LauricAcid Posted September 4, 2005 Report Share Posted September 4, 2005 (edited) Depending on how you arrange the series, you get different answers. Then you should wonder whether the "ways you arrange" the series even make sense. Take [ 9 + {sum of 9/10^x, x from 1 to infinity} ] - [sum of 9/10^x, x from 1 to infinity] This equals 9, since it is of the form (9 + y) - y. (@first x values) 9 + 0.9 - 0.9 = 9 If you mean, for x = 1, and you're talking about the finite sums, then okay. (@second x values) 9 + 0.09 - 0.09 = 9 ad infinitum = 9 If you mean, for x = 2, then we have not your incorrect equation, but the correct equation: 9 + 9/10 + 9/100 - 9/10 - 9/100. But no matter, since (9 + y) - y = 9 still. But you arbitrarily took the 9 out of the series, and that has an effect on the answer. Who took 9 arbitrarily out the series? And what do you even mean? [sum of 9/10^x, x from 0 to infinity] - [sum of 9/10^x, x from 1 to infinity]. (@first x values) 9 - 0.9 = 8.1 For x = 0, there is no evaluation. For x = 1, this is 9 + 9/10 - 9/10, which is 9. (@second x values) 8.1 + 0.9 - 0.09 = 8.91 ad infinitum to 8.999...9991 = 8.999repeating For x = 2, this is 9 + 9/10 + 99/100 - 9/10 - 99/100, which is 9. [9.99 + {sum of 9/10^x, x from 3 to infinity} ] - [sum of 9/10^x, x from 1 to infinity]: (@first x values) 9.99 + 0.009 - 0.9 = 9.099 (@ second x values) 9.099 + 0.0009 - 0.09 = 9.0099 ad infinitum to 9.000...00099 For x = 1 and x = 2, there is no evaluation. For x = 3, this is: 9.99 + 9/1000 - 9/10 - 9/100 - 9/1000, which is 9. But the problem isn't a limit one, and none of those numbers equals the others. 1. To show that two infinite sequences converge to the same limit, we don't have to show or even care whether the sequences match terms for some n. That is, to show that infinite sequences f and g converge to the same point, we are not required to show that, for some n, f(n) = g(n). In fact, the two sequence may have ENTIRELY different ranges, so that it may be that not only do they not match for some n, but there is no n and no k such that f(n) = f(k). 2. Even though it is not relevant that the terms may not match, you did not even show any unmatching terms. 3. See my previous remarks about how you regard this problem. Moreover, not only are you claiming that there is another context, without limits, in which to address this matter, but now, despite that you've been given information that would disabuse you of your misconceptions, you are, ironically, inspired to claim something even stronger, and even more silly, which is that ANY evaluation of this problem in terms of limits is an incorrect one. How ridiculous. You don't even have a coherent theory to present, yet you arrogate that the existing mathematical theory, which dissolves this problem in an instant, is irrelevant. Who made you the determiner of what mathematics may and may not be brought in to correctly address a mathematical question? On a similar note, the limit of 0.333repeating = 1/3, but 0.333repeating doesn't equal 1/3, I believe. I too could have mistaken, though You don't need to worry. You just need to familiarize yourself with some more mathematics. Edited September 4, 2005 by LauricAcid Link to comment Share on other sites More sharing options...
LauricAcid Posted September 4, 2005 Report Share Posted September 4, 2005 (edited) I missed this: Okay the problem, as initially used would mean: 0.999repeating equals the sum of 9/10^x, where x goes from 1 to infinity. This question has been rephrased to mean: 0.999repeating equals the limit of the sum of 9/10^x, where x goes from 1 to infinity. Who said that the original problem was about a sum not a limit of a sequence of sums? Anyway, if someone said that, then they're confused. In the context of infinite decimal expansions, what do you even MEAN by an infinite sum if not the limit of a sequence of finite sums? In the context of infinite mathematical expansions, what is your mathematical definition of an infinite sum that is not a limit? Just because the infinite series uses infinity doesn't make it a limit. What justifies evaluating for a limit is convergence, which is proven for infinite decimal expansions. Now if you take infinite series, a second error occured regularly: x = 0.999repeating 10 * x = 10 * 0.999repeating 10x = 10 * 0.999repeating 10x - x = 10 * 0.999repeating - 0.999repeating 9x = 10 * 0.999repeating - 0.999repeating 9x = 9.99repeating - 0.999repeating 9.99repeating - 0.999repeating = 9 That's the error. The reasoning, I suppose is that 10 * 0.999 repeating - 0.999repeating equals [ 9 + {sum of 9/10^x, x from 1 to infinity} ] - [sum of 9/10^x, x from 1 to infinity]It doesn't, AFAIK.. As far as you know is not far enough. First, you don't know what you're saying in the context of infinite decimal expansions, when you talk about infinite sums that are not defined as limits. Second, there's no error, since the reasoning in the step of the argument you mentioned might as well be as simple as observing that 9.999... = 9 + .999..., so 9.999... - .999... = 9 + 9.999... - .999... = 9. / Self-correction to #95: The proof does not even rely upon there not being a smallest positive real number but only upon the fact that for any positive real number, there is a smaller positive rational number. Edited September 4, 2005 by LauricAcid Link to comment Share on other sites More sharing options...
source Posted September 4, 2005 Report Share Posted September 4, 2005 (edited) NOTE: Quote tags didn't work for some reason, so I used bold tags instead. It is for whomever makes an assertion to support that assertion. Rarely are things that simple. It is for whomever makes a positive assertion to support it. You made a claim that 2 = {{}, {{}}} has a practical application, I said it didn't. I think it more than clear who has the burden of proof. But of course, your whole post is everything but a response to mine. I have asked you a simple question, and all you can say about a concrete formula is to vaguely name the context it is used in. I'm a student of engineering and studying about the digital circuitry and how they are built, transistor by transistor, didn't require me to learn about 2 = {{}, {{}}}. In all my textbooks from mathematics to digital electronics, there is no such formula - and, mind you, I have studied mathematical logic AND set theory. The nearest that came to 2 = {{}, {{}}} was a set of empty sets. But it's another thing to assert as a unqualified generalization, as you did, that a particular formula does not have any practical value. You call my statement a sweeping generalization? I was referring to one concrete formula which I held, and still do, until you prove that I am wrong, to be a figment of your imagination. Your response in a footlong essay did not provide any proof to the contrary. How difficult is it to post a link to a page explaining it, if it is so important and so famous as you claim? You skipped responding to my illustration. You know very well that the implications when speaking of the practical value of a fabric are very different from when speaking of the practical value of a mathematical formula. It's not only that you can't make clothes out of a mathematical formula, in your "illustration" you are actually holding the fabric, so, unlike the formula we speak of, it CAN'T be a figment of anyone's imagination. And if I'm pointing to a formula on a chalkboard, and you say, "That formula has no practical value," then what is the basis of your assertion? None at all. I don't need a basis to not believe you. What is the basis of your assertion that it does? You seem to be lost about who has the burden of proof. When Cauchy invented his integral formula, he didn't ask around to have it proven to him that it had no practical value. In fact, had there not been proof of its validity, nobody would believe that this formula is so important. Your formula seems to be a definition of something. Therefore, a good example of what it's good for should prove that you are right and I am wrong. If you said, "That formula has no practical value," then a reasonable response is, "How do you know that?" No, the reasonable response is "Actually, it does." And then provide the proof. And to say that the formula might have practical value is not saying that the formula probably does, but only that it might. A god might exist. An invisible pink unicorn might exist. If these were my thoughts, then I would certainly say: "Your formula might have a practical value." As it is, it doesn't - not within the context of all my knowledge. No, you've committed a non sequitur. By saying that a certain thing might be true does not entail that one thinks that anything might be true. If I assert that the train might arrive late does not entail that I hold that two might equal four. And you are dropping context, as well as twisting my words. We weren't predicting future, we were talking about a mathematical formula. You claimed the proposition that the formula has no practical value. The burden of proof for that claim is on you for that proposition. If I claim that the formula does have practical value, then the burden of proof is on me for that proposition. And if I claim that the formula might have practical value, the burden, though it be far less, is also on me. 1. No, if I claim a negative, I don't have the burden of proof. Expecting me to provide proof for the negative assertion is like expecting me to provide you with proof of god's nonexistence. The burden of proof is on you, not me. 2. Yes, you do claim that the formula we speak of has practical value. 