it. If you want to make it an infinite number of steps then you must say each takes zero time to accomplish. If it takes 0 time to travel a step you can move an infinite number of steps in 0 time! so it can be argued either way, the rabbit could catch the tortoise in an infinite time or zero time. It's the old trick of trying to give infinity[which is a concept] an absolute value.
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    Actually, you can complete each step in a finite, non-zero time and still complete infinite steps in finite time. For example, if the first step took 1 unit of time, the second 1/2, the third 1/4, etc. then you would complete an infinite number of steps in 2 units of time.
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    Its not as simple as that though Magi.

    Imagine you are giving a coin, and at time t=1/2 you turn it from heads to tails, then time t = 1/4 you turn it from tails to heads etc. What way is the coin facing after 2 time units? You have completed the sequence of actions in finite time, but the time t=2 is causally disconnected from the previous times - which is quite confusing to say the least!
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    Quote Originally Posted by river_rat
    Its not as simple as that though Magi.

    Imagine you are giving a coin, and at time t=1/2 you turn it from heads to tails, then time t = 1/4 you turn it from tails to heads etc. What way is the coin facing after 2 time units? You have completed the sequence of actions in finite time, but the time t=2 is causally disconnected from the previous times - which is quite confusing to say the least!
    I think Magi was (correctly) pointing out that the limit of an infinite series where each element in the series has a non-zero value can give a finite value. Here you seem to be asking a question about the "last element" in an infinite series, which doesn't make sense. If the time needed for the coin to flip approaches zero as t approaches one, then at t=2 it would be rotating infinitely quickly.
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    Quote Originally Posted by river_rat
    Its not as simple as that though Magi.

    Imagine you are giving a coin, and at time t=1/2 you turn it from heads to tails, then time t = 1/4 you turn it from tails to heads etc. What way is the coin facing after 2 time units? You have completed the sequence of actions in finite time, but the time t=2 is causally disconnected from the previous times - which is quite confusing to say the least!
    Do you think that for this old slow brain you could demonstrate that in practice? - I'll supply the coin and if you succeed I'll let you keep it.... :wink:
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    Billco,

    If you care to object to the impossibility of the ever increasing speed of the coin toss, then you must also reject the entire paradox out of hand.

    The whole affair is mental masturbation.

    But it leads to some interesting abstractions.
    The severing of causality demonstrated by the coin toss is one of them.

    I suppose this sort of thing is expected to happen every time one deals with the infiinite?
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  10. #9  
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    Quote Originally Posted by invert_nexus
    Billco,

    If you care to object to the impossibility of the ever increasing speed of the coin toss, then you must also reject the entire paradox out of hand.

    The whole affair is mental masturbation.

    But it leads to some interesting abstractions.
    The severing of causality demonstrated by the coin toss is one of them.

    I suppose this sort of thing is expected to happen every time one deals with the infiinite?
    All I said was I'd like to see it done.

    I'd like to see a shuttle launch as well.
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    Quote Originally Posted by invert_nexus
    The severing of causality demonstrated by the coin toss is one of them.
    How does the coin toss scenario lead to a severing of causality?
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    Quote Originally Posted by Scifor Refugee
    The "paradox" arises because many people incorrectly assume that the sum of an infinite number of discrete values must yield an infinite result. The paradox is usually stated along the lines of:

    "Suppose a tortoise and a rabbit (many other things are often substituted for the rabbit here) are racing. If the rabbit gives the tortoise a slight head start, it should be impossible for the rabbit to ever catch the tortoise - even though the rabbit is faster. The is because before the rabbit could catch the tortoise, it would first have to travel half-way to the tortoise. Then it would have to travel half-way to the tortoise again. And again...an infinite number of times, since there is an infinite number of half-distances between any two points."

    Many people will assume that since you have to travel through an infinite number of half-distances, it must take an infinite amount of time. Zeno's paradox was invented before calculus, which shows that you can sum an infinite set of numbers and get a non-infinite result. So even though you have to pass through an infinite number of discrete distances, it only takes a finite time to do it.
    I have heard another versin of that paradox. It is something like this: "Suppose a tortoise and a rabbit are racing. The tortoise starts from point B and the rabbit starts from point A. When the rabbit has come to the poit B the tortoise has already gone to a poit C. When the rabbit comes to that point the tortoise is already at a poit D and so on.

    But again rabbit catches the tortoise because it is faster. After some time it will be infinitely close to the tortoise which is the same thing than being at the same point. After that the rabbit will pass the tortoise.

    Here is a littele mathematical explanation. r is rabbit and t is tortoise. First the distance between r and t is s1. r's speed is v1 and t's speed is v2. The time it takes for r to go to point B is t1=s1/v1. The next distance r has to travel is s2=t1v2=(s1v2)/v1. Time n is tn=(s1v2)/v1^n and the distance n is (s1v2^n)/v1^n. Those both are zero at the limit where n is infinite because v1>v2. The sum of all times between the pointse where r and t are is (s1v2)/(v1-1) and the distance which they have travelled when they meet is (s1v2)/(v1-v2).

    I'm not 100% sure but I think it's like that. The situation isn't defined after they meet if we define it like I defined above. It can be defined as limit when thy meet but not after the meetingpoint.
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    Scifor Refugee,

    How does the coin toss scenario lead to a severing of causality?
    Because at t=2 there is no method of determining deterministically whether the coin would be at the state of heads or tails.

