Forum Index Register Members List Events Mark Forums Read Help

 International Skeptics Forum Continuation Deeper than primes - Continuation 2

 Welcome to the International Skeptics Forum, where we discuss skepticism, critical thinking, the paranormal and science in a friendly but lively way. You are currently viewing the forum as a guest, which means you are missing out on discussing matters that are of interest to you. Please consider registering so you can gain full use of the forum features and interact with other Members. Registration is simple, fast and free! Click here to register today.
 17th November 2014, 12:43 PM #81 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi If the basis of your proof depends on the existence of a parallel-summation operator, how is it defined mathematically? I can't analyze your proof beyond this point without a meaningful description of a parallel-summation. There can be finitely or infinitely many operators of all kinds on finitely or infinitely many arranged levels that simply are calculated in one step, called parallel-summation. Please think about parallel-summation more in terms of synthesis (more like parallel thinking) and less in terms of analysis (less like serial thinking). __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 17th November 2014 at 12:54 PM.
 17th November 2014, 12:51 PM #82 Dessi Species Traitor   Join Date: Jul 2011 Location: Omaha, NE Posts: 3,615 Doron, I'm seriously interested in your opinion: does your parallel-summation operator (however it is defined) give a different result than the ordinary summation operator Σ? Part of the beauty of mathematics is that many different approaches to a problem can yield the same solution, many different expressions are in fact equivalent. What difference should it make if we add terms together one at a time, or add them up in one operation? As near as I can tell: 0.999... = 0.9 + 0.09 + 0.009 + 0.0009 + . . . = Σ(n = 0, n -> infty) of (9/10)(1/10n) = Σ(n = 0, n -> infty) of akn for a = 9/10, k = 1/10 Using the generalized form, note that the sum S is: S = Σ(n = 0, n -> infty) of akn S = ak0 + ak1 + ak2 + . . . Note that Sk = k(ak0 + ak1 + ak2 + . . .) = ak1 + ak2 + ak3 + . . . We can infer S - Sk = (ak0 + ak1 + ak2 + . . .) - (ak1 + ak2 + ak3 + . . .) S - Sk = ak0 + (ak1 - ak1) + (ak2 - ak2) + (ak3 - ak3) + . . . . S - Sk = ak0 + 0 + 0 + 0 + . . . S - Sk = ak0 S - Sk = a S(1 - k) = a S = a / (1 - k) Substituting our original values, a = 9/10, k = 1/10: S = (9/10) / (1 - (1/10)) S = (9/10) / (9/10) S = 1 Why would you get a different result computing S = Σ(akn) vs S = parallel-summation-operator(akn)? __________________ >^.^< Last edited by Dessi; 17th November 2014 at 12:54 PM.
 17th November 2014, 01:02 PM #83 Dessi Species Traitor   Join Date: Jul 2011 Location: Omaha, NE Posts: 3,615 Originally Posted by doronshadmi Please think about parallel-summation more in terms of synthesis (more like parallel thinking) and less in terms of analysis (less like serial thinking). I'm a native English speaker but I don't understand the statement above. __________________ >^.^<
 17th November 2014, 01:04 PM #84 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi What difference should it make if we add terms together one at a time, or add them up in one operation? When you deal with infinity, using step-by-step (serial) thinking style gives the illusion that some kind of "process" is involved. In order to avoid such an illusion, it is better to use parallel thinking style. __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 17th November 2014 at 01:09 PM.
 17th November 2014, 01:08 PM #85 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi I'm a native English speaker but I don't understand the statement above. The difference between get X at once (parallel thinking) instead of step-by-step (serial thinking). __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
 17th November 2014, 01:12 PM #86 Dessi Species Traitor   Join Date: Jul 2011 Location: Omaha, NE Posts: 3,615 Originally Posted by doronshadmi When you deal with infinity, using step-by-step (serial) thinking style gives the illusion that some kind of "process" is involved. In order to avoid such an illusion, it is better to use parallel thinking-style. Let P = the parallel-summation operator Let S = P(0.9, 0.09, 0.009, . . .) = 0.999... (0.1)S = 0.0999... S - (0.1)S = 0.999... - 0.0999... S - (0.1)S = 0.9 S(1 - 0.1) = 0.9 S(0.9) = 0.9 S = (0.9) / (0.9) S = 1 0.999... = 1 It's the exact same algebra whether we use parallel or serial thinking. Why do you think you get a different result? __________________ >^.^< Last edited by Dessi; 17th November 2014 at 01:17 PM.