3. No, you don't claim that the formula might have a practical value, you claim it does have a practical value. We may not know of a practical use for a particular piece of fabric or formula on a chalkboard, but that does not entail that we might not ever know of a practical use for them, and especially does not entail that other people might not have a use for them. Practical examples of the fallacy of such thinking are all around us. I'll take my parents as an example. They are building a tennis court. I ask them why. They say it's because somehow, some day it may pay off. They build apartments at sea. I ask them why. They say it's because they are hoping to sell them. They borrow money to neighbors, they are building a house for my grandmother. I ask them again why do they do these things, as they make no sense? My dad owns a private company and is making applications for other companies. Yet they do all of the above, they are buying land, planting crops, raising fences. They say that these things are investments in the future, that somehow, some day these things might all pay off. They do these things even if the future of all these investments is uncertain. Not long ago, all of this cashed in. My parents almost went bankrupt. Now, the apartments they were building are left unfinished. Neighbors didn't return money. My old grandmother's house was torn down, but the new one wasn't built. The tennis court is almost ruined, but there is no money to renew it. What they did was that they overextended their investments into an uncertain future. Holding on to ideas only because they "might" work out is not good enough - not in business, not in mathematics, not in philosophy - not anywhere. On all decisions, you must reach certainty - something either is, or it is not. In a business venture, something either will be, or it won't be. Once you have reached this kind of certainty within the context of your own knowledge, only then can you make a decision - will you keep holding on to the idea, will you set it in motion, or will you abandon it. You don't deal with "mights" and "probabilities". This is true in science as well. Holding on to formulas that have no value is irresponsible if it is done for the sole reason that their application might one day be found. I could come up with a thousand useless definitions if I wanted to put my mind to it. If and when any defnition becomes necessary, it will be made, no sooner and no later. Even if it did exist previously, if someone happened to stumble upon it somehow, it didn't have any practical value. Imagine a Neanderthal discovering an oil field. What's it to him? Within the context of what he knows, the oil is valueless. Even more than that, it is a nuisance; junk. If you mean for me to show you a particular machine or piece of technology that uses that fiesign, then I don't know of such a thing, especially since I've never looked for such a thing. But I did give you an answer as to the overall usefulness of this formula as part of a mathematical theory and mathematical thinking that has fed, probably more than anything else has, into the invention of modern computing. Again, this is a vague answer. With this information, I can't even begin a search on google. Actually,b]k for John von Neumann and read through the thousands of pages about him in search for the formula, but why? I don't believe you to begin][...]Anyway, again, my initial point was not even in showing that the formula does have use, but rather to ask you how you had arrived at the conclusion that it does not. I believe I have answered that. Oh please, what a blus/b]e. Not only do I know what consistency is, but I can give you a rigorous mathematical form the principle, and discuss several important theorems about the relations among consistency, satisfiability, provability, cardinality, number systems, etc. Not that I'm such an expert on the subject, but I know enough about it to make your challenge foolish. Actually, I was asking from a purely philosophical standpoint. Nonetheless, it is nice when philosophy and natural sciences go hand in hand like this. Edited September 4, 2005 by source Link to comment Share on other sites More sharing options...
LauricAcid Posted September 4, 2005 Report Share Posted September 4, 2005 (edited) Before my responses, I note that you haven't responded to this point previously made: You claimed that useless formulas die out. But the formula we're talking about has not died out as it thrives approaching a second century of being ensconced as part of the standard representation of natural numbers. Therefore, by your own statement, this is evidence of the formula's usefulness. Quote tags didn't work for some reason There's a limit on the number of quote tags per post. If you go over limit, then none of the quote tags will format. You have to break your post into more than one post when you use of a certain amount of quote tags. It is for whomever makes a positive assertion to support it. Positive or negative, it is for whomever makes an assertion to support it. You made a claim that 2 = {{}, {{}}} has a practical application, I said it didn't. I think it more than clear who has the burden of proof. You're mixed up. FIRST, you made the claim that the formula does not have practical value. THEN, I asked what is your basis for making that claim. I have asked you a simple question Actually, first I asked you a simple question. I'm a student of engineering and studying about the digital circuitry and how they are built, transistor by transistor, didn't require me to learn about 2 = {{}, {{}}}. In all my textbooks from mathematics to digital electronics, there is no such formula - and, mind you, I have studied mathematical logic AND set theory. Then either you're not very observant or you have a poor memory. The von Neumann formulation of the natural numbers is in almost any textbook of set theory. The nearest that came to 2 = {{}, {{}}} was a set of empty sets. There is no set of empty sets, for plural 'sets'. If a set has only empty set(s), then since there is only one empty set, the set has one member, which is the empty set, singular. I was referring to one concrete formula Your generalization about the one formula is that it has NO practical value, which, unless you wish to qualify at this point, is to say that the formula has no practical value to anyone, anywhere, at anytime, and in any context. which I held, and still do, until you prove that I am wrong, to be a figment of your imagination. When you made the claim you had no idea what was in my mind about whether the formula has practical value, since I had not mentioned the formula in this regard. So you could not have rationally held when you first mentioned the formula that I had any particular view in mind about its practicality. Your response in a footlong essay did not provide any proof to the contrary. How difficult is it to post a link to a page explaining it, if it is so important and so famous as you claim? Oy vey, now you're really grasping for straws. No one asked for a link. That I didn't provide a link to supplant my explanation doesn't entail anything of any relevance at all here. If you're asking for a link now about the von Neumann representation of natural numbers, and in particular that 2 = {{} {{}}}, here is not an explanation, but a mention: "In the set-theoretical construction of the natural numbers, 2 is identified with the set {0,1}." [http://en.wikipedia.org/wiki/2_%28number%29] And since, just as the same website mentions, 1 is taken to be {0}, we have that 2 = {0 1} = {{} {{}}}. I haven't searched the Internet further, but if you would like to know more about the von Neumann construction, then I would recommend standard texts such as Herbert B. Enderton's Elements Of Set Theory (with online errata sheet at his website, which is needed especially in this context, since there's a significantly misleading typo in the part about the the representation of natural numbers) or, in paperback at a nice price, Patrick Suppes's Axiomatic Set Theory. Both have nice discussions about the very famous construction in which each natural number is the set of all previous natural numbers (just as in the example for 2). By the way, the Suppes book has the famous quote giving Dedekind's philosophical argument for the existence of infinite sets, which may infuriate your own sensibilities, but is interesting in comparison with the argument Capitalism Forever gave for the existence of the set of natural numbers. You know very well that the implications when speaking of the practical value of a fabric are very different from when speaking of the practical value of a mathematical formula. It's not only that you can't make clothes out of a mathematical formula, in your "illustration" you are actually holding the fabric, so, unlike the formula we speak of, it CAN'T be a figment of anyone's imagination. You're grasping at inessential aspects of the example. It could be small piece of fabric, not something you'd weave into an article of clothing. It could be a pebble. It could be a seemingly meaningless string of words. And you conflate the existence of the piece of fabric with the question of its usefulness. Neither the existence of the piece of fabric nor of the formula are in question. Whether the formula has use does not even bear upon whether the formula exists, as you yourself referred to it as a "concrete formula". [continued next post:] Edited September 4, 2005 by LauricAcid Link to comment Share on other sites More sharing options...
