Infinity a > infinity b?
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8/17/2012 12:33:16 AM Posted: 4 years ago http://www.wimp.com...
This is a quick video on mathematics, and makes the claim that the infinity of decimals is larger than the infinity of whole numbers. While one can easily see why there would be more decimals than whole numbers, I can't understand how infinities can have different sizes. I always thought infinity wasn't a number, it was more of a concept of neverendingness, and it is nonsensical to say one neverending thing is bigger than another neverending thing. In the beginning of the video he explains that there are as many even numbers as total numbers... if the logic was consistent, wouldn't the infinity of even numbers be smaller than the infinity of total numbers? I don't know, mathematics has never been one of my strong points. Beliefs in a nutshell:  The Ends never justify the Means.  Objectivity is secondary to subjectivity.  The War on Drugs is the worst policy in the U.S.  Most people worship technology as a religion.  Computers will never become sentient. 
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8/17/2012 1:11:18 AM Posted: 4 years ago Consider the dimensions of said infinity. Let's say you have a 2 dimensional universe that is infinitely large. Because it's only 2d, it is only infinitely large length/widthwise. Now let's take an infinitely large 3 dimensional universe. It is infinitely large by length, width and depth. They are both technically infinitely large, but the 3d infinity is infinitely larger than the 2d infinity because of its larger scope. That is, each "unit" of the 2d infinite has it's own line of infinity spanning through the 3rd dimension. When viewed in this way it's easy to see how some infinities are larger than others. It comes down to context.
The difficulty found in trying to comprehend infinity is in not knowing how to dissect it. When you only acknowledge whole numbers, your infinity is 2 dimensional (1,2,3,4...). When you allow for decimal numbers, you've given each unit its own infinity, which can be looked at like this; 1.....2.....3.....etc 1.1/ 2.1/ 3.1/ etc 1.2/ 2.2/ 3.2/ etc 1.3/ 2.3/ 3.3/ etc etc/ etc/ etc/ etc ^ makeshift graph ftw When you picture it as all one straight line (1,1.1,1.2,...2,2.1,2.2,...3, etc) it doesn't seem to make any sense because you're dissecting the problem using the logic of conventional mathematics. When dealing with infinity, you must keep in mind that any "units" you make are only theoretical as any actual splitting of an infinity requires limiting said infinity, which makes it not infinite any longer. And because these units are only theoretical, you can branch additional infinities off of them by expanding into a higher dimension. If you're reading this, you're awesome and you should feel awesome. 
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8/17/2012 2:00:06 AM Posted: 4 years ago Yeah, there can be different cardinalities of infinity. The confusion, I think, is that one is this kind of weird limit that defines itself as outside/the threshold of counting, and the other one (e.g., aleph zero) is about how large a set is.

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8/17/2012 2:01:14 AM Posted: 4 years ago One way to explain it, which is probably wrong because I'm not a mathematician or a number theorist, is to take two guys who are running forever, one of whom has just been running longer. Even though both are running continuously, the analogy is that the cardinality of the longerrunning man is higher than the same of the shorterrunning man.

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8/17/2012 2:38:35 AM Posted: 4 years ago At 8/17/2012 12:33:16 AM, R0b1Billion wrote: Not a mathematician, but then again, I don't think anybody who will respond to you will be either. You'd do better go to a math or physics forum. I take it that there isn't any real intuitive way to understand this. Part of the problem stems from the very contradictory notion of infinite sets, as this implies some sort of bounded infinity, as if you got infinity, bottled it up and said "here is a bottle of infinity". Again, I'd say there is no one, intuitive way to understand this. Whatever way you'd like to picture it, or try to root it into a sort of intuitive digestible concept is totally up to you. Perhaps this is the creative aspect of mathematics. Other than that, you simply have to follow where the proof leads you. 
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8/17/2012 2:45:05 AM Posted: 4 years ago At 8/17/2012 1:11:18 AM, RyuuKyuzo wrote: I simplified this explanation here, but I think I'd better straighten something out, less a misunderstanding arise. When I said only using whole numbers is a 2d infinity, that is only true in the context of my explanation. Really, an infinite chain of whole numbers correlates to an infinitely long 1d line. Adding in numbers with 1 decimal creates an infinity large 2d square of numbers like the graph I made above. Adding in numbers with 2 values behind the decimal creates an infinitely large cube of numbers and if you add in numbers with three values behind the decimal you would have an infinitely large tesseract of numbers, which is beyond our ability to graph in a comprehensive way as our human brains can't comprehend objects taking up area in more more than three spatial dimensions. If you're reading this, you're awesome and you should feel awesome. 
