Since it's "infinitely small," it's not there! Why's that? No actually it doesn't. The decimal system is inaccurate. At least, it's not as accurate as fractions. 0.3333.... is technically not exactly the same as 1/3. Since the decimal system is not as accurate, it does make a difference when you convert in some cases, as I've shown. Thank you! Not so, as it has already been shown. Well, 0.3333... isn't really equal to 1/3 because you can never put enough threes, but at the limit of infinity (which is a contradiction in terms) it would be exactly equal. Just like 0.9999... would be equal to 1. How do you figure? 0.3333.... is one third, not one quarter. And that's not the limit you approach. No, seeing as you're travelling by 1/4 the limit is not 1/3. Explain. We're talking math and physics aren't we? They both apply. I've used both math and physics to disprove Zeno's paradox, and even just pure logic. Then perhaps you could explain it? Reason is not flawed, just some people's implementation of it. As you can see, the flaw in the "0.9999...=1" thing was also pointed out by reason. Seeming flaws in reason can be pointed out by further reason. Since no-one is perfect, sometimes it takes someone else to come along and reason it out better. OK, if you have 0.3333.... and you keep trying to add threes, you'll never reach exactly 1/3. But at the limit of infinity it is equal. But since infinity has no limit, it will never be exactly equal, only almost. That's a flaw in the decimal numbering system. It really isn't a math "trick." It is a valid proof that illustrates the fact that if you could have an infinit number of nines after the decimal (at the limit of no limit) it would be equal to one. All this can be shown by reason. Does this help? Xeno, if you want a clearer version of that, read Mathematical Fallacies and Paradoxes. It'll state what he was trying to say in a clearer way, and there's lots of other stuff in there too. I read it yesterday. Well, most of it. I would say that you can't divide infinity by infinity because it's not a number, it's more of a process than anything. You could only divide by it if you used the result of the "process," but since it has no limit, the process never ends, so you'd never get to divide by it.
<b>Merlijn</b>: <i>I seriously doubt that SUM(i=0 -> infinity) [2^-i] = 2</i> Where did that come from? Seems to be irrelevant to the current argument. <i>Still: 0,999... <>1</i> It has been shown that it is, above. Do you have a disproof, or just a gut feeling? <i>...since the universe is of limited size and information is a physical entity, we can never fully write 1/3 as 0,333... </i> You just wrote it! It must be understood, of course, that the ellipsis "..." means the 3s go on forever. You're not the only person here who has failed to grasp that point. When I write 0.333..., I mean, literally, an infinite string of 3s after the decimal point. That is very different to the number 0.3333...3, where the "..." stands for a finite number of 3s. <i>So what it boils down to is: is 0,999.... element of Z (or: 0,999... =1) ?</i> Yes. 0.999... = 1, exactly. If you think otherwise, please provide a disproof. I said: <i>Remembering that r is less that 1, if we make n very big (infinite, in fact) then rn+1 is a very very small number (actually zero if n is infinity).</i> This was in the context of summing a geometric series. In reply, you say: <i>here you're wrong. let W stand for infinitely large. then: W* 1/W = n, whereas W*0 = 0.</i> My statement has nothing to do with transfinite arithmetic, so I don't see how your statement is relevant. <b>alpha</b>: <i>Since it's "infinitely small," it's not there!</i> You have to be careful of statements like that. An infinitessimal still exists. <i>No actually it doesn't. The decimal system is inaccurate. At least, it's not as accurate as fractions. 0.3333.... is technically not exactly the same as 1/3.</i> See my comments above concerning the ellipsis (...). That makes 0.333... precisely equal to 1/3. It is an accepted convention for writing infinite decimal expansions. <i>Since the decimal system is not as accurate, it does make a difference when you convert in some cases, as I've shown.</i> No. The decimal system is as accurate as any other system of writing real numbers. In any base there will be some numbers with infinite expansions. Write 1/2 in base 3 and you get an infinitely long expansion. Write it in base 10 and you get 0.5. Nothing is lost in the conversion from one base to another, if you do it properly.