LauricAcid Posted September 4, 2005 Report Share Posted September 4, 2005 [LauricAcid:]And if I'm pointing to a formula on a chalkboard, and you say, "That formula has no practical value," then what is the basis of your assertion?" None at all. I don't need a basis to not believe you. What is the basis of your assertion that it does? Your making the same mistake again. In this example, no one asserted that the formula has value. In this example, no one asserted anything until you asserted that the formula does NOT have value. And, in this example, no one has asked you to believe ANYTHING. No one asked you to believe that the formula has value. No one even asked you not to be skeptical that the formula has value. No one even asked for you to conclusively demonstrate that the formula has value. All that is being asked is, "What makes you think that the formula does not have value?" And no one is even asking for you to give up your guess that the formula does not have value. Again, the question was just on what basis do you assert that the formula does not have value? When Cauchy invented his integral formula, he didn't ask around to have it proven to him that it had no practical value. And what does that have to do with the price of tea in China? I have no reason to think that von Neumann asked around to have it proven to him that his construction had no practical value. You're not making any kind of point here. In fact, had there not been proof of its validity, nobody would believe that this formula is so important. Now you've got jumble a bunch of concepts. First, a theorem is not proven valid, it is just proven. The proof is what is valid, not the theorem. (The exceptions, of course, are theorems that are themselves valid as they are proven from valid axioms. Mathematical theorems, such as whatever you have in mind in this example, are not of this kind.) Second, proof of a theorem does not itself make a theorem important to anyone. Third, perhaps you mean to express that there was some demonstration that the theorem has practical value. In that regard, I have never disagreed that it's probably safe to say that mathematics that has more immediate technological application tends to be mathematics that is more widely valued, especially among non-mathematicians. But that fact doesn't entail that mathematics that has less immediate technological application is not also valued as important. Your formula seems to be a definition of something. Therefore, a good example of what it's good for should prove that you are right and I am wrong. As I said, my point is not to prove that the formula has some immediate technological application. [LauricAcid:]If you said, "That formula has no practical value," then a reasonable response is, "How do you know that?" No, the reasonable response is "Actually, it does." And then provide the proof. What you mentioned is also a reasonable response. But it is not the reasonable response, since it is not the only reasonable one. A god might exist. That's an irrelevant example since 'god' is an undefined abstraction, while I take it that your own 'has a practical value' is not an undefined abstraction. [LauricAcid:]"No, you've committed a non sequitur. By saying that a certain thing might be true does not entail that one thinks that anything might be true. If I assert that the train might arrive late does not entail that I hold that two might equal four." And you are dropping context, as well as twisting my words. We weren't predicting future, we were talking about a mathematical formula. What context have I dropped? And I haven't twisted your words. We were talking about the claim that a certain mathematical formula has no use. You claimed that my asserting that something might be true leads to "anything goes" in the sense that anything might be true. And that my example above uses future tense has nothing to do with this. I'll give another example without future tense: If I assert that the train might have no dining car does not entail that I assert that two might equal four. Or past tense: If I assert that the train might have arrived late, does not entail that I hold that two might equal four. This is just to illustrate that you are incorrect to say that my asserting that something might be the case entails "anything goes". Expecting me to provide proof for the negative assertion is like expecting me to provide you with proof of god's nonexistence. 1. Again, with your existence of god example when I had already addressed that in a previous post. The existence of god is an undefined abstraction, so of course I would not ask you to prove that an undefined abstraction does not exist. But 'practical value', especially since it is your term, is taken here not to be an undefined abstraction. 2. You weren't asked to conclusively prove the non-existence of the practical value of the formula. Initially, you were just asked for some basis for your assertion. You can give any degree of inductive support you like. And one can give inductive support of a negative statement. If I claim that no oranges have pits, then each orange without a pit I show is inductive evidence that oranges don't have pits. (As to Hempel's paradox, I admit not having a ready-to-post argument about it.) 3. When you made the assertion that the formula has no practical value, I had no knowledge of the extent of your experience finding that the formula had no value. So I could not suppose any inductive findings of yours about this. Notice, this is different from your pink unicorns (put aside the invisible clause, for a moment). We do have inductive evidence that there are not pink unicorns (at least on this planet), since zoologists have scoured the planet looking for undiscovered species and have not found unicorns or even evidence of their past existence. And, now you even admit that you were not even familiar with the formula (while it is quite famous), so there's no reason to think that you had even asked yourself before whether it has practical value, let alone that you made a point to observe that there are no instances in which it does not. So, again, it was quite reasonable of me to ask you on what basis you assert that the formula has no practical value. [continued next post:] Link to comment Share on other sites More sharing options...
LauricAcid Posted September 4, 2005 Report Share Posted September 4, 2005 (edited) Yes, you do claim that the formula we speak of has practical value. I mentioned the connection between the formula and computing, yes. This was extra. My main point was not and still is not to convince you that the formula has practical value. My point was to ask the basis upon which you concluded that it does not. No, you don't claim that the formula might have a practical value, you claim it does have a practical value. You missed the point again, which is that one does not have to commit to claiming that something is the case in order to claim that something might be the case. I'll take my parents as an example. [...] What they did was that they overextended their investments into an uncertain future. Holding on to ideas only because they "might" work out is not good enough - not in business, not in mathematics, not in philosophy - not anywhere. On all decisions, you must reach certainty - something either is, or it is not. In a business venture, something either will be, or it won't be. Once you have reached this kind of certainty within the context of your own knowledge, only then can you make a decision - will you keep holding on to the idea, will you set it in motion, or will you abandon it. You don't deal with "mights" and "probabilities". This is true in science as well. I'm sorry to hear of your family's bad investments. But I don't see much relevance here. No one asked you to hold an idea because it might be true. I just asked on what basis you have concluded that it is true that a particular formula has no practical value. However, your notion of certainty is quite strange. And your notion that science does not deal with probabilities is strange too. And, the primary motivation of the formula is not to solve technological problems, but rather to provide a grounds for unifying abstractions that are the foundation for the intellectual reservoir for mathematics, including applied mathematics. Imagine a Neanderthal discovering an oil field. What's it to him? Within the context of what he knows, the oil is valueless. Even more than that, it is a nuisance; junk. Indeed. Again, this is a vague answer. With this information, I can't even begin a search on google. Actually, I could look for John von Neumann and read through the thousands of pages about him in search for the formula, but why? I don't believe you to begin with. Wow, you really are a true Googletopian. You're seriously grumbling because I can't ensure that the Internet will have quick and easy answers for you, as I have not provided the proper search terms in my post for you to follow-up with? For heaven's sake, I did not intend for my paragraph to provide you with more than a mention of some famous history. If you want to know more about this, then I haven't erred by failing to read your mind that you want links to exact places on the Internet where this famous history is discussed. Maybe the Internet doesn't have this information. That's not my fault. Maybe the Internet does have this information. Then, since you haven't even tried to find about von Neumann (despite that you mention there are thousands of pages about him), then I don't see why I should care in the least what you're motivated to find out about. Actually, I was asking from a purely philosophical standpoint. In the exact context of formal systems you asked if I know what consistency is. I don't know what you mean now by saying that your asking was from a philosophical standpoint. Nonetheless, it is nice when philosophy and natural sciences go hand in hand like this. I have no idea what you have in mind by that in this context. Edited September 4, 2005 by LauricAcid Link to comment Share on other sites More sharing options...
hunterrose Posted September 4, 2005 Report Share Posted September 4, 2005 Moreover, not only are you claiming that there is another context, without limits, in which to address this matter, but now, despite that you've been given information that would disabuse you of your misconceptions, you are, ironically, inspired to claim something even stronger, and even more silly, which is that ANY evaluation of this problem in terms of limits is an incorrect one. How ridiculous. You don't even have a coherent theory to present, yet you arrogate that the existing mathematical theory, which dissolves this problem in an instant, is irrelevant. Who made you the determiner of what mathematics may and may not be brought in to correctly address a mathematical question? ...You don't need to worry. You just need to familiarize yourself with some more mathematics. Whoa. Who stole your Rearden Metal? I don't know what I did or said to offend you, but that wasn't my intent. I'm sorry you misinterpreted me. I'm working on a more fleshed out rationale, but even without seeing the argument, I think you are exaggerating somewhat. Nonetheless, proof is in the pudding, so to work I go. You might find it interesting Link to comment Share on other sites More sharing options...