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8/17/2012 3:07:01 AM Posted: 4 years ago At 8/17/2012 2:38:35 AM, tarkovsky wrote:At 8/17/2012 12:33:16 AM, R0b1Billion wrote: It's actually pretty intuitive when organized correctly  that is, until you get to the point where its complexity exceeds the comprehensive limitations of the human brain, but even at that point you will have the theoretical understand down and you can carry it from there. That's what's so great about theoretical physics. It gives us the ability to carry ideas past shortcomings of our biological hardware. If you're reading this, you're awesome and you should feel awesome. 
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8/17/2012 3:42:53 AM Posted: 4 years ago At 8/17/2012 12:33:16 AM, R0b1Billion wrote: The Fool: A wise man once told me that the infinite is a Fallacy. He was very wise indeed this man. He told me that it was an illusion of mind. It never becomes actual. Its really incomplete in formation. For example we could say that there is an infinite amount of points on a line. And yet say there is no particular point on a line. But they mean the same. We clearly understand what a whole is but yet we can keep breaking it down in to parts. E.g. Picture yourself walking in your mind. You are walking forward and you are following a never ending line. Keep walking and walking and walking. The line will never end. Keep walking. But the fact of the matter is that you MIND IS JUST REPLAYING THE SAME LOOP. OVER AND OVER AGAIN. But when you stop thinking, It stops. It is never infinite but rather always as finite as your thought about it. ILLUSION!! We could never really know infinite. It could never be true knowledge. And that which could never be true is by necessity FALSE!! Well at least this is what wise man one said to me. Its possible that this may have all been a dream. But still an intrestring dream I say. The Fool on the Hill. "The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL 
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8/17/2012 4:16:02 AM Posted: 4 years ago At 8/17/2012 2:01:14 AM, Cody_Franklin wrote: For the man to have been running longer, he would have to have run a finite distance. #UnbanTheMadman "Some will sell their dreams for small desires Or lose the race to rats Get caught in ticking traps And start to dream of somewhere To relax their restless flight Somewhere out of a memory of lighted streets on quiet nights..." ~ Rush 
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8/17/2012 4:17:15 AM Posted: 4 years ago An infinity can't have a value without not being infinite .
#UnbanTheMadman "Some will sell their dreams for small desires Or lose the race to rats Get caught in ticking traps And start to dream of somewhere To relax their restless flight Somewhere out of a memory of lighted streets on quiet nights..." ~ Rush 
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8/17/2012 11:49:29 AM Posted: 4 years ago At 8/17/2012 1:11:18 AM, RyuuKyuzo wrote: Excellently explained I see your point. I was using pretty much a onedimensional numberline for each but the different dimensions makes a lot of sense. Thanks! Beliefs in a nutshell:  The Ends never justify the Means.  Objectivity is secondary to subjectivity.  The War on Drugs is the worst policy in the U.S.  Most people worship technology as a religion.  Computers will never become sentient. 
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8/17/2012 1:50:12 PM Posted: 4 years ago At 8/17/2012 11:49:29 AM, R0b1Billion wrote:At 8/17/2012 1:11:18 AM, RyuuKyuzo wrote: Any time If you're reading this, you're awesome and you should feel awesome. 
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8/17/2012 6:24:41 PM Posted: 4 years ago At 8/17/2012 12:33:16 AM, R0b1Billion wrote: It's certainly counterintuitive to consider that there can be different sized infinities, as far as infinity as a number is concerned, it's typically considered to be an absolute number, its mathematically defined using set theory such that a collection is considered infinite if some of its parts are as large as the whole. So from one point of view, different infinities are considered equal, even when an infinity is a subset of another infinity, but it's also easy to see how the case can logically be made that the infinite set of whole numbers is twice as large as the infinite set of even numbers, but equal to the infinite set of odd numbers. Using relative logic like that, Georg Cantor used set theory and a diagonalization proof to convince the mathematics world to take the concept of different sized infinities seriously, and in doing so he provided a logical foundation for a mathematics of infinite quantities that is useful to mathematicians. In the end, it's probably best not to think too hard about it, Georg Cantor spent a good portion of his life doing so and it literally drove him insane, in the end he was placed in an insane asylum where he lived till he died. "It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater 