 17th November 2014, 01:16 PM #87 realpaladin Master Poster     Join Date: Apr 2007 Posts: 2,585 Originally Posted by Dessi Doron, before you comment on my lack of visual/spacial thinking, I want to show you a subset of my Amazon order history: http://i346.photobucket.com/albums/p...ps8b63eee9.png I encourage you to enjoy some posts I've written on computer science, be amused by my implementation of the cons/car/cdr combinators in Ruby, be more amused by my AVL tree implementation using only lambda combinators, and enjoy my book on functional programming. Trust me, I am a goddamn nerd of the highest order. I'm confident my mathematical, visual, and spatial reasoning are more than adequate to analyze a proof that 0.999... < 1 as long as you provide the details. Yes, I certainly bow to your 1337 math skills (and your taste in comics and cartoons), but, as someone who does security evangelism and who teaches hacking academically, I do frown somewhat on your privacy skills. Juliet, I would not recommend using a mixed private/public repo for images on fora where some, I do not say I count Doron to these, decidedly disturbed people hang out. I'd recommend imgur or somesuch. Other than that: __________________ "All is needed (and it is essential to my definitions) is to understand the actuality beyond the description, for example: Nothing is actually" - Doron Shadmi "But this means you actually have nothing." - Realpaladin ---
 17th November 2014, 01:27 PM #88 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi Let P = the parallel-summation operator Let S = P(0.9, 0.09, 0.009, . . .) = 0.999... (0.1)S = 0.0999... S - (0.1)S = 0.999... - 0.0999... S - (0.1)S = 0.9 S(1 - 0.1) = 0.9 S(0.9) = 0.9 S = (0.9) / (0.9) S = 1 0.999... = 1 It's the exact same algebra whether we use parallel or serial thinking. Why do you think you get a different result? What you wrote above is not considered as a rigorous proof of the discussed subject. But since you have used it, I wish to show you how it looks by using parallel thinking: __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 17th November 2014 at 01:36 PM.
 17th November 2014, 01:35 PM #89 realpaladin Master Poster     Join Date: Apr 2007 Posts: 2,585 Originally Posted by doronshadmi The difference between get X at once (parallel thinking) instead of step-by-step (serial thinking). I think this is visualized by Doron as 'just measure the height' on a 2D graph; instead of putting all values on the X-axis and then following the line, just put them on one point on the X-axis and the values on the Y-Axis. Then the number of your steps on the X-Axis is just 1. Or to put into an analogy (which is not correct, I know, but I try to mediate in communications here): - Imagine you have a really big amount of pebbles and you want to know their total weight. - You can measure each pebble, one by one and add each weight to the total sum. - Doron says: I just put all of the pebbles on my scale and know the total sum immediately. At least, this is what I imagine he thinks. But who's to say? EDIT: And then again...Seeing the post just above this one I am lost again. __________________ "All is needed (and it is essential to my definitions) is to understand the actuality beyond the description, for example: Nothing is actually" - Doron Shadmi "But this means you actually have nothing." - Realpaladin --- Last edited by realpaladin; 17th November 2014 at 01:38 PM.
 17th November 2014, 01:53 PM #90 Dessi Species Traitor   Join Date: Jul 2011 Location: Omaha, NE Posts: 3,615 Originally Posted by doronshadmi What you wrote above is not considered a rigorous proof of the discussed subject. Note that 0.22222...3 = 2*3^-1 + 2*3^-2 + 2*3^-3 + 2*3^-4 + 2*3^-5 = 2/3 + 2/9 + 2/27 + .... And 2.2222...3 = = 2*3^0 + 2*3^-1 + 2*3^-2 + 2*3^-3 + 2*3^-4 + 2*3^-5 = 2/1 + 2/3 + 2/9 + 2/27 + ... 2.2222...3 - 0.2222...3 = (2/1 + 2/3 + 2/9 + 2/27 + ...) - (2/3 + 2/9 + 2/27 + ....) = 2/1 + (2/3 - 2/3) + (2/9 - 2/9) + (2/27 - 2/27) + ... = 2 + 0 + 0 + 0 ... = 2 Where's the missing fraction? Have I misunderstood you? __________________ >^.^< Last edited by Dessi; 17th November 2014 at 02:17 PM.