hunterrose Posted September 4, 2005 Report Share Posted September 4, 2005 Okay, my short answer is that 0.999repeating does equal 1, but it can't be independently proven to equal 1. First, (most?) solution problems: The answers derived (from 9.99repeating - 0.999repeating) were 9, 8.999repeating, and 9.000...00099. If you solely look at the '9' equation, you get your desired answer. On the other hand, since arranging the terms differently gives you answers that don't look equivalent, or at least haven't been proven to be equivalent, you can't necessarily say '9' is the answer without mathematically consolidating the divergent alternate answers. Okay, here I go. As my prior example showed (post # 102 ) "arranging" the series in different ways leads to apparently diverging answers. LauricAcid implied then that the ways I was arranging the series might be sensible, but ignored the fact that his answer involved an "arrangement" itself. Take two examples: the solution that led to an answer of "9", and the solution that led to an answer of "8.999repeating." We'll evaluate these two equations at "steps," where "steps" means that we take the first remaining digit off of the first operand, subtract the first digit from the second operand, and add that to our answer up to that point. For the "9" solution, we'll take out the "9" of the first operand so that the equation goes from the "simple" question of [9.99repeating - 0.999repeating] to the much simpler [9 + 0.999repeating - 0.999repeating]. "9" solution: x = 9 + 0.9 - 0.9 = 9 = 9 + 0.09 - 0.09 = 9 = 9 + 0.009 - 0.009 = 9 This will obviously give us our desired answer. On the other hand "8.999repeating" solution: x = 9 - 0.9 = 8.1 x = 8.1 + 0.9 - 0.09 = 8.91 x = 8.91 + 0.09 - 0.009 = 8.991 -> 8.999repeating. At best the answer merely "appear" different. However, if these two solutions are equal, that'd have to be proven. If you can't resolve the 8.999repeating, you can't resolve the 0.999repeating. On the other hand, if there is some satisfactory way of resolving that, feel free to contribute it. How do you prove the 8.999repeating answer is wrong? [sum of 9/10^x, x from 0 to infinity] - [sum of 9/10^x, x from 1 to infinity]. (@first x values) 9 - 0.9 = 8.1 For x = 0, there is no evaluation. For x = 1, this is 9 + 9/10 - 9/10, which is 9. LauricAcid's saying here that the proper way to evaluate the equation is not to evaluate the answer by the first digit of each operand like I did above, but by evaluating the answer at the same x for each equation, that is, line up the subtraction to subtract values at the same decimal place. Thus, x = 9 - 0.9 = 8.1 would be inaccurate, because I'm subtracting the value of the tenths place from the value of the ones place. His example relies on taking the 9 out of the first series, and then proceeding to operate on similar decimal places. While doing it this way is convenient, it may not be best, especially if doing it other ways leads to different answers. Depending on how you arrange the series, you get different answers. Then you should wonder whether the "ways you arrange" the series even make sense. Who took 9 arbitrarily out the series? And what do you even mean? Suppose we have two woodcutters who are cutting wood from their individual blocks, and putting their cutting into their individual baskets. The baskets are set on a scale such that the scale's value reads as the difference between the weight of the first woodcutter's basket and the second's. The first woodcutter A starts out with a block of 10 kilograms. The second woodcutter B starts out with a block of 1 kilogram. With each cut, a woodcutter cuts off 9/10 of the remaining block, and puts it in his basket. We'll assume that the woodcutters are cutting at the same consistent rate... but that we can allow one or both cutters to put a number of cuts into the basket before we evaluate them at synchronized cuts. If we just allow both cutters to put wood in the baskets without allowing any freebies, you get the 8.999repeating solution. If you allow A to put a cut in the basket, and then synchronize the woodcutters, you'll get your self-fulfilling solution of 9. You can have A put in a million cuts before synchronizing, the scale will always read a bit more than 9. You can have B put in a million cuts before synchronizing, the scale will always read a bit less than 9. Because you arranged the way you evaluate, it appears that the scale should read nine, which it does if you prearrange it. It's greatly simplifying to have the cutters putting an equal mass of wood into their baskets between each evaluation, but that doesn't mean it's the right way to go about. However you want to consider it, you have to work a little harder to get this clean and tidying answer of 9. If it's possible. Link to comment Share on other sites More sharing options...
hunterrose Posted September 4, 2005 Report Share Posted September 4, 2005 Second, there is another concept of what 0.999repeating can mean. On the other hand, you just asserted that .999... not= 1, and you haven't even give a proof; moreover, you haven't even given a definition of .999... that makes sense... The definition of 0.999repeating could essentially be what Bryan said in post #18: the quantity of one with an infinitesimally small part taken away. This quantity cannot be equal to one, as you are subtracting a positive non-zero value from one. If x and y are defined such that x > 0 and y = 1 - x then (for any x) y < 1 But real numbers are not infinite sequences, while they are limits of infinite sequences. ...if you are speaking outside the context of limits, then you haven't said in what sense an infinite sequence is a real number. All real numbers aren't necessarily limits of infinite sequences, if I'm right. From what I can offhand dig up, any number is a real number if it is between the values of negative infinity and positive infinity. E.g. 0.9 < 0.999repeating < 1.5 Let's pretend you have 1 cup of coffee. You take the smallest sip of it that you possibly can. You now have less coffee in the cup than you had before. We'll say that you now have .999~ cups of coffee. That there is no smallest possible sip is exactly why .999... = 1. The poster has it backwards: .999... exists all right, as it is the limit of a sequence and that limit is 1; and this relies upon, rather than contradicts, that there is no "smallest sip" or smallest real number to subtract from 1. Self-correction to #95: The proof does not even rely upon there not being a smallest positive real number but only upon the fact that for any positive real number, there is a smaller positive rational number. That this "smallest possible sip" isn't measurable or possible in any physical capacity doesn't eliminate the concept, any more than the concept of infinity could be eliminated by 'A is A.' If if they both have no physical (i.e. matter) implementations, they are still both valid concepts for our intents. The problem with your corrected proof: For any positive real number x, there is a smaller positive rational number y. By this I take it you mean there is a rational number z, such that 0.999repeating < z < 1 I have no problem with z < 1. However, that does not prove that for any positve real numbers x and y, there is a positve rational number z such that, x < z < y. Quite simply, any positive rational number < 1 is going to be less than or equal to 0.999repeating (as I'm defining it), but not greater than. Link to comment Share on other sites More sharing options...
hunterrose Posted September 4, 2005 Report Share Posted September 4, 2005 Finally! Last analogy: Suppose a scientist defined the phrase "red bird" to scientifically mean a cardinal (the bird.) Thus, any time you use the scientists' term "red bird," you'd be referring to a cardinal. Furthermore, suppose scientists defined any phrase involving a bird and its red color to mean a cardinal. You'd now have no way of referring to other red birds, e.g. a red hummingbird. Though the red hummingbird would be objectively considered as much a red bird as the cardinal, according to science, only the cardinal would be considered a "red bird." In fact, if terms involving red and bird were the only way a bird could be referred to, there would be no scientific phrase to refer to the red hummingbird! That's the situation this question falls into. There is no mathematical way to represent the number { 1 minus an infinitesimally small amount} even though this would be a real number. As terms such as 0.999repeating are defined as the limit, the answer doesn't even need any proof (all of which rely on using the unprovable definition) anyway. Now, there may be a reason this definition is mathematically necessary, but I haven't seen it. Does anyone know if 0.999repeating has to be defined as equalling one? 3. See my previous remarks about how you regard this problem. Moreover, not only are you claiming that there is another context, without limits, in which to address this matter, but now, despite that you've been given information that would disabuse you of your misconceptions, you are, ironically, inspired to claim something even stronger, and even more silly, which is that ANY evaluation of this problem in terms of limits is an incorrect one. How ridiculous. You don't even have a coherent theory to present, yet you arrogate that the existing mathematical theory, which dissolves this problem in an instant, is irrelevant. Who made you the determiner of what mathematics may and may not be brought in to correctly address a mathematical question? Such arrogance. There is another context from which to evaluate the term 0.999repeating. That'd be the definition of 1 minus an infinestimally small part. No limit necessary. Have I missed any misconceptions? I'm not sure I said any evaluation of the problem using limits is incorrect. My thoughts on this have been a work in progress, so it's possible, though. What I mean is that, in the way limits have been used for solutions, you get the desired answer because you define certain terms to be limits, whether they have an alternate interpretation or not. I've tried to present my theory coherently, if I haven't, my bad. This theory you say dissolves my point by arbitrarily defining a term to mean one of its possible meanings. In other words, you're right because the math Powers that Be have decided in your favor, not because your answer is verifiable by proof. You can bring in limits to the equation, but once you define 0.999repeating to equal one, or something similar, the "solution" is self-fulfilling: 0.999repeating equals one because we say so. The only way to prove that 0.999repeating is necessarily the limit of 0.999repeating, is to use calculations that involve the questioned limit. You have to rely on 0.999repeating being a limit to "prove" 0.999repeating is a limit. Self-fulfilling mathematics. First, you don't know what you're saying in the context of infinite decimal expansions, when you talk about infinite sums that are not defined as limits. That may have been true, but is there any reason why 0.999repeating has to be defined as one, especially considering there is no way to prove 0.999repeating equals one?? Or I could just be wrong Link to comment Share on other sites More sharing options...
source Posted September 4, 2005 Report Share Posted September 4, 2005 You claimed that useless formulas die out. But the formula we're talking about has not died out as it thrives approaching a second century of being ensconced as part of the standard representation of natural numbers. Therefore, by your own statement, this is evidence of the formula's usefulness. My exact words were: "However, the longevity of a system based on arbitrary axioms is determined by its practical use - if there is none, it won't survive. It will be dismissed as junk." This statement holds. And since, just as the same website mentions, 1 is taken to be {0}, we have that 2 = {0 1} = {{} {{}}}. To be frank, I have never seen this formulation of natural numbers. I did study set cardinality, so cardinality of set {0} is 1 and of {0, 1} is 2, but the actual formulation of numbers 1 = {{}} or 2 = {{}, {{}}} I have never seen. And no, none of my textbooks has it. However, I doubt that this particular notation has much meaning in design of digital computers. As I said, I've been studying engineering and have even designed several chips myself and I never needed this. Not even when I wrote assembly programs did I need it. ...your notion of certainty is quite strange.How so? You're seriously grumbling because I can't ensure that the Internet will have quick and easy answers for you, as I have not provided the proper search terms in my post for you to follow-up with? I don't know how else you prove a point you were making, unless you are willing to type it all here, which I don't doubt you would were the circumstances right. If I was asked for proof of something on these boards (note that I'm only speaking of proving things here in the forums), I would provide a link to a trusted site, rather than type it all myself. So, my comment was only meant to say that you hadn't even bothered to do that. I grant you that your goal isn't proving anything, however, I am curious as to how you would have someone prove to you a nagative assertion? If I was still convinced that 2 = {{}, {{}}} has no practical value, how do you suggest I would go about proving it doesn't? It is impossible - which is why only positive assertions need proof. That is the way I understand it. In the exact context of formal systems you asked if I know what consistency is. I don't know what you mean now by saying that your asking was from a philosophical standpoint. I think (one of) our misunderstanding is in that we are not speaking in the same context. If you remember, I replied to Capitalism Forever about defining 2={{}, {{}}}. I was speaking about formal systems, but not in the context of formal systems. My first post was only intended as an interesting side-note. Link to comment Share on other sites More sharing options...