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8/17/2012 9:08:18 PM Posted: 4 years ago At 8/17/2012 4:17:15 AM, bossyburrito wrote: The Fool: But infinity is contradictory in this sense. If there is such thing as an infinity then any particular value in is 0. That is no value at all. Because one in infinity equal 0. In fact anything with in an infinite is 0. Therefore if there is infinite existence and we are 1 in and infinite, it follows by necessity that we don't exist. But on the other hand it is our idea of infinite. That is the notion itself to be any form of knowledge in the first place Depends us knowing, it. Thus either infinite is exist or we don't exist. Now as foolish as it may sound I will take the later as true. For that is the necessary conditions to be a fool in the first place. Or at least that is the word on the hill. Take it for what it is worth. "The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL 
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8/18/2012 8:50:50 AM Posted: 4 years ago At 8/17/2012 12:33:16 AM, R0b1Billion wrote: That's an accepted, noncontroversial, mathematical claim. While one can easily see why there would be more decimals than whole numbers, I can't understand how infinities can have different sizes. It was a shock to everybody when the concept was introduced. I always thought infinity wasn't a number, it was more of a concept of neverendingness, and it is nonsensical to say one neverending thing is bigger than another neverending thing. Imagine dropping a rock into pile X at the rate of one rock per second, forever. And pile Y gets two rocks per second, forever. Y is twice as big as X after one second, and after 100 seconds, and after a billion seconds, and forever, right? There is never a point at which Y suddenly shrinks back to become the same size as X. So, even though both piles go on building endlessly, Y is always twice as big as X.  The above is wrong, but I shared it anyway in order to get across the idea of different sized infinities. It's wrong because both piles are "countable," so they would, in the system used by mathematicians, have the same cardinality. A mathematician would say that X is the same size as Y. But, see below ... In the beginning of the video he explains that there are as many even numbers as total numbers... if the logic was consistent, wouldn't the infinity of even numbers be smaller than the infinity of total numbers? You'd think. But they've got a system that works for them. Negative numbers were thought to be nonsensical until someone figured out a way to apply them to the real world. And "irrational" numbers. Imaginary numbers are absolutely impossible, right? Except that transistors are based on imaginary numbers. They describe, in some sense, some aspect of reality. And there's a workable system of transfinite numbers too. In this acceptedbymathematicians system, X and Y are actually the same size infinities, even though it seems to me that one is twice as big as the other. When mathematicians call Y bigger than X is not when it is twice as big as X, but when it is infinitely bigger. That is, if you put and infinity of rocks in Y for every single rock you put in X, then Y is a bigger infinity than X. Consider fractions and whole numbers: If you start from zero, how many fractions do you have by the time you get a single whole number? Infinity. So the infinite number of fractions is bigger than the infinite number of whole numbers. If you look up "transfinite numbers" in Wikipedia, you'll probably get some good info. 
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8/18/2012 10:17:46 AM Posted: 4 years ago At 8/17/2012 12:33:16 AM, R0b1Billion wrote: Did you watch the video all the way until the end? The concept of varying sizes in infinities was introduced by Georg Cantor, but that man literally believed that it wasn't true  his theory was that infinity a = infinity b, although he could demonstrate that aligning the two infinities (let's say , infinity a and infinity b) would produce a larger quantity for one of them, and a smaller quantity for the other. The presented the example of irrational numbers compared with rational numbers. The fact is that irrational numbers repeat forever, therefore, it is impossible to make a complete list of all the irrational numbers that exist, as all one would need to do is simply create another irrational number that extends one more digit. That's basically what they were explaining, although they made it a little more complicated by using a binary cypher using 1's and 2's, to make it more easily illustrated, I suppose. Although Georg Cantor didn't believe this actually proved the existence of varying infinity sizes, he couldn't prove it. Kurt Godel asserted that it couldn't be disproven (that there are not varying sizes of infinity), while Paul Cohen later asserted that it couldn't be proven, either. Accordingly, it's considered one of Math's unanswerable questions  an anomaly in the system that essentially renders it incomplete and imperfect, currently. However, Wiploc, that was still an excellent explanation. I love your only slightly irrelevant asides. They're quite witty. ^_^ 
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8/18/2012 11:25:14 AM Posted: 4 years ago Thanks.

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8/18/2012 12:40:04 PM Posted: 4 years ago Objections and replies.