 17th November 2014, 02:09 PM #91 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi Note that 0.22222...3 = 2*3^-1 + 2*3^-2 + 2*3^-3 + 2*3^-4 + 2*3^-5 = 2/3 + 2/9 + 2/27 + .... And 2.2222...3 = = 2*3^0 + 2*3^-1 + 2*3^-2 + 2*3^-3 + 2*3^-4 + 2*3^-5 = 2/1 + 2/3 + 2/9 + 2/27 + ... 2.2222...3 - 0.2222...3 = (2/1 + 2/3 + 2/9 + 2/27 + ...) - (2/3 + 2/9 + 2/27 + ....) = 2/1 + (2/3 - 2/3) + (2/9 - 2/9) + (2/27 - 2/27) + ... = 2 + 0 + 0 + 0 ... = 2 Where's the missing fraction? Dear Dessi, You are using 2.2222...3 - 0.2222...3 and get 2, and then you ask me where is the missing fraction? I like your humor Please look very carefully at my page in http://www.internationalskeptics.com...6&postcount=88. Thank you. ---------------------- Also please look at http://www.internationalskeptics.com...0&postcount=79 for some parallel thinking training. I think that after some training you can get http://www.internationalskeptics.com...7&postcount=73 without any problems. Also please do not ignore that fact that 0.999...10 = 1 if the real-line is observed from |N| level (as explained in that link), so understanding the different levels of cardinality is essential to my theorem, where parallel thinking is the best way to thing about infinite cardinality, if one wishes to avoid the illusion of "process". __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 17th November 2014 at 02:58 PM.
 17th November 2014, 02:28 PM #92 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi Let P = the parallel-summation operator There is no such thing like parallel-summation operator, as I explicitly wrote in http://www.internationalskeptics.com...0&postcount=81. __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
 17th November 2014, 02:38 PM #93 realpaladin Master Poster     Join Date: Apr 2007 Posts: 2,585 Originally Posted by doronshadmi Also please look at http://www.internationalskeptics.com...0&postcount=79 for some parallel thinking training. I think that after some training you can get http://www.internationalskeptics.com...7&postcount=73 without any problems. JSFisher called it. But be consoled by the fact that you set a new speed record on getting added to the 'you don't get it' club. __________________ "All is needed (and it is essential to my definitions) is to understand the actuality beyond the description, for example: Nothing is actually" - Doron Shadmi "But this means you actually have nothing." - Realpaladin ---
 17th November 2014, 03:24 PM #94 Dessi Species Traitor   Join Date: Jul 2011 Location: Omaha, NE Posts: 3,615 Originally Posted by doronshadmi Originally Posted by Dessi Note that 0.22222...3 = 2*3^-1 + 2*3^-2 + 2*3^-3 + 2*3^-4 + 2*3^-5 = 2/3 + 2/9 + 2/27 + .... And 2.2222...3 = = 2*3^0 + 2*3^-1 + 2*3^-2 + 2*3^-3 + 2*3^-4 + 2*3^-5 = 2/1 + 2/3 + 2/9 + 2/27 + ... 2.2222...3 - 0.2222...3 = (2/1 + 2/3 + 2/9 + 2/27 + ...) - (2/3 + 2/9 + 2/27 + ....) = 2/1 + (2/3 - 2/3) + (2/9 - 2/9) + (2/27 - 2/27) + ... = 2 + 0 + 0 + 0 ... = 2 Where's the missing fraction? Dear Dessi, You are using 2.2222...3 - 0.2222...3 and get 2, and then you ask me where is the missing fraction? I like your humor I'm sorry, I don't understand your comment. I could not infer an error from your diagram because, unfortunately, you don't show a proof for the statement we discover the fundamental mistake in this theorem, because the omitted purple part 0.222... is definitely not the result of 2x/2 (the green part). If you say there is an error in my proof, tell me in clear, precise language where I've made an error. Quote: Also please look at http://www.internationalskeptics.com...0&postcount=79 for some parallel thinking training. Note that addition in that notation sums elements one by one, not in a single step. __________________ >^.^<
 17th November 2014, 03:34 PM #95 Dessi Species Traitor   Join Date: Jul 2011 Location: Omaha, NE Posts: 3,615 Originally Posted by doronshadmi There is no such thing like parallel-summation operator, as I explicitly wrote in http://www.internationalskeptics.com...0&postcount=81. I disagree, you did not make that statement explicitly in any manner. That aside, your proof for 0.999... < 1 depends on the existence of some process or operation which sums up all the elements in a sequence in a single step. This operation is the parallel-summation operator. If your parallel-summation operator doesn't exist, then parallel-summation is undefined. __________________ >^.^< Last edited by Dessi; 17th November 2014 at 03:39 PM.