hunterrose Posted September 4, 2005 Report Share Posted September 4, 2005 (edited) Wrap ups LauricAcid, did you mean if 0.999repeating and 1 are real numbers, that there is a real number z, such that 0.999repeating < z < 1? I'm learning as I go on this and came across the fact that there are an infinite number of real number in between any two real numbers. Obviously this would prevent my concept {1 minus an infinitesimally small amount} from existing separate from one. I'm assuming that there's likely some long chain of mathematics that depends on these definitions, so it's necessary that my number not exist for consistency's sake. Or something like that. That being said, I'm still finding it hard to come to grip with the whole thing Does this then mean that the infinitesimally small doesn't exist, even conceptually, wheras the infinitesimally large does exist conceptually? Could we say that the answer to the function of 1/x equals 0? Not just the limit, but the actual function? The red bird example assumes the "red bird" phrase were the only scientific way to refer to a bird that was red. I still stand by the fact that you can't prove that 0.999... equals its limit without relying on [all such numbers equal their limit.] That is, the "proofs" of the premise seem dependent on the premise being true I'm assuming, based on what I'm reading and LauricAcid's said, that [all such numbers equal their limit] is a property of some subset of numbers. Better late than never, eh? Learn something new everyday Edited September 4, 2005 by hunterrose Link to comment Share on other sites More sharing options...
LauricAcid Posted September 5, 2005 Report Share Posted September 5, 2005 (edited) hunterrose, You seem like a nice person, and I admire your verve and do-it-yourself spirit, as well as you seem pretty smart to me. And I'm not saying that to be patronizing. But you really do need to step back from your introverted notions of how you think math works. Basically, right now you don't know what the "f " you're talking about. (And from your last post, I glean that you're starting to realize that.) Do-it-yourself is fine to an extent, but you're beyond that extent now. By not informing yourself of mathematics, you're groping in the dark here and your ruminations have led to a morass. My ad hoc comments on your own ad hoc misformulations are only patches, while what's really needed is to stem the source of the your intellectual hemorrhaging, which is your lack of understanding of how mathematics, mathematical definitions, and mathematical theorem proving even work. If you have questions about my responses below, then I hope I'd have some time to answer. But for the sake of not spreading further confusion and misinformation, please, don't continue to declare about these things that you just don't know about. Notational convention: lim SUM(k = 1 to n) 9/(10^k) will be abbreviated as .9* 0.999repeating does equal 1, but it can't be independently proven to equal 1. That is meaningless, since 'independently proven' has no definition. Meanwhile, .9* = 1 is proven, as I posted a proof. you can't necessarily say '9' is the answer without mathematically consolidating the divergent alternate answers. One does not "consolidate divergent answers". If a theorem is proven, then there is nothing that can change that. Now, if you prove a theorem that is the negation of a previously proven theorem, then you've proven that the axioms are inconsistent. However, even though you seem to think you've proven a contradicting theorem (as you say, a "divergent alternative answer"), you're mistaken, and thus no inconsistency in the axioms has been shown. LauricAcid implied then that the ways I was arranging the series might be sensible, but ignored the fact that his answer involved an "arrangement" itself. I said that if your "arrangements" lead to contradictions, then you should ask yourself whether your "arrangements" are correct. And there's nothing special about the "arrangement" in the proof I posted. Actually, whatever you mean by "arrangement", I can't even address mathematically since it is not mathematically defined. You need to drop this "arrangement" business, since it's undefined and leading you to incorrect conclusions and, at best, nowhere. After you've studied a text on the relevant mathematics here, then you can go back to look at your "arrangement" idea to see what you were doing wrong and even to salvage whatever correct mathematical instincts you might have had working in it. If you can't resolve the 8.999repeating, you can't resolve the 0.999repeating. The proof that 8.999... = 9 is the same proof, mutatis mutandis, that .999 = 1. While doing it this way is convenient, it may not be best, especially if doing it other ways leads to different answers. How absurd. If you perform calculations incorrectly, then you'll get different answers, of course. This doesn't require us to "consolidate" the answer we arrived at by performing the calculation correctly with answers arrived at by performing calculations incorrectly! Suppose we have two woodcutters who are cutting wood Woodcutters cutting or woodchucks chucking, I have the feeling that you don't understand my previous remark that two functions that have ranges disjoint from each other may converge to the same limit. The definition of 0.999repeating could essentially be what Bryan said in post #18: the quantity of one with an infinitesimally small part taken away. In the real number system there ARE no "infinitesimally small parts to take away." Please, understand this: You don't do mathematics by just waving your arm and saying, "Tada! Infinitesimal small parts! Here they are! Get 'em while their hot!" If you want infinitesimals, then you can have them if you have axioms for them. Yes, you can wave your arm and say, "Axioms for infinitesimals! Here they are!" (but you can't say "Get 'em while they're hot!" ). But that requires that you do have AXIOMS, which is different from just blindly using these undefined things called 'infinitesimals' that behave however you want them to behave for any ad hoc argument whatsoever. It turns out that there are theories that have infinitesimals, especially non-standard analysis (which, actually we first arrive upon through models, not axioms, but that's another story) but we're not talking about those theories here. We're talking about the real number system. Sure, in some theory with infinitesimals, you can prove things other than you can in standard real analysis, but non-standard analysis or other theories with infinitesimals aren't on the table right now. No one disputes that .9* might not equal 1 in other theories. All I am saying, and all any mathematician would say, is that in (standard) real analysis, it is a theorem that .9* = 1. And unless anyone mentions otherwise, it is presupposed that such discussions are about standard real analysis. And, in this context, it would not even make sense to consider non-standard analysis, since you didn't even know, until right now, that it exists, let alone what it is, or how to go about stating axioms for it, or even what it means to state mathematical axioms. All real numbers aren't necessarily limits of infinite sequences, if I'm right. You're wrong. It is a theorem that every real number is the limit of some sequence of rational numbers. This is not the definition of a real number, but rather a consequence of the definition. (Actually, there are at least two common ways of defining 'real number', but that's another story.) From what I can offhand dig up, any number is a real number if it is between the values of negative infinity and positive infinity. That's a terrible definition. It's the kind they give in 10th grade algebra books. It's a definition just to keep your orientation until you study the actual mathematics. 1. There are no points that are negative infinity and positive infinity in the real number system. Even when mathematicians refer to these points, these are locutions for expressing other actually defined concepts such 'increasing without bound' and 'decreasing without bound'. 2. 'Between' refers to a relation, which in this case is a certain ordering. But the set upon which the ordering is defined comes before the definition of the ordering. So it is circular to define the set of reals in terms of an ordering that is then defined in terms of the set. 3. Other number systems, such as integers and such as rationals, are also ordered "between negative infinity and positive infinity" (accepting for a moment, those terms), so this definition does not characterize the reals as opposed to the integers or as opposed to the rationals. That this "smallest possible sip" isn't measurable or possible in any physical capacity doesn't eliminate the concept, any more than the concept of infinity could be eliminated by 'A is A.' If they both have no physical (i.e. matter) implementations, they are still both valid concepts for our intents. When we talk about the real number system, we're not talking about some Theory Of Every Possible "Valid" Concept. In the REAL NUMBER SYSTEM it is proven that there is no "smallest sip". You can't bring concepts into theories by just saying they are "valid concepts". If you want the concept of a smallest positive number expressed in your theory, then you have to state axioms for that theory (or show a model such that the theory is the set of sentences true in the model, but that's another story). [continued next post] Edited September 5, 2005 by LauricAcid Link to comment Share on other sites More sharing options...