I would say it is very relevant, since you posted Zeno's first "dichotomy" paradox of motion, and the Achilles' Paradox is essentially the same paradox, as stated in 137's link, which is comparing two bodies in motion (with no end to the motion) vice one body with reference to a fixed point (end of motion). Keep reading the Achilles' link I sent. It may take a week to "get", and this doesn't mean you're dumb. I'm no math whiz, but after first reading and rereading it a year or so ago, along with Rucker's book, I got little "eureka" moments of clarity that, unfortunately, fade with distance from the material. I have to go back when the mood strikes me and recoat those neurons from time to time. Gothard writes at the end of the article's first page in question, "The next installment of my article will show that Cantor's theory of the transfinite cardinal number c, the Aleph of the continuum, provides a genuine solution to Zeno's paradox." I can't verify this, and maybe the other matheaux here can comment, but it is the closest thing that I know of that can answer your quest(ion), as it satisfied mine some time ago. John Le Coq
JamesR Well... it is precisely what this is all about. since Zeno's paradox is about the equation D= 0,5*SUM(i=0 -> infinity) [2^-i] = 1 ? I just left out the scalair. This again is totally relevant! An infinite large amount of zero's still add up to zero. Thus, infinately small is larger than 0. In the x>0, the > will resemble a = but is will never be. AND HEEEE LOOK WHAT WE'VE GOT HERE: Since when are you on our side now? Please Register or Log in to view the hidden image! That is exactly my point! Strange enough, the very next thing you say is the direct opposite of the infinitessimal you brought up.... get your story straight Please Register or Log in to view the hidden image! STILL: you're wrong about the decimal system. the conversion of an n-base system to an m-base system is no problem. HOWEVER, bringing in the dot (or comma) notation is a totally different conversion. And its inherently inaccurate. Remember your infinitissimal! I think it is you time to disprove that. I think I have done my share of giving evidence for now.
Merlijn, I think I'm beginning to follow your argument but not entirely. Let me show you my thought processes to your post and you can show me where I'm wrong... ok, simple, understood yeah you can't do that, can't divide anything by zero still can't do that, can't divide something by zero. yes, can't do this either, can't divide infinity by infinity, get the fat E (error) yes, neither is correct, anything divided by zero gets the fat E (unless infinity is a special case, which doesn't seem rational) Ok, W/0<>0, I agree. It doesn't equal anything but the fat E to me. I would say W*0 = 0. So now what I need now is the logic that makes this disprove that 0.9999...<>1. Let me know if I got any of the above wrong. Thanks for taking the time to clarify some of this.
Alpha, Multiply it out. If something 1/4's its distance, and then 1/4's that and then 1/4's that , you get .3333.... Use a calculator O is your beginning, 1 is your end point 1 * 1/4 = 0.25 0.25 * 1/4 = 0.0625 0.0625 * 1/4 = 0.015625 0.15625 * 1/4 = 0.00390625 etc. keep doing this until your calculator can't divide anymore Now add up all your answers, this is the distance traveled 0.25 + 0.0625 + 0.015625 + 0.00390625 + etc. = .3333.... or some might say 1/3 This object will either be infinitely approaching 1/3 or make it to 1/3. In either case, it doesn't come close to the end point, which is 1; it'll only make it, no matter how much or how fast it moves, at best, about 1/3 of the way there. The point is, it doesn't matter whether or not 0.9999...=1, it doesn't solve Zeno's paradox when something moves in 1/4's rather than 1/2's. -Xenu
Thanks for the recommendation for the book Alpha, I appreciate it. Please Register or Log in to view the hidden image!
No problem. You might find it interesting. OK, my bad. Actually it does. The same argument I used before still applies. Eventually the minimum distance you can travel will be more than one quarter the distance (or whatever fraction you choose). So it doesn't matter how small the fraction is. Eventually you will have to break the limit. And guys, once again: 0.3333... isn't really equal to 1/3 because you can never put enough threes, but at the limit of infinity (which is a contradiction in terms) it would be exactly equal. Just like 0.9999... would be equal to 1.