 17th November 2014, 03:51 PM #96 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,458 Originally Posted by doronshadmi What you wrote above is not considered as a rigorous proof of the discussed subject. Actually, it is sufficient. One might demand an explicit reference to summations, but since that is implicit in the meanings of 0.999... and 0.0999..., the added detail is unnecessary. As with many things, more than one, equally rigorous proof is possible. The most direct, I suppose, would rely on the meaning of 0.999..., itself, as an infinite series then show that the limit of the corresponding partial summation sequence must be 1. Then again, limit proofs are more difficult to grasp for most, so Dessi's approach has the advantage. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 17th November 2014, 04:15 PM #97 realpaladin Master Poster     Join Date: Apr 2007 Posts: 2,585 Originally Posted by jsfisher Then again, limit proofs are more difficult to grasp for most, so Dessi's approach has the advantage. We already did that last year or the year before (my, how time flies), then we went on to the 'two islands' thought experiment debacle, skipped over to metaconsciousness, skipped back here and are now, in extreme fastforward, repeating 7 years of argumentation. The prediction I make is that Dessi will find out that being right merely will disqualify her as someone using the wrong 'set of glasses', 'view, like |N| vs |R|' or 'just don't be getting it'. Doron controls the horizontal and the vertical here, so it's no use trying to adjust your set. __________________ "All is needed (and it is essential to my definitions) is to understand the actuality beyond the description, for example: Nothing is actually" - Doron Shadmi "But this means you actually have nothing." - Realpaladin ---
 17th November 2014, 04:48 PM #98 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,458 Originally Posted by realpaladin We already did that last year or the year before (my, how time flies), then we went on to the 'two islands' thought experiment debacle, skipped over to metaconsciousness, skipped back here and are now, in extreme fastforward, repeating 7 years of argumentation. The prediction I make is that Dessi will find out that being right merely will disqualify her as someone using the wrong 'set of glasses', 'view, like |N| vs |R|' or 'just don't be getting it'. Doron controls the horizontal and the vertical here, so it's no use trying to adjust your set. She may never get to enjoy a close encounter with Doron's exclusive "direct perception", then. How sad for her. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 18th November 2014, 12:46 AM #99 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi I disagree, you did not make that statement explicitly in any manner. Here is the explicit statement: Originally Posted by doronshadmi There can be finitely or infinitely many operators of all kinds on finitely or infinitely many arranged levels that simply are calculated in one step, called parallel-summation. Once again, parallel-summation is not some particular operator, it is simply using parallel approach in order to get a result in one step by not being influenced by the possibly complex structure of mathematical operations and their related variables\constant values, upon finitely or infinitely many arranged levels. Originally Posted by Dessi That aside, your proof for 0.999... < 1 depends on the existence of some process or operation which sums up all the elements in a sequence in a single step. Dear Dessi, At this stage you are simply forcing serial (step-by-step) thinking style, which is involved with some kind of process. This is explicitly not the case if one using parallel thinking style of the considered subject. As long as this is the case, there can't be any useful communication between us about this fine subject. __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 18th November 2014 at 01:13 AM.
 18th November 2014, 01:12 AM #100 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi If you say there is an error in my proof, tell me in clear, precise language where I've made an error. By using parallel thinking style, you are able to know in one step that the green and purple prats in the diagram are (at least) two different levels. By using 2.222...3 - 0.222...3 we actually get rid of all levels below the floating point (marked by purple rectangle), and from now on we work only with the values at the level above the floating point and get the result 2X/2 = 1 that has nothing to do with with 0.222...3, because we already got rid of 0.222...3 by using 2.222...3 - 0.222...3, and from now on it is not used anymore as a factor of our conclusions. __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 18th November 2014 at 02:02 AM.
 18th November 2014, 01:42 AM #101 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by jsfisher Then again, limit proofs are more difficult to grasp for most, so Dessi's approach has the advantage. Limits are definitely not difficult to grasp if one observe the real-line by distinguish between one level of cardinality observation of the real-line, and more than one level of cardinality observation of the real-line. By doing that, for example, 0.999...10 = 1 from only |N| observation. But 0.999...10 < 1 by observing |N| from |P(N)|, exactly as done in http://www.internationalskeptics.com...7&postcount=73. __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 18th November 2014 at 01:44 AM.
 18th November 2014, 01:50 AM #102 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi Dear Dessi, Please note that writing/reading http://www.internationalskeptics.com...6&postcount=16 step by step does not eliminate the ability to get all what is written in one step, and this is exactly what happens if at least visual_spatial AND verbal_symbolic brain skills are used. Things become important when we deal with infinite collections, for example: 2 -> 1 4 -> 2 6 -> 3 8 -> 4 ... We do not need to write down all even and natural numbers or to think step-by-step, in order to know in one step that there is a bijection from all even numbers to all natural numbers (we are using parallel thinking in order to conclude that both collections have cardinality |N|). Please do not forget http://www.internationalskeptics.com...9&postcount=68, and your reply to it in http://www.internationalskeptics.com...6&postcount=69. Thank you. __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 18th November 2014 at 02:20 AM.