LauricAcid Posted September 5, 2005 Report Share Posted September 5, 2005 (edited) By this I take it you mean there is a rational number z, such that 0.999repeating < z < 1 No! You've got it backwards. There is NOT a rational number between .9* and 1. prove that for any positve real numbers x and y, there is a positve rational number z such that, x < z < y. Yes, is proven, as you'll find in virtually any textbook on the subject. I stated a related theorem in the proof of .9* = 1. Quite simply, any positive rational number < 1 is going to be less than or equal to 0.999repeating (as I'm defining it), but not greater than. I don't know about your "definition", but it is true that any positive rational number < 1 is less than or equal to 0.999repeating. You know why? Because 1 IS .9*, and anything less than 1 is, a fortiori, less than or equal to 1. And, thus, not greater than 1. if terms involving red and bird were the only way a bird could be referred to, there would be no scientific phrase to refer to the red hummingbird! Hummingbird, shmuningbird, you need to learn about the theory of mathematical definitions. You have a countable number of predicate symbols and a countable number of function symbols (you don't even need the function symbols) to make up definitions to your heart's content. There is no mathematical way to represent the number { 1 minus an infinitesimally small amount} even though this would be a real number. No, it wouldn't be a real number. PROVABLY, it could NOT be a real number. If it were a real number, then it would violate the very properties of real numbers. Just because you can imagine the concept does not entail that the mathematical theory provides for your imaginings. If you want infinitesimals, then you need another theory in which to have them. Anyway, in set theory, you can define the property of being infinitesimal with respect to the reals, but just defining a property does not prove that there is anything that HAS that property. So, you could say, "x is infinitesimal iff x > 0 and for all p, if p is a positive real number, then x < p." But just stating that definition does not prove that there exists an x such that x is infinitesimal. As terms such as 0.999repeating are defined as the limit, the answer doesn't even need any proof (all of which rely on using the unprovable definition) anyway. Oy vey, you're really running me ragged. 1. That .9* = 1 DOES require a proof. I posted a proof. The definition (perhaps better called 'a notational convention') is of infinite decimal expansions. Given the definition for infinite decimal expansions, we still need to prove that the infinite decimal expansion .9* is 1. 2. Definitions are not proven. They wouldn't be definitions if they were proven. So it makes no sense even to say that one has "relied on using an unprovable definition". Now, there may be a reason this definition is mathematically necessary, but I haven't seen it. Does anyone know if 0.999repeating has to be defined as equalling one? No definition is mathematically necessary. If you want to redefine the terms, then go ahead. But then you're talking about different things than virtually all other mathematicians are talking about. Look, if you don't like the standard definition for infinite decimal expansions, then don't use it. The proof I posted just means, IF we are talking about this definition, THEN the theorem is proven. But it happens that virtually all mathematicians are talking about that definition, so, in that context, the 'IF, THEN' is superfluous. Get it? And, again, you're conflating the definition of infinite decimal expansions in general with a RESULT (viz. that .9* = 1) of that definition. This theory you say dissolves my point by arbitrarily defining a term to mean one of its possible meanings. In other words, you're right because the math Powers that Be have decided in your favor, not because your answer is verifiable by proof. All mathematical definitions are arbitrary, in the sense that, ultimately, mathematical definitions are given not for concepts, but rather for symbols and notation. These symbols and notations do have both formal interpretations and also informal conceptual understanding. The natural language words we use, such as 'infinite', 'limit', 'real number' are convenient tags for what are actually formal predicate symbols. If these natural language tags did not fit the informal conceptual understanding we glean from the formulas, then we probably wouldn't use the particular natural language tags. And, ultimately, we don't have to use these natural language tags. They're just convenient, and are NOT part of the actual mathematics. In other words, the mathematics does not depend upon what you and I conceptualize as 'real number' or 'limit' or 'infinity' (though we will use these conceptualizations, per a particular theory, as long as the conceptualizations are "supported" by the particular theory). The mathematics does not allow us to use just any concept from our imagination. If a concept is not supported by a particular theory, then we have to create a different theory. If one cannot glean one's own conceptualizations in the formulas or in the formal interpretations of a theory, then one is free to set up one's own theory with axioms, primitive terms, and definitions. And, when speaking with other mathematicians, as long as they understood that your use of natural language tags such as 'real number' are in reference to your own definitions, then there'd be no harm. But for most purposes, mathematicians are happy with the way the available theories voice the mathematicians' conceptualizations. Yet mathematics is always looking for improved theories - ones that are more expressively economical and give even sharper expression of certain concepts and even voice new concepts. So there is never an embargo on new theories. But new theories are submitted to criteria, including that they can be formalized, that they are consistent (or show a track record of consistency), that they are expressively economical, etc. once you define 0.999repeating to equal one, or something similar, the "solution" is self-fulfilling: 0.999repeating equals one because we say so. No, we have to prove that .9* = 1. The only way to prove that 0.999repeating is necessarily the limit of 0.999repeating No, we don't prove that. You're still mixed up about this. By definition, .9* is notation for lim SUM(k = 1 to n) 9/(10^k). THEN we have to PROVE that .9* = 1. [continued next post] Edited September 5, 2005 by LauricAcid Link to comment Share on other sites More sharing options...
LauricAcid Posted September 5, 2005 Report Share Posted September 5, 2005 (edited) You have to rely on 0.999repeating being a limit to "prove" 0.999repeating is a limit. Self-fulfilling mathematics. In general, we have to prove that, for any sequence for a decimal expansion, the sequence converges (to a real number, of course). And this general result is proven (see just about any textbook in real analysis). But not only is the general case proven, in the particular case of 9/10, 99/100, 999/1000.., the proof of the convergence of the sequence is part of the proof that it converges to 1. We don't just show that it converges, but rather we show that it converges to 1, so that, a fortiori, it converges, so the limit exists. Here's the setup line in the proof: Show that, for all e > 0, there exists n such that, for all k > n, |f(k) - 1| < e. And the proof makes good on that. But instead of recognizing that, your instead falsely claim that the proof presupposes the existence of a limit not proven to exist. You're just not even close to being on the page here. As to 'self-fulfilling mathematics', it is true that one adopts axioms to prove the sentences one wants to be theorems. For example, if for some number system stated by its own axiom system, we want it to be a theorem that every number has an additive inverse, then we have to adopt axioms that prove that every number has an additive inverse. Either we adopt the existence of additive inverses as an axiom itself or we adopt other axioms that make the existence of additive inverses provable from our axioms. Another concept of axiomatics is that we adopt only axioms that are seen to be true in certain fundamental senses, such as that they cannot be rationally doubted or that our intellect receives them with immediate awareness of their truth. While we might like to adhere that, it does not seem that anyone has so far been able to find such axioms that will account for mathematics at least including real analysis, unless one accepts the axioms of set theory in this sense, while it seems that there are not many people who do expect that the set theory axioms can be accepted this way. This is to say that we can't have our cake and eat it too. Something's gotta give: either we have undoubtable axioms but insufficient theorems or we have sufficient theorems but doubtable axioms. Or, some knight on a white horse may come along to show us how to have both of the good opposites, but, as they say, don't hold your breath for it. (And there's the second incompleteness theorem, which would seem to be an unslayable dragon for any knight on a white horse anyway.) is there any reason why 0.999repeating has to be defined as one, especially considering there is no way to prove 0.999repeating equals one?? Again, this is so fundamentally mixed up about what proof is and what definitions are that the remedy can only be "Get thee to a logic book." And, again, .9* = 1 is proven. did you mean if 0.999repeating and 1 are real numbers, that there is a real number z, such that 0.999repeating < z < 1? No, the opposite! There is no real number between .9* and 1, since .9* IS 1 and there is no real number between 1 and 1. there are an infinite number of real number in between any two real numbers. Obviously this would prevent my concept {1 minus an infinitesimally small amount}from existing separate from one. Yes, there are an infinite (actually, an uncountably infinite) number of real numbers between any two real numbers. But that's not the crux of why .9* = 1. The crux is, in rather general terms, that it is a theorem that between any two real numbers there is a rational number. That basic idea, in a slightly different form, was used in this line of the proof: Let e > 0. Then there exists n such that, 1/(10^n) < e. Obviously this would prevent my concept {1 minus an infinitesimally small amount} from existing separate from one. Yes. Now, you're getting somewhere! I'm assuming that there's likely some long chain of mathematics that depends on these definitions, so it's necessary that my number not exist for consistency's sake. Or something like that. That being said, I'm still finding it hard to come to grip with the whole thing You don't have to be confused. If you approach the subject systematically, then there are junctures at which you will be confused, but as you keep at it and then go back to revisit the points at which you were confused, you'll find that things clear up. Eventually, you can see the whole sequence, from the start, with clarity. It's okay, informally, to call this a 'chain', but 'chain' has another special meaning in mathematics. So what we have is not really a chain, but rather a sequence that starts with the axioms of set theory (couched in first order predicate logic), or axioms of a set theory (since there are variations). The real numbers are developed in this theory. Meanwhile, the real number system can be stated as its own axiom system (the axioms for a complete ordered field) which can be thought of as subsumed by set theory. But even taking the reals as their own system, there are things that we want to express and prove about the reals that require that we step back into the larger theory, which is set theory. Yes, one way of showing that infinitesimals DON'T exist in the reals is to prove that assuming that infinitesimals exist in the reals leads to a contradiction (this a more to the point way of saying your 'for consistency's sake'). Meanwhile, we hope that any attempt to prove that infinitesimals DO exist in the reals will fail, since otherwise set theory would be proven inconsistent. (This is different from showing non-standard models such as give rise to non-standard analysis.) Does this then mean that the infinitesimally small doesn't exist, even conceptually, wheras the infinitesimally large does exist conceptually? Could we say that the answer to the function of 1/x equals 0? Not just the limit, but the actual function? 