nope So you cannot comprehend the difference between a mathematical infinite geometric series and the adjunct tool of the calculus of limits? The sum halvseys will always and forever be in the form of (n-1)/n. This cannot ever reach one, mathematically. Putting limits on what infinitesimal quantity, and below, is worth throwing away is a separate issue from the proposed mathematical paradox. A human mind developed a separate tool and artificially ASSIGNS a limit; without this artificially assigned limit, the series marches on in a spiral of smallness with no end. David Berlinkski in a Tour of Calculus [fantastic book by the way] has some good illustration of Zeno's paradox. To paraphrase, the solution of limits does not satisfy the Paradox, but it allows the unmanageable to be managed. Where Zeno's paradox intersects the real world, of course there is a limit where our separation between the wall and our self becomes negligible. A human inability to comprehend a never ending decrease, does not change the fact that it never ends. Why the inability for some folks? Because a decrease of distance between physical objects HAS a limit. It is counter-intuitive to our experience. BUT, in the playground of mathematics, there are formulas which do not have real world correspondents. Why do you think the Pythagorians romped in the world of mathematics as their mystical world and source of the infinite divine mystery. Is pi really 3.14? Have you found the end of pi? Given, this is a different order of mathematical usage, but the hunt goes on and on. We put a limit on pi for practical usage, but the same mathematical brainiacs who make a living studying the endless non-repetitive nature of pi's trailing decimal numbers would laugh you out of the room if you 'proved' by limits that pi has been proven to = 3.14. Alpha, you have been omega-ed. 137
I have some questions... it seems that our matemathics are not very good... Please Register or Log in to view the hidden image! Well... and if we do the same thing for 1/4? And 1/5? And 1/6? And 1/2485928460976? And 1/W???? Please Register or Log in to view the hidden image! Will it ever get to the same paradox?? I mean... if we divide a number, any number, "infinite" times, or multiply a number "infinite" times, will we get to a limit? Is there a limit?????:bugeye: :bugeye: Please Register or Log in to view the hidden image!
Truthseeker, You could use all of those fractions ( not entirely sure about the 1/W though) and they would go on forever. Just like 137 said right above the post of yours, a limit is something created to make infinite numbers into finite concepts, so that we can deal with them in mathematical equations. Limits are basically "close enoughs". So, no, there are no true limits, mathematically speaking. This however is pure mathematics which doesn't necessarily apply to the "real"world. Alpha keeps spouting about Planck lengths being the minimum possible length (which still doesn't apply to a mathematical problem, I might add), which may or may not be true, so maybe in "real" life there is a limit. I would personally say that infinities exist everywhere, but you can't rationally debate about them because rationality can't handle infinities too well, as shown by this whole debate over Zeno's paradox.
Xenu, Rationally, we are unable to discern about things without using limitations. See how we categorized and ordered all living beings of this planet. When you see them in their natural habitat, are they ordered? No. We created an order just to make it easier for us to observe and to understand. However, it limits our reality to our own little perspective. Besides that, mathematics are the bases of all our physics, chemistry, and even to buy something in the market we use mathematics. If mathematics don't aply to the "real" world, why do we use them to determine things in the "real" world!?!?:bugeye: If we can't explain infinity in a rational manner, this is an evidence that our rationalism limits itself and can't handle with the task of discovering what is(are) the basic(s) Truth(s) of the universe.
Do you realize how many tortroises the Ancient Greeks went through before the gave up on the half then half then half (or hthth) premise? You fire an arrow at a tortoise. The arrow not only has to cover half the remaining distance for eternity but also has to make up the additional distance the tortoise has moved.
Rationally, we are unable to discern about things without using limitations. All things have limitations which we are able to observe. See how we categorized and ordered all living beings of this planet. When you see them in their natural habitat, are they ordered? No. What about an ant colony ? There exists one of most ordered of natural habitats known to man. We created an order just to make it easier for us to observe and to understand. However, it limits our reality to our own little perspective. We do not observe the order, we observe the reality. Our perspective is that of reality, nothing more, nothing less. Besides that, mathematics are the bases of all our physics, chemistry, and even to buy something in the market we use mathematics. If mathematics don't aply to the "real" world, why do we use them to determine things in the "real" world!?!? Mathematics does apply to the real world. Our universe follows mathematics. Mathematics was derived from our knowledge of the universe and our surroundings. Mathematics is the one true universal language that all may speak and understand. If we can't explain infinity in a rational manner, this is an evidence that our rationalism limits itself and can't handle with the task of discovering what is(are) the basic(s) Truth(s) of the universe. The concept of infinity can be explained quite easily and in a rational manner. Perhaps it is your inability to understand the explanations that is causing you to question the concept.