 18th November 2014, 05:52 AM #103 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,458 Originally Posted by doronshadmi ... But 0.999...10 < 1 by observing |N| from |P(N)|, exactly as done in http://www.internationalskeptics.com...7&postcount=73. So, Doron, you are now asserting the the Continuum Hypothesis as fact? That's a bold claim, even for you. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 18th November 2014, 06:11 AM #104 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by jsfisher So, Doron, you are now asserting the the Continuum Hypothesis as fact? That's a bold claim, even for you. The fact that |N| < |P(N)| has nothing to do with CH. __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
 18th November 2014, 07:10 AM #105 jsfisher ETcorngods survivorModerator     Join Date: Dec 2005 Posts: 22,458 Originally Posted by doronshadmi The fact that |N| < |P(N)| has nothing to do with CH. I didn't say it did. You might want to review, though, your own sequence of posts and their consequence. __________________ A proud member of the Simpson 15+7, named in the suit, Simpson v. Zwinge, et al., and founder of the ET Corn Gods Survivors Group. "He's the greatest mod that never was!" -- Monketey Ghost
 18th November 2014, 07:22 AM #106 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 My theorem in http://www.internationalskeptics.com...7&postcount=73 is very simple. Exactly as some finite series that is observed from a sequence with |N| values < some given limit value, so is the case about some countably infinite series that is observed from a sequence with |P(N)| values, etc. ad infinitum ... where my conclusion is that no collection is accessible to the non-composed 1-dimensional space (which is not necessarily a metric space) or in other words, it is the inaccessible limit of all possible collections. Moreover this "points"\non-composed "line" association is without loss of generality (can be expanded to any association among collection of lower spaces with a given higher space). __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 18th November 2014 at 07:44 AM.
 18th November 2014, 07:46 AM #107 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by jsfisher I didn't say it did. So please tell me what is the purpose of your question about CH? (It has to be stressed that any number system that its members are explicitly defined and symbolized, has at most |N| members). __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 18th November 2014 at 08:01 AM.
 19th November 2014, 07:31 AM #109 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi In other words, you reject algebraic equivalence in principle. This equation a = b cannot be equivalent to any of the following in your system: a/k = b/k, for all k ka = kb, for all k a + k = b + k, for all k a^k = b^k, for all k f(a) = f(b), for any f Dear Dessi, I certainly do not reject algebraic equivalence, I simply do not ignore the different levels of values that are used in some algebra, for example: X = 0.999... 10X = 9.999... 10X - X = 9.999... - 0.999 = 9 (this is the critical operation, where we get rid of X (which is some value at the level of fractions) and what is left is the (positive, in this case) value at the level of whole numbers. From now on X can't be but some positive whole number, which has nothing to do with the original value of X, as was given by X = 0.999... It is well known that given initial values can't be changed in the middle of some argument if we wish to provide some valid conclusion about the given initial values. You have missed the critical operation exactly because you are using only serial observation (step-by-step thinking style) of the considered framework. Actually there is no problem to use serial_only observation of some finite framework in case of summation, but this is not the case if we deal with infinite summation in terms of process, because from this point of view the process can't be stopped and no exact result can be provided. So, in this case we are using the brilliant notion of Cantor's transfinite cardinality that is definitely based also on parallel thinking, as explained in http://www.internationalskeptics.com...&postcount=102. The use of finite cardinality, countably infinite or uncountably infinite cardinality is essential to my theorem in http://www.internationalskeptics.com...7&postcount=73, where parallel thinking can't be avoided if we don't wish to find our framework stuck in some endless process, when we deal with countably infinite or uncountably infinite cardinality. Moreover, the whole idea of, for example, the accurate value of |N| is possible only if we transcend some endless process, and this is done exactly by using parallel thinking that captures a given collection by using one step. Also please be aware that (0.999...10 = 1) OR (0.999...10 < 1 by 0.000...110), or in other words, there is no contradiction because they are solutions in two different levels of the real-line. ------------------- Please do not forget http://www.internationalskeptics.com...9&postcount=68, and your reply to it in http://www.internationalskeptics.com...6&postcount=69. It is crucial for valid communication between us. Thank you. __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 19th November 2014 at 08:58 AM.