    Yes. I know you said earlier that it doesn't make sense to ask this question as the coin would actually be spinning infinitely fast. But, suppose that the coin has to stop spinning at time t=2. That this were a part of the experiment's guidelines.
    Then, there would be no method of determining whether it would be heads or tails at time t=2.
    Causality is severed.
    Chaos ensues.
    Worlds collapse.
    Mathematicians feast on the blood of the living...


    Billco,

    All I said was I'd like to see it done.
    Ah.
    Well then.
    It would most certainly be a dazzling display, aye?
    One to write the grandchildren about.
    Especially the feasting on the blood part after the chaos ensuing part.
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    Quote Originally Posted by Scifor Refugee
    I think Magi was (correctly) pointing out that the limit of an infinite series where each element in the series has a non-zero value can give a finite value. Here you seem to be asking a question about the "last element" in an infinite series, which doesn't make sense. If the time needed for the coin to flip approaches zero as t approaches one, then at t=2 it would be rotating infinitely quickly.
    I am not asking about the last element - though after 2 seconds you would have finished the coin game so it is meaningful to ask what position the coin was in after the game was finished. At no point in this game does the coin rotate infinitely quickly.

    To rephrase the idea, imagine instead that you create a series of mirrors such that the distance between them is halved everytime. Then shoot a laser pulse into the apparatus, as this light pulse would have a finite transist time through the mirror maze you can ask which direction is it pointing (up, or down for example) once it exits - problem is you cannot say. Each one of these actions are called supertasks (Hilberts hotel is another example) - and no one is really sure if they are impossible or not. Normal motion is normally thought of as a supertask (which is zeno's paradox in its essence), but supertasks seem like they should be impossible....

    Every one of zeno's paradoxes was asking in effect what happens if we assume space and time is either continuous or discrete - and in each of the four cases you can come up with a paradox.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    Quote Originally Posted by invert_nexus
    Yes. I know you said earlier that it doesn't make sense to ask this question as the coin would actually be spinning infinitely fast. But, suppose that the coin has to stop spinning at time t=2. That this were a part of the experiment's guidelines.
    Then, there would be no method of determining whether it would be heads or tails at time t=2.
    This seems no different from asking whether the sum of every integer from 1 to infinity is even or odd, then claiming you have a paradox because the answer is “neither.” Evenness or oddness is simply not a property of the limit of the sum of every number from 1 to infinity. Similarly, a distinct heads or tails state is not a property of the coin at t=2.

    It seems to me that you are simply creating a paradox by demanding that two contrary things occur at the same time. You first set up a scenario where the coin will be rotating infinitely quickly at t=2 then demand to know whether it would be heads or tails at t=2, implicitly insisting that it has a distinct heads or tails property at t=2.

    It seems like asking "if a house was made of 100% wood but was also made partly of metal, how much metal does it have?" Either the house is 100% wood or it contains some amount of metal. Either the coin is rotating infinitly quickly at t=2 or the coin has a distinct heads/tails state at t=2. Either you have the sum of every integer from 1 to infinity or you have some other number that posesses an even/odd charictaristic. In any of those examples you can create a paradox by insisting that both are true, but it's only a paradox because you have presupposed something that is impossible and then pointed out that it is impossible.
    To rephrase the idea, imagine instead that you create a series of mirrors such that the distance between them is halved everytime. Then shoot a laser pulse into the apparatus, as this light pulse would have a finite transist time through the mirror maze you can ask which direction is it pointing (up, or down for example) once it exits - problem is you cannot say.
    Here you are simply confining the light to a motionless, dimensionless point and then asking what direction it is traveling in one of the dimensions that you have eliminated. Again, it seems that you are only creating a paradox by insisting that a dimension be eliminated and then insisting that the light must be traveling in that dimension. Based on the setup that you describe, at the limit of the system the light would not be traveling up or down because the up and down dimensions have been eliminated. It would only be a problem if there was some finite distance before the limit where you couldn’t define the direction of the light’s travel.
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    Ok.
    Seeing as how 1.999.... is equivalent to 2 in the real number system.
    Let's shake things up a bit.

    What about t=2.01?
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    Quote Originally Posted by invert_nexus
    Ok.
    Seeing as how 1.999.... is equivalent to 2 in the real number system.
    Let's shake things up a bit.

    What about t=2.01?
    Edit: I erased my first post here because I didn't read yours carefully enough.

    Asking about t=2.01 is no different from asking whether infinity + .01 is an even or odd.
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    I think you're thinking too much.
    I'm just moving past the state where the coin would be spinning infinitely fast.
    Because 1.99... is the same as 2 per definition by the real number system, then the coin must be spinning infinitely fast at t=2. Therefore, my supposition that the game should end so as to be able to read the coin at t=2 makes no sense.

    So.
    I moved the inspection of the coin to a moment after t=2. To t=2.01. Which is NOT the same as 1.999... and therefore is perfectly amenable to being in a state of non-infinite spin.

    So.
    Is there a way to determine whether the coin will read heads or tails at t=2.01?

    Note: This whole thing is physically impossible. It demonstrates an essential fact about the reals. That they aren't really all that real. Ironic.

    Anyway.
    Nope. I wasn't saying anything about infinite zeros before the 2 or anything else like that. I frankly have no idea where you got that from.


    Edit: An added thought. Much of this is semantic, perhaps. I mentioned a severing of causality as I had just read River Rat's post which mentioned causality and the term seemed apt at the time. But, perhaps it is misleading. More like determinacy is severed.

    It's true that the world we are living in is complex and thus determinacy is not as easily aquired as once thought by the Newtonians, but with full knowledge of the complexities involved, there is the possibility that determinacy could be regained.
    We don't know enough yet to determine this...