The Fool: I admit to be a complete fool to such matters but I do strive the best I can to understand. So I raise these objection, so that I may understand more clearly. I mean this as positive critism. But these are the reasons I can't accept that they exist base at least from you examples. At 8/18/2012 8:50:50 AM, wiploc wrote:
The Fool: Thus we should try are very best to understand it. With as much PoC possible. But it is certainly no more true then the earth being flat via this method. While one can easily see why there would be more decimals than whole numbers, I can't understand how infinities can have different sizes. The Fool: I don;t think it is A number. I highlilght A so it can be recognize A=1 as in I thought it 1 particluar number. Imagine dropping a rock into pile X at the rate of one rock per second, forever. And pile Y gets two rocks per second, forever. The Fool: You see my problem here i that I can't even Imagine 'Forever'. Can anyone really imagine Forever. In the same sense that forever can't never be reached. If it could it would be Finite. Even if we make a hypothesis that if the later was to continue it would be forever. But that it continues does not include the notion forever. Secondly this hypothesis could never be validated. And that could never be true is by necessity false. Y is twice as big as X after one second, and after 100 seconds, and after a billion seconds, and forever, right? The Fool: A few problems here, it all make sense untill the and forever part. Firstly one unit what so every becomes 0 when subsumed under infinity. So this analogy is problematic. Either is false that any value under infinite is 0 or infinite is made up of finite units. And you can't get infinite from a set of finite units. For I am sure its very clear to most that [if] we can picture another finite set of seconds [then] it would rational to assume a larger pile. But this is again just another hypothesis. If I do can I picture but I cant picture for ever, honesly. There is never a point at which Y suddenly shrinks back to become the same size as X. So, even though both piles go on building endlessly, Y is always twice as big as X. The above is wrong, but I shared it anyway in order to get across the idea of different sized infinities. The Fool: Right because to take 'account' of the concept I must be able to count. That is the very literal sense of account. But I cannot account for infinity. In the beginning of the video he explains that there are as many even numbers as total numbers... if the logic was consistent, wouldn't the infinity of even numbers be smaller than the infinity of total numbers? The Fool: Awsome I would like to know this system, can you explain? Negative numbers were thought to be nonsensical until someone figured out a way to apply them to the real world. The Fool: yeah its called take the abolute value, There is still no actua correspondence to negate value in a real sense. Negative refers to relativly opposite, which we can take an absolute value. Secondly that is a poor justfication to accept the truth of something else. Aka it happens with other concept therefore it will happen again with this concept????? And "irrational" numbers. Imaginary numbers are absolutely impossible, right? The Fool: They still are absolutely impossible. We rationalize the value. aka we never and it would be impossible to take into account the popular notion of Pi being infinite.. By rationalizing the value I mean we use 3.14 or another longer but finite version, to calculate not the actual irrational number. For that would be irrational. In the exact sense where being rational come from. Except that transistors are based on imaginary numbers. The Fool: Based.. does not say much at all. Again the irrational number is not used in its irrational form. Why? Because it would be impossible. We could never have the complete number. They describe, in some sense, some aspect of reality. The Fool: Hell no! Description depends on describing experience you could never experience never ending ness. And there's a workable system of transfinite numbers too. The Fool: What is ment here by workable system. What is and unworkable system. I am a fool to tranfinite numbers but I will check themout. Maybe it is foolish of me but I don;t recognize the demonstration. In this acceptedbymathematicians system, X and Y are actually the same size infinities, even though it seems to me that one is twice as big as the other. The Fool: Saying something is true by virtue of acceptence is the worst possible argument you can give. Its like saying come on its true, why? Everybody whos cool is doing it. When mathematicians call Y bigger than X is not when it is twice as big as X, but when it is infinitely bigger. The Fool: which begs the question in what the hell is infinitly bigger, that doesn't render the size of a lesser thing nonexitent, therefore what is it bigger than? That is, if you put and infinity of rocks in Y for every single rock you put in X, then Y is a bigger infinity than X. The Fool: RIght. If and only if, we could put and infinite amount of rocks. Ofcourse there doesn't even exist an infinite amount of rocks. Secondly if there was it could never be reach to form any conclusion. Because infinite is unconclusive by nature. There could never be an actual conclusion. Consider fractions and whole numbers: If you start from zero, how many fractions do you have by the time you get a single whole number? The Fool: you can't start anying from zero. Zero is nonexistence of a value, you can't start from that. You have to assume there is a unit in which they are fraction of or it makes nosense(literally) at all. Infinity. So the infinite number of fractions is bigger than the infinite number of whole numbers. The Fool: Again this would render 0 units therefore 0 numbers. Bigger is a magitudal difference. Lets say it was actually possible to conscieve of infinit in a way that is not simply a hypothetical conception. How then could you from the perspective of THE INFINITE. Identify that they one is more INFINITE THEN then the other. I don't think we could tell the differnce because it is no particular value. If you look up "transfinite numbers" in Wikipedia, you'll probably get some good info. The Fool: I wil check that out. But I can't recognize anything going on here. I am curious if you could refute my claims, in a rational matter. "The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL 
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8/18/2012 9:41:58 PM Posted: 4 years ago The video is very good. There are two interesting examples of using the principle of correspondence to find the order of infinity.