 19th November 2014, 07:41 AM #110 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Let's improve what was written in http://www.internationalskeptics.com...&postcount=106, as follows: My theorem in http://www.internationalskeptics.com...7&postcount=73 is very simple. Exactly as some finite series that is observed from a convergent sequence with |N| values < some given limit value, so is the case about some countably infinite series that is observed from a convergent sequence with |P(N)| values, etc. ad infinitum ... where my conclusion is that no collection is accessible to the non-composed 1-dimensional space (which is not necessarily a metric space) or in other words, it is the inaccessible limit of all possible collections. Moreover this "points"\non-composed "line" association is without loss of generality (can be expanded to any association among collection of lower spaces with a given higher space). (B.T.W convergent sequences with transfinite cardinality |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ... etc. ad infinitum ... are simply proper subsets of the sets that have these transfinite cardinality, for example: The values of <0.9, 0.09, 0.009,...> are the same values of {0.9, 0.09, 0.009,...}, which is a proper subset of Q, where |{0.9, 0.09, 0.009,...}| = |Q| = |N|, and the same principle holds among collections with greater cardinality (but in this case we can't explicitly symbolize their values, since they are uncountable)). __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 19th November 2014 at 08:07 AM.
 19th November 2014, 10:05 AM #111 Dessi Species Traitor   Join Date: Jul 2011 Location: Omaha, NE Posts: 3,615 Originally Posted by doronshadmi Dear Dessi, I certainly do not reject algebraic equivalence, I simply do not ignore the different levels of values that are used in some algebra, for example: X = 0.999... 10X = 9.999... 10X - X = 9.999... - 0.999 = 9 (this is the critical operation, where we get rid of X (which is some value at the level of fractions) and what is left is the (positive, in this case) value at the level of whole numbers). From now on X can't be but some positive whole number, which has nothing to do with the original value of X, as was given by X = 0.999... Yes, it has everything to do with the original value of X = 0.999... . The value of X is constant in your equation; 10X is the constant 9.999..., 10X - X is also constant, and our original X = 9/9 = 1 is also constant. Every value in the equation is a constant. So why does X = 0.999... become X = 1? Because 0.999... is an algebraically identical representation of 1. Try plugging a few different start values of X into your equation, and perform the same operations. Code: ```X = 0.999... X = 1 10X = 9.999... 10X = 10 10X - X = 9.999... - 0.999... 10X - X = 10 - 1 9X = 9 9X = 9 X = 1 X = 1 X = √2/2 X = 12 10X = 10√2/2 10X = 120 10X - X = 10√2/2 - √2/2 10X - X = 120 - 12 9X = 9√2/2 9X = 108 X = √2/2 X = 12 X = 0.11111... X = 0.12345 12345 12345... 10X = 1.1111... 10X = 1.2345 123451 23451... 10X - X = 1.11111... - 0.11111... 10X - X = 1.23451 23451 23451... - 0.12345 12345 12345... 9X = 1 9X = 1.11106 11106 11106... X = 1/9 X = 0.12345 12345 12345... X = 0.99999999 (finite) X = n 10X = 9.9999999 10X = 10n 10X - X = 9.9999999 - 0.99999999 10X - X = 10n = n 9X = 8.9999999 9X = 9n X = 0.99999999 X = n``` Every X = (10X - X)/9 = X. It starts and ends with the same value. Why wouldn't it? Why is X = 0.999... the singular exception? As near as I can tell, you object to the X = 0.999... case because 9.999... - 0.9999... removes all of the trailing digits. Indeed, it does, 10X - X = 9, it has no other value. I can prove the algebra works out by showing that if X = 0.999..., then 10X - X = 9, therefore 9X = 9*(0.999...) = 9: 9 * .999... = 8.999... = Σ(n = 0, n -> ∞) (81/10)(1/10n) = (81/10) / (1 - 1/10) [see this identity] = (81/10) / (9/10) = 9 The math works beautifully. From this, one can conclude that 0.999... is an equivalent representation of 1, for the same reason that 0.1111.... is an equivalent representation of 1/9. Quote: Actually there is no problem to use serial_only observation of some finite framework in case of summation, but this is not the case if we deal with infinite summation in terms of process, because from this point of view the process can't be stopped and no exact result can be provided. This statement is incorrect. The example above, showing 9*(0.999...) = 9, demonstrates one way of the many ways people can analyze infinite series. Let me explain: Let's say we have a sequence {a1, a2, a2, a2, . . . }, the nth partial sum Sn is the sum of the first n terms of the sequence: Sn = Σ(k = 1, k -> n) ak This series converges if the sequence of its partial sums, { S1, S2, S3, . . . } converges. In other words, the series converges if