    But, we are able to determine the position of the coin right up until t=2 at which point the coin is spinning infinitely fast. After this, it is impossible to determine the state of the coin.

    Thus a severing.

    And the feasting on blood may commence.
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    This seems no different from asking whether the sum of every integer from 1 to infinity is even or odd, then claiming you have a paradox because the answer is “neither.” Evenness or oddness is simply not a property of the limit of the sum of every number from 1 to infinity. Similarly, a distinct heads or tails state is not a property of the coin at t=2.
    You cant compare summing up all the positive integers (which doesnt make any reasonable sense) to a definite state of a coin. Nothing is rotation here - the coin has two distinct states and is limited to those states. It is either heads or tails - no spinning or rotation required. If you dont like a coin then substitute an electron's spin, or a light bulb.

    It seems to me that you are simply creating a paradox by demanding that two contrary things occur at the same time. You first set up a scenario where the coin will be rotating infinitely quickly at t=2 then demand to know whether it would be heads or tails at t=2, implicitly insisting that it has a distinct heads or tails property at t=2.

    It seems like asking "if a house was made of 100% wood but was also made partly of metal, how much metal does it have?" Either the house is 100% wood or it contains some amount of metal. Either the coin is rotating infinitly quickly at t=2 or the coin has a distinct heads/tails state at t=2. Either you have the sum of every integer from 1 to infinity or you have some other number that posesses an even/odd charictaristic. In any of those examples you can create a paradox by insisting that both are true, but it's only a paradox because you have presupposed something that is impossible and then pointed out that it is impossible.
    That would be fine if the coin no longer existed after the experiment - and the fact that the time between the two states gets arbitrarily small (but never zero!) has no baring on the question. The coin still exists after 2 seconds - so questions about its orientation make sense. The distinct head or tails property does not magically disappear.

    Here you are simply confining the light to a motionless, dimensionless point and then asking what direction it is traveling in one of the dimensions that you have eliminated. Again, it seems that you are only creating a paradox by insisting that a dimension be eliminated and then insisting that the light must be traveling in that dimension. Based on the setup that you describe, at the limit of the system the light would not be traveling up or down because the up and down dimensions have been eliminated. It would only be a problem if there was some finite distance before the limit where you couldn’t define the direction of the light’s travel.
    You seem to be confusing the fact that the distance between the mirrors gets arbitrarily small with the idea that they become zero. If mirror has a nonzero distance between it and its neighbour - and thus the idea of up and down makes sense at every junction. No dimensions have been factored out - so the question still stands.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    Or perhaps I should say it's like asking if infinity minus infinity is even or odd.
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    Or perhaps I should say it's like asking if infinity minus infinity is even or odd.
    Or perhaps something like if continuum minus aleph nought is even or odd?
    Or maybe if continuum is equal to aleph one?

    Or maybe nothing of the sort...
    Heh.
    Just an odd thought.
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    Edit: I erased my first post here because I didn't read yours carefully enough.

    Asking about t=2.01 is no different from asking whether infinity + .01 is an even or odd.
    Heh.
    Ok.

    Anyway.
    Yeah. I agree pretty much agree.
    The thing is part of my original point.
    When infinity intrudes, things sorta get weird.

    But, this does come back to the whole continuum hypothesis, doesn't it?

    That's the thing about math, and especially when dealing with reals, we sorta fall into this abstract state of divorce from reality.
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    Quote Originally Posted by river_rat
    That would be fine if the coin no longer existed after the experiment - and the fact that the time between the two states gets arbitrarily small (but never zero!) has no baring on the question. The coin still exists after 2 seconds - so questions about its orientation make sense. The distinct head or tails property does not magically disappear.
    Yes, actually, if the time between states is constantly cut in half in the way that you describe then at t=2 the time between state changes will be exactly zero. If the time between state changes is "arbitrarily small" then you have not yet reached t=2.
    You seem to be confusing the fact that the distance between the mirrors gets arbitrarily small with the idea that they become zero. If mirror has a nonzero distance between it and its neighbour - and thus the idea of up and down makes sense at every junction. No dimensions have been factored out - so the question still stands.
    Does your mirror device require the light to bounce off an infinite number of mirrors before the light exits? If so, then the only way for light to take a finite amount of time to exit is if the mirrors get infinitly close together as the light progresses through the device. If the mirrors don't get closer together then the light will never exit the device - its path length will be infinitly long.

    If if the light is bouncing off a finite number of mirrors, then there is no problem.
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    Quote Originally Posted by invert_nexus
    That's the thing about math, and especially when dealing with reals, we sorta fall into this abstract state of divorce from reality.
    Yes, I see what you mean. Sorry I jumped on you.
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    Yes, actually, if the time between states is constantly cut in half in the way that you describe then at t=2 the time between state changes will be exactly zero. If the time between state changes is "arbitrarily small" then you have not yet reached t=2.
    Where is time t=2 ever mentioned in the setting up of the problem, it only comes in as the series limit (but every step occurs before that time). You dont ever get to say this step took 0 seconds to do, and time t=2 is not even a part of the time when the experiment is running! If it is, give me the step at which you do something there (and remember the steps are labelled by natural numbers).

    Does your mirror device require the light to bounce off an infinite number of mirrors before the light exits? If so, then the only way for light to take a finite amount of time to exit is if the mirrors get infinitly close together as the light progresses through the device. If the mirrors don't get closer together then the light will never exit the device - its path length will be infinitly long.