The easy one is proving that there are same number of decimals between 0 and 1 as between 1 and infinity. For every number x greater than 1, it's reciprocal 1/x is between 0 and 1. The clever one is proving that there are as many points in a unit square as points on the line between 0 and 1. Every point in a unit square on the origin has coordinates (x, y) with 0 < x < 1 and 0 < y < 1. x and y each can be expressed as decimals. Interleave the digits in the x coordinate with digits in the y coordinate. For example, if x = .9999 ... and y = .1111 ... then the interleaved number will be .91919191 .... By interleaving, every point in the square corresponds to a point on the line between 0 and 1. In math and modern physics, intuition is not worth much. You have go with what the math says. 
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8/20/2012 12:42:44 PM Posted: 4 years ago I think Roy has an excellent approach to equating infinities, but to say one is greater than another is bollocks because adding additional elements doesn't affect the quality of infinity. Adding an infinity to an infinity doesn't change the quality either. The set of whole numbers and the set of even numbers both have infinite elements, and each element in the first can be doubled to correspond to each unit in the second, meaning that both infinities, despite one containing another, are equal.
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8/20/2012 1:54:47 PM Posted: 4 years ago At 8/20/2012 12:42:44 PM, wjmelements wrote: The video gives the method. Two infinities are of the same order iff a one toone correspondence can be established between the elements of the two sets. Adding two sets doesn't change the order. The video notes that the set of all subsets of the integers has a higher order of infinity than the integers. The set of all the the subsets of that set is still higher, and so forth. 
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8/20/2012 2:27:36 PM Posted: 4 years ago The Fool: You can't have a SET [ ...] of infinity.
"The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL 
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8/20/2012 3:03:42 PM Posted: 4 years ago At 8/20/2012 2:27:36 PM, The_Fool_on_the_hill wrote:Please learn set theory, troll. WOS : At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote: : Without nothing existing, you couldn't have something. 
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8/20/2012 4:37:06 PM Posted: 4 years ago At 8/20/2012 3:03:42 PM, tBoonePickens wrote:At 8/20/2012 2:27:36 PM, The_Fool_on_the_hill wrote:Please learn set theory, troll. The Fool: You are using it as a form of abuse.. I am saying its wrong by definition of a set.. I am saying that part of the theory is wrong. [........ In that if it was true then you should be able to bracet one side. Because that would indicate that it is finite. "The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL 
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8/20/2012 4:38:17 PM Posted: 4 years ago The Fool: My main argument being that Infinite is not conclusive.
"The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL 
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8/20/2012 4:40:17 PM Posted: 4 years ago At 8/20/2012 4:37:06 PM, The_Fool_on_the_hill wrote:Again, please learn set theory.At 8/20/2012 3:03:42 PM, tBoonePickens wrote:At 8/20/2012 2:27:36 PM, The_Fool_on_the_hill wrote:Please learn set theory, troll. http://en.wikipedia.org... WOS : At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote: : Without nothing existing, you couldn't have something. 
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8/20/2012 5:00:20 PM Posted: 4 years ago your source: set theory can be used in the definitions of nearly all mathematical objects.
Definition The Fool: infinite is undifined. Aka NOTdefined. I know its very populare. But that what I mean by critical thinking for youself and not just authority. "The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL 
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8/20/2012 5:03:45 PM Posted: 4 years ago At 8/20/2012 4:38:17 PM, The_Fool_on_the_hill wrote: Your main argument is that you don't know what you are talking about. And that's always your main argument. "It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater 
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8/20/2012 5:06:42 PM Posted: 4 years ago At 8/20/2012 5:03:45 PM, Sidewalker wrote:At 8/20/2012 4:38:17 PM, The_Fool_on_the_hill wrote: The Fool: Now that is exactly what the definition of an Internet Troll is. It refers to people to only come into distrupt, harass, forums, with giving sincere contrabutions. I am assuming you never actually checked what it means. RIght? "The bud disappears when the blossom breaks through, and we might say that the former is refuted by the latter; in the same way when the fruit comes, the blossom may be explained to be a false form of the plant's existence, for the fruit appears as its true nature in place of the blossom. These stages are not merely differentiated; they supplant one another as being incompatible with one another." G. W. F. HEGEL 
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8/20/2012 5:08:15 PM Posted: 4 years ago At 8/20/2012 5:00:20 PM, The_Fool_on_the_hill wrote: This is a Cretinous spelling and grammar fallacy. Inane comments and making up different definitions of words isn't critical thinking, it's just being a troll. "It is one of the commonest of mistakes to consider that the limit of our power of perception is also the limit of all there is to perceive." " C. W. Leadbeater 