there exists an L such that for any arbitrarily small positive number x > 0, there exists a large N so that for all n >= N, | Sn - L | <= x where |expr| is the absolute value function. If a series converges (and there are many tests for convergence), then as x -> 0, Sn - L -> 0 and Sn -> L. A formal proof of this property closely resembles this explanation. I know it doesn't seem "intuitive" that an infinite series converges to anything, but that intuition is wrong. Your insistence that we need a different way of analyzing infinite series isn't based on anything, and in fact many ot of the same techniques for analyzing finite series hold for infinite series. I don't think you can articulate any counterargument to this point. Quote: So, in this case we are using the brilliant notion of Cantor's transfinite cardinality that is definitely based also on parallel thinking, as explained in http://www.internationalskeptics.com...&postcount=102. The use of finite cardinality, countably infinite or uncountably infinite cardinality is essential to my theorem in http://www.internationalskeptics.com...7&postcount=73, where parallel thinking can't be avoided if we don't wish to find our framework stack in some endless process, if we deal with countably infinite or uncountably infinite cardinality. Moreover, the whole idea of, for example, the accurate value of |N| is possible only if we transcend some endless process, and this is done exactly by using parallel thinking that captures a given collection by using one step. Doron, again, I view you as a very polite person with a serious passion for mathematics, but whatever you mean by "parallel thinking" is undefined, whatever operation you describe to compute an entire series in a single step without intermediate calculations is undefined. You aren't able to explain your to undefined concepts to anyone, so your proofs can't be analyzed for correctness. As near as I can tell, you don't have a proof nor anything novel to say about cardinality. Since you mentioned Cantor, I encourage you to study the Cantor set and its analysis, as it has some fascinating and unexpected properties. The Wiki article on the Cantor set is actually very accessible to readers like you who have at least some familiarity with series, sets, and numbers in other bases. Lastly, I note that you must be a computer programmer or have some programming background from the statement "we find our framework stack in some endless process". Thinking in terms of call stacks, frameworks, and serial processes immediately tells me that you imagine operations on infinite series as a computer program that can never halt. I infer that "parallel thinking" is a very informal description of a hypothetical Turing machine which, by some miracle, instantaneously reads an infinite tape of inputs and halts with an output. This process happens inside a "black box"; we don't know how the black box works, we just know that it does. Let's call this model a Super Duper Turing Machine, its what you call "parallel-computation". While the expression (2/1 + 2/3 + 2/9 + 2/27 + ...) - (2/3 + 2/9 + 2/27 + ....) will never halt on an ordinary Turing machine, the Super Duper Turing Machine, by some black box miracle, always halts with the answer 2. There are some inputs which even our powerful Super Duper Turing Machine cannot compute, jsfisher provided one such example, Busy Beaver numbers are another example. An interesting question would be whether we can determine the halting behavior and output of our Super Duper Turing Machine, if such a machine existed? Yes, in fact we can; the entire study of calculus, infinite series, and differential geometry does exactly that. Conventional analysis techniques like the ones in this thread are Turing equivalent to computations on a Super Duper Turing Machine, both give the same answers to the same questions, there's no reason to think we would get a different result. __________________ >^.^< Last edited by Dessi; 19th November 2014 at 12:05 PM.
 19th November 2014, 01:35 PM #113 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi Every value in the equation is a constant. So why does X = 0.999... become X = 1? Because 0.999... is an algebraically identical representation of 1. No Dear Dessi, The initial X = 0.999... is omitted form 9.999... , and what is left is 9 that is a whole number that has nothing to do with fractions, and so is the case of 9X/9 = 9/9 = 1 that is also a whole number that has nothing to do with fractions. The rest of your post is based on your indistinguishably between fractions and whole numbers. Moreover, you are still missing the fact that, for example, |N| is undefined if we get the natural numbers only in terms of endless process. __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 19th November 2014 at 01:54 PM.