    If if the light is bouncing off a finite number of mirrors, then there is no problem.
    Where did i say that the mirrors dont get arbitrarily close? Thats the point, but arbitrarily close and having no seperation is not the same thing - the sequence 1/n gets arbitrarily close to 0 but is never zero! You cant point to two mirrors and say "look, those two have no seperation" for the same reasons explained above.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    If it is, give me the step at which you do something there (and remember the steps are labelled by natural numbers).
    I don't see how this can be set up as natural numbers. Yes. The times are labelled at t=1, t=2, etc..
    But the very premise of the paradox is that it is dividing the time between 1 and 2 seconds infinitely.
    Thus, t1=1, t2=1.5, t3=1.75, etc...
    All the way up to 1.9999...
    Which, per definition, is the same as 2.

    Therefore, t=2 is the time at which the coin is spinning infinitely fast.

    Zeno's Paradox is inextricably embedded within the real number system as far as I can tell. That's why it is such an interesting puzzle for today's introductory calculus lessons.
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    1.9999.... is not less then 2 i_v - so is not in the region you are looking at. You cant treat 1.999.... as being just smaller then 2 as you are doing. We are also not talking about a rate here, but about discrete states. But even if we where (and i used discrete states so i could avoid this problem) what you are doing is supposing that the rate is continuous at t=2 (which is not obvious at all!) and thus declaring it to be its limit. But the experiment has stopped at t=2 so you cant do that.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    1.9999.... is not less then 2
    ???
    Hmm?
    I thought I expressly said that 1.999...=2?
    Coulda swore I said that a bunch of times, in fact.

    You cant treat 1.999.... as being just smaller then 2 as you are doing.

    /.../

    But the experiment has stopped at t=2 so you cant do that.
    If the experiment has stopped at t=2 then it must also be stopped at t=1.99...? Yes?
    Then the experiment has never actually gone to completion?

    In which case, it seems to me that the state of the coin could be determined by.... Well. Now we fall into another conundrum. What is the time immediately prior to t=1.99...? And would it be considered equal to 1.99..?

    Damn these reals!

    As per the usual inculcation of the method of limits, we can determine the state of the coin at any time as far as we're willing to go into the problem.

    Physical reality limits the use of reals to a finite point, however.
    Planck length and Planck time are the lower boundaries at present. I'm unsure what might constitute an upper boundary.

    Roger Penrose, in his Emperor's New Mind (a few years out of date, true), claims a range of 10<sup>42</sup> to 10<sup>60</sup> which is pretty good as far as things go. But the Zeno paradox surpasses this range by a huge degree, yes? To an infinite degree?

    and i used discrete states so i could avoid this problem
    I don't get what you're talking about with discrete states, I'm afraid. Either we're dealing with a phenomenon which is of a coin flipping from head to tails at ever increasing speed or we're... not?
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    Hmm?
    I thought I expressly said that 1.999...=2?
    Coulda swore I said that a bunch of times, in fact.
    Yes you have said it a couple of times, but you continue to treat it as if it is somehow less then 2 (which you have done below again)

    If the experiment has stopped at t=2 then it must also be stopped at t=1.99...? Yes?
    Then the experiment has never actually gone to completion?

    In which case, it seems to me that the state of the coin could be determined by.... Well. Now we fall into another conundrum. What is the time immediately prior to t=1.99...? And would it be considered equal to 1.99..?
    Your first line is a tautology (it stopped at 2 = 2?) and then you use 1.99999.... as if it is smaller then 2 (or else your question about it coming to an end is meaningless). If you want the time prior to t=2 you're stuck as there isn't one (if time is continuous that is). Feel free to well order the interval (0, 2) if you want though

    As per the usual inculcation of the method of limits, we can determine the state of the coin at any time as far as we're willing to go into the problem.

    Physical reality limits the use of reals to a finite point, however.
    Planck length and Planck time are the lower boundaries at present. I'm unsure what might constitute an upper boundary.

    Roger Penrose, in his Emperor's New Mind (a few years out of date, true), claims a range of 10<sup>42</sup> to 10<sup>60</sup> which is pretty good as far as things go. But the Zeno paradox surpasses this range by a huge degree, yes? To an infinite degree?
    Now you have changed the playing field though (by making both space and time discrete, though there is a paradox for that as well)

    I don't get what you're talking about with discrete states, I'm afraid. Either we're dealing with a phenomenon which is of a coin flipping from head to tails at ever increasing speed or we're... not?
    Ok, close your eyes - flip the coin, look its heads. Then close your eyes, flip the coin and then look - its tails. The only states that are important are the heads and tails, the coin flips between those two states and thats all you are worried about. If you want imagine instead of a coin you have a light bulb and a switch, then all you are worried about is if the light is on or off. You are not worried about the continuum of states between heads or tails (on or off) and you can pretend they do not exist. Or assume the change of state occur immediately. But the change is not important, focus instead on the discrete state (i.e. head/tail, on/off, 0/1, red/blue etc)
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    Yes you have said it a couple of times, but you continue to treat it as if it is somehow less then 2 (which you have done below again)
    Semantics.
    You, of course, refer to my use of 'also'. But this doesn't mean that I don't accept that the two numbers are actually one number. (See. I just did it agin if you want to get technical. The problem being one of nomenclature. The real language system is not constructed to deal properly with the real number system.... And neither are truly adequate for reality. But they suffice if their respective limitations are kept in mind.)

    Your first line is a tautology (it stopped at 2 = 2?) and then you use 1.99999....
    Exactly.
    Which means that the coin is spinning infinitely fast at t=2.
    Because it is spinning infinitely fast at t=1.99...
    Which is saying exactly the same thing.