From a basic mathematical stance: let d[p] be distance in meters from start point let d[y] be distance in meters from start point + 1 meter let t be time in elapsed seconds let x be d[y]-d[y] at t+ >plank time (t(d[y] - d[p])) / 2 = x No matter how many times you increase s, you never reach a point where x = 0 Why?
I agree with this. I see where you are going with this, but I don't entirely agree. Reason, Mathematics, etc. may not describe things completely, but they are one of the best tools we have in describing the universe. We didn't evolve it for no reason. I like to describe it as a model. Reason is a good model, but all models don't describe what they are describing completely. I believe in a balance between Reason and the things that we can't describe with reason. But by all means, Reason should not be eradicated. This is all my viewpoint. (Q), This is a direct perception standpoint. I will be starting a rather lengthy set of threads in the near future, and part of this will be showing how this isn't necessarily true. Is this so? I would like to be enlightened.
Merlijn: <i>SUM(i=0 -> infinity) [2^-i] = 2</i> I'm sorry. I agree with this. I didn't read it carefully enough. I didn't notice that i was the index you were using, and thought it must have been i = sqrt(-1). As written above, the statement is true. <i>An infinite large amount of zero's still add up to zero.</i> I agree. <i>Thus, infinately small is larger than 0.</i> Yes. I agree. <i>STILL: you're wrong about the decimal system. the conversion of an n-base system to an m-base system is no problem. HOWEVER, bringing in the dot (or comma) notation is a totally different conversion. And its inherently inaccurate. Remember your infinitissimal! I think it is you time to disprove that. I think I have done my share of giving evidence for now.</i> This is easily settled. I say that I can write an exact decimal expansion of any rational number. You presumably dispute that by saying decimal is "inherently inaccurate". So, all you need to do is to provide an example of a rational number for which I cannot write an exact decimal. Remember that I've already explained my use of the ellipsis, so for example, 0.333... is an exact decimal expansion of 1/3. So? Your counter-example?
Thanks James R, you just defeated yourself: Thus we can conclude: 0,999... < 1 0,999... / 3 = 0,333... AND 1 / 3 = 1/3 Therefor we can conclude 1/3 - 0,333... > 0 The overall conclusion is that Zeno's paradox still stands. This is not a real surprise is it? After thousands of years, suddenly a group of sciforums users solve the paradox, and even in a very simple manner. And now for Phase Two. Why don't we differntiate the functions first, and then re-examine the problem? You start Please Register or Log in to view the hidden image! I'm going to take a shower (one of my favorite things in life) PS... I was not very certain which notation to use.... Why are the math symbols not standard? my opteions were SUM (2^-i) = 2 (leaving out the index, I decided I did not want confusion about i.... the one you ran into) SUM(2^0 ... 2^- infinity) = 2 (here I found the - and the infinity a bit unclear, so I opted for: ) SUM(i=0 -> infinity) [2^-i]=2 (which I thought to be closest to the original mathematical notation) Sorry to have caused confusion.
Xenu, I agree with that. However we should try to find a better model... a model that transcends the limitations of reason.
Re: nope LOL! That's not what I'm saying at all! I can see you don't understand. Perhaps you should re-read the thread. Question, why do you keep using a comma for a decimal point? 0.3333.... is equal to 1/3 only at the limit. Since you could never reach the limit, you would never get it to equal 1/3. Sure, we can type a representation of it at the limit (with a bar or elipses), but then you're defining the limit of infinity which is a contradiction. And if you accept a limit of infinity, where it does equal 1/3, then at the same limit for 0.9999... it equals 1. (!) OK, lemme try this again, arguing from a math standpoint. If you travel towards your destination you have to travel half the distance. (Or whatever fraction, lets say half so we're not altering the parameters). Well, if you have to travel half the distance first, and you think of that as your new destination, then you must first travel half the distance to that halfway point, but then with that as your new destination, you must first travel half the distance to there, and so on, and so on.... Essentially, you can't move! Because any distance away from where you are now that you can state, (no matter how close) will always be further than half that distance!
