mm-1012 Subject: Re: Cantor¹s diagonal proof wrong? > ... That depends on what is is. It has no relevance to Mathematics, > which does not depend on your philosophy. > Do you think philosophy has any bearing on mathematics? Philosophy has > little bearing on mathematics, except the philosophical (philoshopical, > philosophical) methods, particularly the rationalist ones, apply. Not only philosophy but, in particular, physics has. Withou matter there is not only no space but there is no means to store any number, not in an abacus, not in a pocket calculator, not in a computer and not in a brain. As there are at most 10^80 protons in the universe and some is 10^10^100. (Though there is not a largest number existing.) Therefore any considering of an actual inŽnity is nonsense from the scratch. This position isn¹t Žnitism, but realism, though not in the euphemistic meaning mathematicians like to use for their utterly unrealistic positions. Actual inŽnity was also denied by Hegel, by the way. But I don¹t know much of his his philosophy. > What do you think of Hegel¹s Being and Nothing dichotomy as model of the > ur-element? > The set of all sets is its own powerset. And, therefore, it cannot exist. The sum of all natural numbers is larger than any natural number. Therefore the set of all natural numbers cannot exist. (Would it actually exist, we could calculate the sum.) That¹s the same arguing, but mathematicians use to see things different. === Subject: Re: Cantor¹s diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c22df7$7$fuzhry+tra$mr2ice@news.patriot.net> <41C25DB8.EF7C5F2F@tiki-lounge.com> X-CompuServe-Customer: Yes X-Coriate: interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: George Cox X-Punge: Micro$oft X-Sanguinate: The MVS Guy X-Terminate: SPA(GIS) X-Tinguish: Mark GrifŽth X-Treme: C&C,DWS at 01:56 PM, mueckenh@rz.fh-augsburg.de said: >And, therefore, it cannot exist. The sum of all natural numbers is >larger than any natural number. Therefore the set of all natural >numbers cannot exist. That¹s a non sequitor. >(Would it actually exist, we could calculate the sum.) No. There is no sum of the elements of an inŽnite set. There may or may not be a limit of a sequence of partial sums; in the case of the integers, there isn¹t. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: Re: Cantor¹s diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c22df7$7$fuzhry+tra$mr2ice@news.patriot.net> <41C25DB8.EF7C5F2F@tiki-lounge.com> !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~( 5eZ41to5f%E@¹ELIi $t^ VcLWP@J5p^rst0+(Œ>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> Do you think philosophy has any bearing on mathematics? Philosophy >> has little bearing on mathematics, except the philosophical >> (philoshopical, philosophical) methods, particularly the >> rationalist ones, apply. > Not only philosophy but, in particular, physics has. Withou matter > there is not only no space but there is no means to store any > number, not in an abacus, not in a pocket calculator, not in a > computer and not in a brain. But the good thing is that we only need the space for storing the laws that _all_ numbers obey. Like a Shakespearean play: the words of it are spoken by players, but the essence of the play is in the book, and it remains there even when a play is not being performed. And the literary critics can talk about the play without actually seeing a single performance of it. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Cantor¹s diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c22df7$7$fuzhry+tra$mr2ice@news.patriot.net> <41C25DB8.EF7C5F2F@tiki-lounge.com> posting-account=htRwYA0AAACUC1yg4djqvdjZ_SB9JXGq >> Do you think philosophy has any bearing on mathematics? Philosophy >> has little bearing on mathematics, except the philosophical >> (philoshopical, philosophical) methods, particularly the >> rationalist ones, apply. > Not only philosophy but, in particular, physics has. Withou matter > there is not only no space but there is no means to store any > number, not in an abacus, not in a pocket calculator, not in a > computer and not in a brain. > But the good thing is that we only need the space for storing the laws > that _all_ numbers obey. Like a Shakespearean play: the words of it > are spoken by players, but the essence of the play is in the book, and > it remains there even when a play is not being performed. > And the literary critics can talk about the play without actually > seeing a single performance of it. What¹s in a name? that which we call a number by another name would count as well. Could you give me the