    Hmm.
    Rereading:
    "Where is time t=2 ever mentioned in the setting up of the problem, it only comes in as the series limit (but every step occurs before that time). You dont ever get to say this step took 0 seconds to do, and time t=2 is not even a part of the time when the experiment is running! If it is, give me the step at which you do something there (and remember the steps are labelled by natural numbers). "

    I can only assume that the way you are pushing aside the rate and the limit means that. like Achilles, t=2 can never be achieved.
    There will always be a discrete state following whatever state you might be observing at present.

    Heads.
    Tails.
    Heads.
    Tails.
    Heads.
    Tails.
    Heads.
    Tails.
    Heads.
    Tails.
    Ad infinitum.

    However.
    Physical reality intervenes and t=2 does occur no matter how long your rational thinking may be stuck on pondering the infinite states of heads and tails.

    And.
    What is the state of the coin at this time?

    The REAL answer would be exceedingly complex, I think.
    It would have to take account of the planck units as well as relativity.
    Hmmm.
    Actually.
    No. It would probably be more of an engineering problem.
    At some point the coin would be destroyed.

    Therefore, the answer to the question is that the coin would be at neither heads nor tails. IT would be obliterated due to physical force beyond it's capability of enduring.

    Muah.

    Now you have changed the playing field though (by making both space and time discrete, though there is a paradox for that as well)
    Well. This is reality as is presently observed.

    However, as I said before, these boundaries wouldn't actually be reached before the experiment went up in a puff of smoke.

    Ok, close your eyes - flip the coin, look its heads. Then close your eyes, flip the coin and then look - its tails. The only states that are important are the heads and tails, the coin flips between those two states and thats all you are worried about. If you want imagine instead of a coin you have a light bulb and a switch, then all you are worried about is if the light is on or off. You are not worried about the continuum of states between heads or tails (on or off) and you can pretend they do not exist. Or assume the change of state occur immediately. But the change is not important, focus instead on the discrete state (i.e. head/tail, on/off, 0/1, red/blue etc)
    The problem with this is that the discrete states are still limited to the time between 0 and 2 seconds. It is an integral part of the problem.
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    Semantics.
    You, of course, refer to my use of 'also'. But this doesn't mean that I don't accept that the two numbers are actually one number. (See. I just did it agin if you want to get technical. The problem being one of nomenclature. The real language system is not constructed to deal properly with the real number system.... And neither are truly adequate for reality. But they suffice if their respective limitations are kept in mind.)
    Its not semantics - its bad mathematics. Language deals with the reals quite well, that is what mathematics is all about. You seem to be refering to some hocus pocus of the real system, but the reals are constructed using normal language and are handled using normal language. Intuition may fail you, but language should not.

    Exactly.
    Which means that the coin is spinning infinitely fast at t=2.
    Because it is spinning infinitely fast at t=1.99...
    Which is saying exactly the same thing.
    It does not mean that, once again the time t=2 is never a part of the expirement and also the coin never "spins", it merely flips between discrete states.

    Hmm.
    Rereading:
    "Where is time t=2 ever mentioned in the setting up of the problem, it only comes in as the series limit (but every step occurs before that time). You dont ever get to say this step took 0 seconds to do, and time t=2 is not even a part of the time when the experiment is running! If it is, give me the step at which you do something there (and remember the steps are labelled by natural numbers). "

    I can only assume that the way you are pushing aside the rate and the limit means that. like Achilles, t=2 can never be achieved.
    There will always be a discrete state following whatever state you might be observing at present.
    No, i mean that by the time you reach t=2 you have finished the experiment. You arrive at time t = 2 without hassle, you just are no longer flipping any coins. Coins are only flipped at times t<sub>n</sub> = 2 - 2<sup>1 - n</sup> where n is a natural number, which means you flip coins arbitrarily close to the time t=2 but at t=2 you are finished.

    However.
    Physical reality intervenes and t=2 does occur no matter how long your rational thinking may be stuck on pondering the infinite states of heads and tails.

    And.
    What is the state of the coin at this time?

    The REAL answer would be exceedingly complex, I think.
    It would have to take account of the planck units as well as relativity.
    Hmmm.
    Actually.
    No. It would probably be more of an engineering problem.
    At some point the coin would be destroyed.

    Therefore, the answer to the question is that the coin would be at neither heads nor tails. IT would be obliterated due to physical force beyond it's capability of enduring.

    Muah.
    Cute, though this is a thought experiment so our coins are indestructable and our time is a continuum

    Well. This is reality as is presently observed.

    However, as I said before, these boundaries wouldn't actually be reached before the experiment went up in a puff of smoke.
    Be careful between stating "this is how our theories think reality is" to "this is what is presently observed". We have no idea what reality really is, and reifying the maths behind QFT and co is a dangerous philosophical game which drives you to "absurdities" like the multiverse or asking if an electron is a particle and a wave (its an electron, deal with it!)

    The problem with this is that the discrete states are still limited to the time between 0 and 2 seconds. It is an integral part of the problem.
    And? That is my point, and note that your experimeters actions are only defined for times t < 2. Thats why you cant say what happens at t=2. If you want, place the equipment in a black box, and open after 2 seconds - which way is the coin facing? Are supertasks possible?
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    Its not semantics - its bad mathematics. Language deals with the reals quite well, that is what mathematics is all about. You seem to be refering to some hocus pocus of the real system, but the reals are constructed using normal language and are handled using normal language. Intuition may fail you, but language should not.
    It is semantics.
    I.e. I don't have the proper semantic background in this area to use language in an effective manner to describe the strangeness of the reals.

    I'm not a math guy. Just dip into it from time to time for fun.