1. Whatever exists conceptually for you is between you and your ontological conscience. 2. Mathematical theories provide for existence only per proof from axioms. 3. There is no infinitesimally large real number. First, 'infinitesimally large' is a contradiction in terms. Second, even if you mean 'infinitely large', then there is no infinitely large real number in the sense of the standard ORDERING on the reals. There is no real number that is greater than all other real numbers. But, in set theory, there are infinite sets, and they have cardinalities, which are called 'infinite cardinals', as well as there are infinite ordinals. But neither the infinite cardinals nor the infinite ordinals are real numbers. A funny thing is that each real number (based on either the equivalence class method of construction or the cut method of construction) is itself an infinite set and therefore has an infinite cardinality. But this is not reckoned in the standard ordering on the reals. Also, in non-standard analysis there are objects that are larger in the ordering than all real numbers, and the reciprocal of such objects are infinitesimals. 4. The function 1/x never takes the value 0. 5. Technical qualification: Usually mathematicians say that division by 0 is undefined. But one can say, and may need to say depending on one's handling of the problem of undefined terms, that the result of division by 0 is not undefined, but rather has the value 0 (or whatever value is agreed upon). This is called for since formal languages do not allow function symbols (including constants such as '0') to be undefined, since the usual semantics of the language would not work otherwise. But note that set theory needs only two symbols (the membership symbol and the identity symbol) plus a quantifier, a logical connective, and a countable number of variables. (I think even the identity symbol can be dispensed.) Basically, this means that with just two primitives - identity and membership - virtually all of mathematics can be expressed. Virtually everything in mathematics can be defined in terms of just these two. That is, every math formula can be, in principle, expressed with just quantifiers, connectives, variables, and the identity and membership symbol. So even the symbol '0' can be eliminated, in which case, division by 0 is not even an issue since the primitive formulation will have built in some provision for the situation. [all such numbers equal their limit] is a property of some subset of numbers. No, numbers do not have limits. Functions have limits. What is a theorem is that if r is a real number, then there exists an infinite sequence f of rational numbers such that r is the limit of f. Moreover, if r is a real number, then there are a countably infinite number of infinite sequences of rational numbers such that r is the limit of them. Moreover, among those is an f that is of the form of an infinite decimal expansion (or, for that matter, an infinite expansion of any base, from 2 on up). And, conversely, if S is an infinite decimal expansion (or, for that matter, an infinite expansion of any base, from 2 on up), then there exists a real number that is the limit of S. Edited September 5, 2005 by LauricAcid Link to comment Share on other sites More sharing options...
hunterrose Posted September 5, 2005 Report Share Posted September 5, 2005 But for the sake of not spreading further confusion and misinformation, please, don't continue to declare about these things that you just don't know about. I don't know the declarations are wrong until they're proven wrong. However, I'll try to be more conciliatory. Okay, I think I've got answers to all my questions on the "0.999... = 1" topic except this one. Aren't the "one third" and "9x = 9" "proofs" similar to the "terrible [rationales] they give in 10th grade algebra to keep your orientation?" I accept all of the other answers, but these two proofs can't be sufficient, can they? 1/3 = 0.333... 3 * (1/3) = 3 * 0.333... 1 = 0.999... Doesn't this rely on the fact that 1/3 = 0.333...? I can't declare it but what answer do you give to someone who considers that as dubious as 0.999... = 1? "I'm the expert, shut up?" The "9x = 9" proof seems similar. 9.99... - 0.999... = 8.999... right? Now if I get 8.999... as an answer, that doesn't get me any further than I started out at, it seems. If you're saying I commited an error in getting 8.999... as an answer, what was the error? Don't misinterpret me: I understand the other proofs; I can't declare these two former proofs to be inaccurate, but they seem to be "orientation" proofs. Link to comment Share on other sites More sharing options...
source Posted September 5, 2005 Report Share Posted September 5, 2005 (edited) Doesn't this rely on the fact that 1/3 = 0.333...? I can't declare it but what answer do you give to someone who considers that as dubious as 0.999... = 1? "I'm the expert, shut up?" If you really don't think that 1/3 = 0.333~ then try dividing 1 by 3 on paper. If you really think that 1/3 != 0.333~, then you expect that in your calculation you might find that it is something else. Do you still not believe me? Keep dividing. Still don't believe me? Keep dividing. Ad infinitum. Eventually you'll give up and admit that indeed 1/3 = 0.333~ and subsequently that 1 = 0.999~. Don't misinterpret me: I understand the other proofs; I can't declare these two former proofs to be inaccurate, but they seem to be "orientation" proofs. If you have studied higher mathematics, you may find my proof very useful. It's in post #53. Edited September 5, 2005 by source Link to comment Share on other sites More sharing options...
Capitalism Forever Posted September 5, 2005 Report Share Posted September 5, 2005 LauricAcid, In order to keep my posts from growing geometrically, let me just focus (for now) on the most crucial point you've made: The codifications are means to arrive at number systems that are isomorphic with the way both you and I think about these numbers in their more everyday sense. [...] That is the point of set theoretical developments: to uphold the essential properties of the number systems while also unifying them so that we don't need a separate axiom system for each one while we can talk about functions and other comparisons among them - all within one theory. So what we have is: A system of ideas in our minds, which you called "the way both you and I think about these numbers in their more everyday sense" ; let me just refer to it as E. A formal system, F, that is isomorphic with E. E is based in reality, while F is based on axioms. The axioms are chosen so as to ensure that F is isomorphic with E. We prove theorems using F, since only F has the rigor required for proofs. We engineer our technology (and count our money, etc.) using E, as E is the system that is "plugged into" reality. This "two-system solution" has been working; as you have pointed out, it has enabled many technological innovations, including the very computer I use to tell you this. The reason it has been working is that F is isomorphic with E, and E is based in reality. I have been proposing to replace this isomorphy with an identity. Instead of having two systems, one in our mind and one on paper, one based in reality and one on a couple of seemingly arbitrary postulates--why not have just one reality-based system whose propositions can be expressed on paper as well as mentally grasped. My motivation for this is that, although the "two-system solution" has been wielded successfully for engineering purposes, it has been causing philosophical confusion. Given that F, the only one of the two systems that people ever get to see, has a foundation that appears (to a naive person) to be just a set of arbitrary postulates, naive people can easily be led to question the validity of mathematics and, by transitivity, reason. Since E, the only system connected to one's senses and practical decisions, remains unformalized, sense perception and action can easily become divorced from logic in people's minds. Since the isomorphy between E and F has not been explored and validated in a widely publicized way, many people are unaware of it, so the duality between the "theoretical" system (F) and the "practical" one (E) helps promote the deadly mind-body dichotomy. It engenders rationalism in the classroom and irrationalism outside it. From the facts that E exists and is isomorphic with a formal system, it follows that E itself is formalizable. A formalization of E would include all of F except for the "acrobatics" that F uses to achieve isomorphy with E. Instead of those, the formalization of E would contain a formalized expression of the corresponding relationships within E. For example, instead of 2 = {{},{{}}}, it would say something like: a != b -> |{a,b}| = 2 Addition of natural numbers would be defined in terms of a union of disjoint sets: A intersect B = {} -> |A union B| = |A| + |B| (The commutativity of addition would thus follow from the commutativity of set union.) And so on. Of course, these formalized relationships at the base of E would appear as axioms in the sense used with formal systems. You say: if S adds axioms to T, then S may be inconsistent even if T is consistent. This is true--unless we know that all our axioms are expressions of a fact of reality. A formalization of E would contain many axioms, but all the axioms would be consistent with reality, and that, together with the fact that reality is internally consistent, would guarantee E's internal consistency. A consistency with reality is THE goal when formalizing E's axioms; a system that is internally consistent but not consistent with reality would amount to a failure of this endeavor. Link to comment Share on other sites More sharing options...