 19th November 2014, 01:45 PM #114 Dessi Species Traitor   Join Date: Jul 2011 Location: Omaha, NE Posts: 3,615 Originally Posted by realpaladin Dessi, are you able to (I am not, I admit) disprove why his wished-for result can never be achieved? I don't know what Doron's desired objective is, but in our exchanges I've learned a lot about his thought process. I notice there is an internal logic to his entire reasoning process: he has a fascination with concepts of infinities and infinitesimals, but he struggles with them conceptually. He only thinks of mathematical expressions in terms of a computation that runs in a computer program; this model model understandably breaks down on expressions involving infinite and infinitessimal quantities because a summation algorithm can never, in principle, halt on an endless stream of summands. Thinking like a computer programmer, Doron wonders how one would one go about computing an endless stream of inputs in a computer program? The answer is so obvious, so intuitive: by introducing a hypothetical computer that, by some miracle, sums the whole stream at once, processing every input in parallel -- parallel-summation! Amdahl's law be damned! Let's call our hypothetical processor a Doron Machine ^_^ Doron intuits that his Doron Machine will give different answers to mathematically expressions than we normally get with analysis techniques, for reasons related to limitations in floating point representations of numbers, meaning some real numbers have no fixed binary representation inside a Doron Machine, only infinitely precise approximations. I personally think infinite precision is pretty good, but can we do better? Yes we can. I'm an software engineer too, and today has been a really slow day in the office. So, I decided to build a better, less buggy parallel-summation machine: The Doron Machine 2.0 Super Deluxe Ultra. It's an extension of a normal Doron Machine with a better implementation of numbers which, by some miracle, doesn't store approximate quantities but rather stores quantities exactly. It also makes lattes and has a decent text editor. But the important thing is that any two quantities that are algebraically equal on paper really are equal in the Super Deluxe Ultra, and vice versa. This works nicely, because any sort of infinite and infinitesimal numerical analysis we can compute on paper, we can compute in the Super Deluxe Ultra, and vice versa. We infer that numerical analysis on paper, while a little slower, is computationally equivalent to any computable expression on the Super Deluxe Ultra. Even in Doron's very limited model of mathematical computation, the Doron Machine, the equivalence of 0.999... = 1 is inescapable. Intuition be damned. __________________ >^.^< Last edited by Dessi; 19th November 2014 at 02:24 PM.
 19th November 2014, 01:57 PM #115 Dessi Species Traitor   Join Date: Jul 2011 Location: Omaha, NE Posts: 3,615 Originally Posted by doronshadmi No Dear Dessi, The initial X = 0.999... is omitted form 9.999... , and what is left is 9 that is a whole number that has nothing to do with fractions, and so is the case of 9X/9 = 9/9 = 1 that is also a whole number that has nothing to do with fractions. It might seem unintuitive at first glance, but the expression 0.999... does not have a fractional part. The infinite series representation, Σ(n = 0, n -> ∞) (9/10)(1/10n), converges to a whole number. A proof of this series convergence is given here. Quote: The rest of your post is based on your indistinguishably between fractions and whole numbers. Rational numbers are any a/b, where a and b are natural numbers; whole numbers are rational numbers for any a = b or a modulo b = 0. There is no meaningful distinction between whole rationals and fractional rationals, they are treated in the exact same way. __________________ >^.^< Last edited by Dessi; 19th November 2014 at 02:14 PM.
 19th November 2014, 02:02 PM #116 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi While the expression (2/1 + 2/3 + 2/9 + 2/27 + ...) - (2/3 + 2/9 + 2/27 + ....) will never halt on an ordinary Turing machine, the Super Duper Turing Machine, by some black box miracle, always halts with the answer 2. Also in this case dear Dessi, you simply eliminate infinitely many values < 1 AND > 0 by using one step with cardinality |N|, and what is left is some whole number. __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.
 19th November 2014, 02:06 PM #117 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi It might seem unintuitive at first glance, but the expression 0.999... does not have a fractional part. Actually 0.999...10 is a factional number, and this simple fact is not involved with any intuition's problem. __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video. Last edited by doronshadmi; 19th November 2014 at 02:08 PM.
 19th November 2014, 02:08 PM #118 Dessi Species Traitor   Join Date: Jul 2011 Location: Omaha, NE Posts: 3,615 Originally Posted by doronshadmi Actually 0.999...10 is a factional number. Show a proof please. __________________ >^.^<
 19th November 2014, 02:11 PM #119 realpaladin Master Poster     Join Date: Apr 2007 Posts: 2,585 Originally Posted by Dessi Rational numbers are any a/b, where a and b are whole numbers; whole numbers are rational numbers for any a = b. There is no meaningful distinction between whole rationals and fractional rationals, they are treated in the exact same way. Shouldn't that be "any a = xb, x being an integer and non-zero"? EDIT: I don't know if 0 is considered a whole number and am too lazy too look it up at this moment. __________________ "All is needed (and it is essential to my definitions) is to understand the actuality beyond the description, for example: Nothing is actually" - Doron Shadmi "But this means you actually have nothing." - Realpaladin ---
 19th November 2014, 02:15 PM #120 doronshadmi Penultimate Amazing     Join Date: Mar 2008 Posts: 12,943 Originally Posted by Dessi There is no meaningful distinction between whole rationals and fractional rationals, they are treated in the exact same way. Wrong dear Dessi, you will never find a whole rational that is < 1 AND > 0 (which is the "home" of fractional rationals). __________________ That is also over the matrix, is aware of the matrix. That is under the matrix, is unaware of the matrix. For more details, please carefully observe Prof. Edward Frenkel's video from https://youtu.be/PFkZGpN4wmM?t=697 until the end of the video.

International Skeptics Forum