    No, i mean that by the time you reach t=2 you have finished the experiment. You arrive at time t = 2 without hassle, you just are no longer flipping any coins. Coins are only flipped at times tn = 2 - 2<sup>1 - n</sup> where n is a natural number, which means you flip coins arbitrarily close to the time t=2 but at t=2 you are finished.
    Sorta run headlong into Turing's Halting Problem here, yes?
    Put the Turing Machine in a box... and it never stops?
    t=2 is never reached.

    I.e. the problem is uncomputable.

    Cute, though this is a thought experiment so our coins are indestructable and our time is a continuum
    See?
    Divorce from reality.

    Be careful between stating "this is how our theories think reality is" to "this is what is presently observed". We have no idea what reality really is
    Theories are based on observations.
    I actually originally started to write something about how this is what present theories state, but preferred the statement on present observations better.
    Both imply the same thing, however.
    Neither deal with actual reality, but rather with our perception of it.
    So, thanks for the warning, but I already had it in mind...

    which drives you to "absurdities" like the multiverse
    A Copenhagen man, are you?
    Deutsch disagrees.

    or asking if an electron is a particle and a wave (its an electron, deal with it!)
    And bold as well.

    If you want, place the equipment in a black box, and open after 2 seconds - which way is the coin facing? Are supertasks possible?
    Not using present-day computation methods.
    It seems that a 'supertask' is not computable in polynomial time.
    Perhaps quantum computers might be up to task some day?

    I predict that when the box is opened, Pandora will no longer be the scapegoat for the sins of the world. Mathematicians will.
    And the feasting on the blood will commence.
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    River Rat: There seems to be a lot of confusion here, which is probably at least partly my fault. Let’s go back to your original post:

    Imagine you are giving a coin, and at time t=1/2 you turn it from heads to tails, then time t = 1/4 you turn it from tails to heads etc. What way is the coin facing after 2 time units? You have completed the sequence of actions in finite time, but the time t=2 is causally disconnected from the previous times - which is quite confusing to say the least!
    We can model this by adding an infinite sequence of ones. Let’s say that we start with 1, which is an odd number, so our number being odd means that the coin is “heads”. Now at t=1/2 we add another 1 to it, giving us 2. Tails will be denoted by even numbers. At t=1/4 we add 1 again, giving us 3; an odd number, so the coin is back to “heads”. If we keep adding up ones in this manner then the number will start to approach infinity as t approaches 1. If we stop adding at any point arbitrarily close to t=1, then we will have a non-infinite number that will be either even or odd. If we continue in adding until we reach t=1, then the result of our summation will be infinity, which is neither even nor odd. Similarly, if you have a coin instantaneously flipping between heads and tails and the time between flips is exactly zero, the coin no longer possesses a distinct heads or tails state, just as infinity no longer possesses a distinct even or odd state.

    You are creating a paradox by insisting that the coin must have a distinct heads or tails state, then also insisting that we follow this scenario where the time between flips goes to zero. You can’t have it both ways. If you insist that the coin must be either heads or tails, then it is impossible for us to carry out your experiment all the way to t=1, where the time between flips will be zero. If it is possible for the time between flips to equal zero, then it must also be true that it is possible for a coin to no longer have a distinct heads or tails state. Saying "A coin must be either heads or tail. Allow the time between flips to go to zero." Is like saying "The car is completely red. The car is completely black." Do you see what I mean?
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    It is semantics.
    I.e. I don't have the proper semantic background in this area to use language in an effective manner to describe the strangeness of the reals.

    I'm not a math guy. Just dip into it from time to time for fun.
    My apologise then but, as we have seen here, as soon as the language gets a bit fuzzy then people start talking cross channel.

    Sorta run headlong into Turing's Halting Problem here, yes?
    Put the Turing Machine in a box... and it never stops?
    t=2 is never reached.
    Nope, no halting problem in sight. We do have a supertask though, and if we allow time to be a continuum then this entire operation is even observable (throw yourself into a black hole for example and watch what happens to someone doing this experiment from the outside as you cross the event horizon is the normal guise i think)

    I.e. the problem is uncomputable.
    Paradoxical yes but only uncomputable in the sense that the coins "value" is not determined by having all the information of the previous 2 seconds.

    See?
    Divorce from reality.
    But that is what mathematics is, very divorced from reality. Its a miracle that any of it is even usable!

    Theories are based on observations.
    I actually originally started to write something about how this is what present theories state, but preferred the statement on present observations better.
    Both imply the same thing, however.
    Neither deal with actual reality, but rather with our perception of it.
    So, thanks for the warning, but I already had it in mind...
    Theories are based on observation, and as such are impotent to say anything about the reality you base them on. Perceptions start to drag you towards psychology, science is about predictions - how it obtains it predictions may be thought of as mathematical magic.

    A Copenhagen man, are you?
    Deutsch disagrees.

    And bold as well.
    I prefer quantum decoherence, and a positivist view towards the mathematics of QFT. For an analogy, you can do fourier analysis on the sound of a trumpet and break it down to its basic tones. But it is crazy to then extrapolate and say that you have an infinite number of "pure" trumpets and we experience the "sum" of these experiences. Physicists do not make good metephysicists as they tend to reify their maths too often for my comfort.


    Not using present-day computation methods.
    It seems that a 'supertask' is not computable in polynomial time.
    Perhaps quantum computers might be up to task some day?