Felix Posted September 5, 2005 Report Share Posted September 5, 2005 x = x Since '^1/2' is another way of saying square rooted... x = (x^2)^1/2 Split up 'x^2' into '-x' times '-x'. x = ((-x)(-x))^1/2 Separate the two numbers under the radical. x = ((-x)^1/2)((-x)^1/2) You now have '(-x)^1/2' times itself, or '(-x)^1/2' squared. x = ((-x)^1/2)^2 If something is square-rooted, and you square it, they cancel out. x = -x Divide both sides by 'x' 1 = -1 Try to find the error in this one. The first step is wrong. (x^2)^1/2 can be both x and -x. That's the trick of it. Link to comment Share on other sites More sharing options...
Felix Posted September 5, 2005 Report Share Posted September 5, 2005 The whole problem with this infinity-shit is that infinity doesn't exist. It is a mathematical invention. If you have an infinite amount of natural numbers: 1,2,3,4,... then you have an infinite number of square numbers: 1,4,9,16,... Both amounts of numbers are equal only if you take it to infinity. If you set a limit to the numbers you look at you see: If you look at the numbers ranging from 1 to 100 you have 100 natural numbers and 10 square numbers. This doesn't get better if you enlarge the range. But somehow once you look at infinity magically both groups are of the same magnitude. This is just plain stupid and it's the reason why I believe that infinity is an inconceivable concept. Therefore it cannot be reasonably used in mathematics. Cantor worked on infinities a lot and he ended up as a nutcase. There is a branch of mathematics that took out infinity completely and it could repeat all the mayor proofs even in the field of quantum mechanics. Link to comment Share on other sites More sharing options...
hunterrose Posted September 5, 2005 Report Share Posted September 5, 2005 If you really don't think that 1/3 = 0.333~ then try dividing 1 by 3 on paper. If you really think that 1/3 != 0.333~, then you expect that in your calculation you might find that it is something else. Do you still not believe me? Keep dividing. Still don't believe me? Keep dividing. Ad infinitum. Eventually you'll give up and admit that indeed 1/3 = 0.333~ and subsequently that 1 = 0.999~. Well, it's not that I don't believe it; it's just that I don't think you can prove 0.999... = 1 by such methods. It seems reasonable, but still seems to depend on accepting another infinite expression. Inevitably, using these methods, you have come to a point that you say it's "obvious", but not proven, that 0.999... = 1, 8.999... = 9, or 0.333... = 1/3. That's not to say your proof on #53 is the same. I do acknowledge and agree with all the other proofs, particularly the ones based on the principles behind (real) numbers. Thanks to you, LauricAcid, and many others that have shown them Link to comment Share on other sites More sharing options...
source Posted September 5, 2005 Report Share Posted September 5, 2005 The whole problem with this infinity-shit is that infinity doesn't exist. Your choice of words doesn't add up to your argument, Felix. It is a mathematical invention.Do you perhaps have some basis for this claim? I submit that the term existed before it was imported into the field of mathematics. If you have an infinite amount of natural numbers: then you have an infinite number of square numbers: There is no "infinite amount" of anything. Your problem is that you are trying to grasp the concept of infinity as something finite; something that ends. Infinite is that which has no end! But somehow once you look at infinity magically both groups are of the same magnitude.Again your notion of infinity being finite. There is no "magnitude" of infinity, and you can't compare that which is infinite. There is no magic to it either. What doesn't end, doesn't end. This is just plain stupid and it's the reason why I believe that infinity is an inconceivable concept. You clearly don't know much about mathematical infinity... Therefore it cannot be reasonably used in mathematics....hence this statement. However, there is one curiosity which I would like to know about before saying more: what do you mean by "reasonably used" here? Cantor worked on infinities a lot and he ended up as a nutcase. If you are suggesting that it was his work that led him to depression, perhaps you have some proof to back this up? Link to comment Share on other sites More sharing options...
Hal Posted September 5, 2005 Report Share Posted September 5, 2005 (edited) Cantor's mental state wasnt particularly helped by the fact he was persecuted by people who found his work on the infinite to be unsettling due to their deeply entrenched philosophical prejudices (Kronecker for instance). Both amounts of numbers are equal only if you take it to infinity.Indeed, however the amount of real numbers and the amount of integers arent equal even if you 'take it to infinity'. But this has nothing to do with the discussion because Cantor's work on transfinite arithmetic is independent of infinity as used as a 'limiting concept' in calculus, and the theory of limits in general. Again your notion of infinity being finite. There is no "magnitude" of infinity, and you can't compare that which is infinite. There is no magic to it either. What doesn't end, doesn't end.I'm not sure what you mean. You can compare the cardinality of the set of real numbers and the set of rational numbers for instance, and conclude that the former is greater. You can write correct statements like "2^aleph_0 > aleph_0", and ask whether "2^aleph_0 = aleph_1" Edited September 5, 2005 by Hal Link to comment Share on other sites More sharing options...
source Posted September 5, 2005 Report Share Posted September 5, 2005 (edited) I'm not sure what you mean. You can compare the cardinality of the set of real numbers and the set of rational numbers for instance, and conclude that the former is greater. You can write correct statements like "2^aleph_0 > aleph_0", and ask whether "2^aleph_0 = aleph_1" Yes, but what you are comparing is not how big they are, but rather how dense they are. Both the set of real numbers and the set of rational numbers spread into infinity both ways. You can't ask how far do they go into infinity, because infinity means non-ending. There is no end to either rational or real numbers. What you can ask is how dense they are. It is because of density that the cardinality of real numbers (which is more dense) is greater. For set A to have greater density than B means that you can't form a bijection (transformation one-to-one and onto) from B to A. So a set with greater density has greater cardinality. For the two sets proposed by Felix, there IS a bijection from one to another. It is f(x) = x^2. So, this means same density and same cardinality, even though it may seem that the set containing the squares is not as dense as the set of natural numbers, for there are numbers in between. However, between 4 and 9, there are a finite number of natural numbers. Between any two closest squares, there are finite number of natural numbers. Finity is negligible in comparison (for lack of a better word) to infinity. So, both sets are equally dense and have the same cardinality, which is why they are "of the same magnitude" as Felix claims. When you are comparing the cardinality of two infinite sets, you are basically comparing their density, not their size, and when you say that one infinite set's cardinality is greater than the other's, you are saying that its density is greater, that there is no bijection from the second to the first. In effect, you are not comparing infinities. There is no mathematical theorem that will allow you to compare infinities. Edited September 5, 2005 by source Link to comment Share on other sites More sharing options...
LauricAcid Posted September 5, 2005 Report Share Posted September 5, 2005 1/3 = 0.333... 3 * (1/3) = 3 * 0.333... 1 = 0.999... Doesn't this rely on the fact that 1/3 = 0.333...?The "9x = 9" proof seems similar. Yes, that argument has a premise that 1/3 = .333... And if that premise had not been previously proven, then the argument does not give us basis to accept its conclusion. 9.99... - 0.999... = 8.999... right? Now if I get 8.999... as an answer, that doesn't get me any further than I started out at, it seems. If you're saying I commited an error in getting 8.999... as an answer, what was the error? I don't recall the specifics of your argument, but if your conclusion was that 9.99... - 0.999... = 8.999 then you arrived at a correct conclusion (whether your argument for arriving at it were correct or not, I don't recall). So perhaps your error is in thinking that that correct conclusion somehow contradicts that .999... = 1. It doesn't. Look, we can prove all of these: .999... = 1 9.999... = 10 8.999... = 9 And they don't contradict one another. "orientation" proofs. 'orientation proofs'. I like that phrase. Maybe 'orientation argument' is even better. Yes, even post #1 in this thread is an orientation argument. It's okay for what it is, but it relies on the shakey assumption that we can manipulate infinite strings in rows and columns like we manipulate finite strings in rows and columns. In this case, no harm is done, since the manipulations do end up working correctly. But since what's actually represented by these infinite strings are limits, to proof anything about them correctly, we really can't assume they'll work like finite strings. Meanwhile, the proof I posted is a solid mathematical, not just orientation, proof. If you really don't think that 1/3 = 0.333~ then try dividing 1 by 3 on paper. If you really think that 1/3 != 0.333~, then you expect that in your calculation you might find that it is something else. Do you still not believe me? Keep dividing. Still don't believe me? Keep dividing. Ad infinitum. Eventually you'll give up and admit that indeed 1/3 = 0.333~ and subsequently that 1 = 0.999~. That's an orientation argument. It's okay as far it goes. Link to comment Share on other sites More sharing options...
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