    I predict that when the box is opened, Pandora will no longer be the scapegoat for the sins of the world. Mathematicians will.
    And the feasting on the blood will commence.
    Go read up on infinity machines - the philosophy of the idea is quite interesting. Anyway, these are machines that can preform supertasks (i think bifurcated supertasks to be precise) and thus can convert almost any problem on the naturals (like the goldbach conjecture, or twin prime conjecture) to one solvable in "polynomial time".
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    We can model this by adding an infinite sequence of ones. Let’s say that we start with 1, which is an odd number, so our number being odd means that the coin is “heads”. Now at t=1/2 we add another 1 to it, giving us 2. Tails will be denoted by even numbers. At t=1/4 we add 1 again, giving us 3; an odd number, so the coin is back to “heads”. If we keep adding up ones in this manner then the number will start to approach infinity as t approaches 1. If we stop adding at any point arbitrarily close to t=1, then we will have a non-infinite number that will be either even or odd. If we continue in adding until we reach t=1, then the result of our summation will be infinity, which is neither even nor odd. Similarly, if you have a coin instantaneously flipping between heads and tails and the time between flips is exactly zero, the coin no longer possesses a distinct heads or tails state, just as infinity no longer possesses a distinct even or odd state.
    I agree with that (it shows that knowing what happens for all time t < 2 tells you nothing about t = 2). Thats the entire point of the experiment. Think of it as a list of things (infinite of course) that the person conducting the experiment must do.
    1. Start with a coin in the heads position
    2. At time t = 1 turn it from heads to tails
    3. At time t = 3/2 turn it from tails to heads
    4. continue as is, flipping coins, halving the time interval
    5. at time t=2 (which we have not specified) stop!



    You are creating a paradox by insisting that the coin must have a distinct heads or tails state, then also insisting that we follow this scenario where the time between flips goes to zero. You can’t have it both ways. If you insist that the coin must be either heads or tails, then it is impossible for us to carry out your experiment all the way to t=1, where the time between flips will be zero. If it is possible for the time between flips to equal zero, then it must also be true that it is possible for a coin to no longer have a distinct heads or tails state. Saying "A coin must be either heads or tail. Allow the time between flips to go to zero." Is like saying "The car is completely red. The car is completely black." Do you see what I mean?
    I havent done that though, go through the list carefully. Every action is possible, and at no time is the time available to flip the coin zero (which is where your objection comes from - if something can be done instantaneously then your objection comes into play.) However, at no time does the experimenter have to flip the coin in zero time - at every point he as a finite, non-zero amount of time to flip the coin.
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  36. #35  
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    Quote Originally Posted by invert_nexus
    Ok.
    Seeing as how 1.999.... is equivalent to 2 in the real number system.
    Let's shake things up a bit.

    What about t=2.01?
    It's slowing down at that point as you have burnt the motor out...... :wink:
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    Quote Originally Posted by river_rat
    I havent done that though, go through the list carefully. Every action is possible, and at no time is the time available to flip the coin zero (which is where your objection comes from - if something can be done instantaneously then your objection comes into play.) However, at no time does the experimenter have to flip the coin in zero time - at every point he as a finite, non-zero amount of time to flip the coin.
    You are contradicting yourself. First you say that we aren't supposed to stop flipping the coin until t=2, then you say that "at no time is the time available to flip the coin zero." Which is it? Because the very moment at which t=2 is also the moment where the time between turns is exactly zero. If you don't want the time between flips to be zero then we must stop flipping the coin before t=2, not at t=2.

    Edit:Perhaps this analogy would be helpful.

    Suppose I have a car that begins at rest and accelerates at 1 m/s^2. If I need to stop the car before it reaches a speed of 10 m/s, then I must stop it at any time before (but arbitrarily close to) t=10 seconds. If I wait until exactly t=10 seconds to stop the car, I will have failed at my task because I will be stopping the car when it is traveling at exactly 10 m/s (rather than under 10 m/s). Similarly, if you don't want the time between flips of the coin to be zero when the person stops flipping the coin, then you must stop flipping it before t=2, because t=2 is the exact moment when the time between flip will be exactly zero – just as t=10 seconds is the exact moment when the car will be traveling at 10 m/s. The only difference between the two situations is that the car's speed increases linearly while the time between coin flips decreases exponentially.
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    You are contradicting yourself. First you say that we aren't supposed to stop flipping the coin until t=2, then you say that "at no time is the time available to flip the coin zero." Which is it? Because the very moment at which t=2 is also the moment where the time between turns is exactly zero. If you don't want the time between flips to be zero then we must stop flipping the coin before t=2, not at t=2.
    Look carefully Sci_For, you never get to flip a coin at time t=2 (so how can the time be zero?) Every time point has t < 2, by the construction of the paradox. You do stop flipping before time t=2, but you are always flipping coins arbitrarily close to t = 2 but never at t = 2. You want to flip a coin at time t=2, but that is not part of the game - all the flips happen at times t<sub>n</sub> = 2 - 2<sup>1-n</sup> < 2 forall n in Naturals!

    Suppose I have a car that begins at rest and accelerates at 1 m/s^2. If I need to stop the car before it reaches a speed of 10 m/s, then I must stop it at any time before (but arbitrarily close to) t=10 seconds. If I wait until exactly t=10 seconds to stop the car, I will have failed at my task because I will be stopping the car when it is traveling at exactly 10 m/s (rather than under 10 m/s). Similarly, if you don't want the time between flips of the coin to be zero when the person stops flipping the coin, then you must stop flipping it before t=2, because t=2 is the exact moment when the time between flip will be exactly zero – just as t=10 seconds is the exact moment when the car will be traveling at 10 m/s. The only difference between the two situations is that the car's speed increases linearly while the time between coin flips decreases exponentially.
    Now state that t < 10, is your velocity ever 10m/s? This is also a deceiving analogy for two reasons, for one we have a (perhaps false) intuition about the continuity of all motion and we are dealing with discrete actions here not a continuum of velocities. As another example, imagine achilles chasing the tortise - how do you show that at time t = 2 that archilles is actually at the same point that the hare is - just using what is given to you by the paradox?
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    Quote Originally Posted by river_rat

    You do stop flipping before time t=2, but you are always flipping coins arbitrarily close to t = 2 but never at t = 2.
    Oh, so I stop flipping the coin before t=2? Fine, tell me how soon before t=2 I should stop and I will be able to tell you whether the coin is heads or tails.

    There is no difference between “infinitely close to t=2” and “t=2”, just as 0.999… and 1 are exactly the same value.

    I'm not sure who know what "arbitrarily close" means. If I flip until t is "arbitrarily close" to t=2, then there is some non-zero difference between when I stop and t=2. You can use that difference to compute whether the coin will be heads or tails. If the time difference between when I stop and t=2 is zero (or "infinitly small" if you prefer), then I have reached the point where the time between flips is zero.
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    Oh, so I stop flipping the coin before t=2? Fine, tell me how soon before t=2 I should stop and I will be able to tell you whether the coin is heads or tails.
    Rather tell me at which step you are at time t = 2 if every step occurs at t<sub>n</sub> = 2 - 2<sup>1-n</sup>? No time step occurs at time t = 2, if it does please show me which one!

    There is no difference between “infinitely close to t=2” and “t=2”, just as 0.999… and 1 are exactly the same value.
    There is a difference though slight, firstly i am talking about arbitrarily close not infinitely close (which seems to imply limits). the interval (0, 1) has elements arbitrarily close to 0, but 0 is not in (0, 1). Every irrational has a rational arbitrarily close to it, but that does not mean every irrational is rational. The time t=2 has experiment time points arbitrarily close to it, but t=2 is never an experimental point. The experiment is over by the time the clock strikes t = 2.

    I'm not sure who know what "arbitrarily close" means. If I flip until t is "arbitrarily close" to t=2, then there is some non-zero difference between when I stop and t=2. You can use that difference to compute whether the coin will be heads or tails. If the time difference between when I stop and t=2 is zero (or "infinitly small" if you prefer), then I have reached the point where the time between flips is zero.
    A set has elements which are arbitrarily close to a point x means (at least for the reals due to them being first countable) that for every natural number n, there exists a point y in the set such that |x - y| < 1/n. You dont have a lower bound on the distances between though that is non-zero, but zero is not in the set of distances between t=2 and t<sub>n</sub>
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    Quote Originally Posted by river_rat

    There is a difference though slight, firstly i am talking about arbitrarily close not infinitely close (which seems to imply limits).
    This is not what you described. You described an experiment where the time between flips approaches zero as t approached 2, and said that the coin does not stop flipping until t=2.

    Answer this question, please: Is the difference between when you stop flipping the coin and t=2 zero, or is it non-zero? You seem to constantly switch back and forth, first saying that you stop flipping at t=2 and then claiming that you stop flipping before t=2. If the difference between when you stop flipping and t=2 is zero, then at t=2 the time between coin flips will also be zero. If the difference between when you stop flipping and t=2 is any finite, non-zero value then you can compute the state of the coin. Let me repeat that: for any arbitrary time before t=2, you can computer the state of the coin. So if you are merely getting arbitrarily close (as you now seem to be claiming), rather than infinitely close (as would occur in your original description of the experiment) then there is no problem because for any arbitrary value that is chosen the coin will have a definite state.

    And please, for the love of god don't tell me that the difference between t=2 and when you stop flipping is both non-zero and infinitely small.
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    Answer this question, please: Is the difference between when you stop flipping the coin and t=2 zero, or is it non-zero? You seem to constantly switch back and forth, first saying that you stop flipping at t=2 and then claiming that you stop flipping before t=2. If the difference between when you stop flipping and t=2 is zero, then at t=2 the time between coin flips will also be zero. If the difference between when you stop flipping and t=2 is any finite, non-zero value then you can compute the state of the coin.
    You keep on claiming that i do that, i have gone to great lengths to explain that at no time do you flip at time t=2. When you stop is anyones guess, but it is definitely before time t=2!

    You are demanding that the sup{ t<sub>n</sub> : n in naturals} (which happens to be two) be one of the members of the above mentioned set - that is not the case. The least upper bound is 2, so what? It means that you can get as close to two as you would like, but you never are at 2.

    Let me repeat that: for any arbitrary time before t=2, you can computer the state of the coin. So if you are merely getting arbitrarily close (as you now seem to be claiming), rather than infinitely close (as would occur in your original description of the experiment) then there is no problem because for any arbitrary value that is chosen the coin will have a definite state.
    Getting arbirtarily close has been what i have been claiming the whole time! How does t<sub>n</sub> = 2 - 2<sup>1-n</sup> not show that i am talking about getting arbitrarily close? You seem to claim that there is a natural for which the above equation is exactly two - please demonstrate that natural. You are playing a silly semantic game here now.

    And please, for the love of god don't tell me that the difference between t=2 and when you stop flipping is both non-zero and infinitely small.
    Where have I stated that? Im not actually sure you are reading and replying to my posts. Ive never once used the word infinitly small (yet you accuse me of jumping between ideas) and now this?

    I said it is bounded below by 0, but is never zero. If that frightens you then i just dont know, every irrational number must have the same effect on you as this properties holds for rational approx. to irrational numbers! They get arbitrarily close but are never zero!
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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