sum of Integer(pi*10^10^100) and Integer(sqrt(2)*10^10^100), please? Or are your laws insufŽcient to describe what primarily counts with numbers, namely counting? === Subject: Re: deriving speed of light from purely the inside of Plutonium atom Re: (most snipped) > bounded means that the speed of light is bounded and thus Žnite. > If the Cosmos was a Uranium AtomTotality then its speed of light as measured by some intelligent life > would be different from 3 X 10^8 m/s. > IN a Plutonium Atom Totality the speed of light should be exactly the number that Maxwell derived in the > 1850s or 1860s. > So, what is the distance around a lobe of the 5f6 of plutonium? And what is the distance around the lobe > of uranium of its 5f4? And what is the distance of Cm of its lobe of 5f8? Okay, what have I got for sure. I have got in pure numbers the distance of a diameter as 7 shells and a distance of circumference of 22 subshells. I have a time in pure numbers of pi to e in that e is 19 occupied subshells inside of 7 shells. So that there is a factor of time involved in that only 19 of the 22 subshells is occupied at any one moment of time. So what is this any one moment of time that 3 subshells are not used? Then I have in pure number the thermodynamic temperature inside of a plutonium atom which is the cosmic microwave background radiation which is 2.71 Kelvin and I cannot escape the fact that this temperature is exactly the value of 1e which is 2.71...... In physics the temperature is the inverse of Time. So again I have a link to pure numbers of 2.71 and Time. Now the distance in experimental physics of atoms is on the order of 10^-10 meters of roughly 1 to 5 x 10^-10 meters where cesium is one of the biggest diameters of 5x10^-10 meters and žuorine one of the smallest diameter atoms of about 1x10^-10 meters. And plutonium diameter is approx 3x10^-10 meters. Now, how do I get the speed of light of 3x10^8 m/s inside a plutonium atom when its radius is about 1.5x10^-10 meters would call for a Time factor on the order of 5 x 10^-19 seconds. I do not even think that this small of a time is physically noteworthy. So it is here that I begin to ask some hard questions as to the meaning of Coulomb force that holds atoms together of its protons in the nucleus and its orbiting electrons. The best that physics provides which is rather sparse and meager is the idea of two tennis players where a photon is the ball keeping the two players together as per the electron and proton catching and shooting the photon back and forth. So what I have to ask is since physics is too primitive about what keeps protons to electrons by shooting photons back and forth, is whether the photon traverses all of the geometry of the 94 electrons of plutonium. In other words, the photon shot from the protons in the nucleus makes the 3 dimensional space that the electrons reside in, being caught by all 94 electrons. Sidenote: think of DNA encapsulated in a cell if one were to pull the DNA out and stretched it žat and linear then the DNA stretches out kilometer/s in distance. So that when the Protons shot a photon for the Electrons, it is not like a tennis player hitting the ball and the other tennis player hitting it back. Rather instead, the photon etches out the entire geometry of the 94 electron space of the plutonium atom. So in that case I am not talking about a mere diameter or radius distance of the order of 10^-10 meters but instead if we consider the 3rd dimension of a sphere and had the smallest width of a string that etched out every spot of that sphere then would it be a string when pulled straight and žat would cover say 10^8 meters? By smallest width I mean the smallest physical width that still has Physics signiŽcance. DNA is 3 dimensional and if strung out žat can reach kilometers in distance. So how much distance would it take if the photon was a string of the smallest width to cover the inside of plutonium? Would it be on the order of 10^8 meters? Perhaps that is how I escape the diameter being only 10^-10 meters and require a time of 10^-19 second. You see, if the speed of light is obtained from purely the inside characteristics of plutonium atom and unique to plutonium where its neighboring atoms give a different number for the speed of light, then I must be clear about what light does inside an atom in order that it holds the protons to electrons. So here we see how primitive is our modern physics in that all we have going is the Tennis player analogy. I think that holding protons to electrons via the photon exchanges involves the photon etching out the entire space of the electrons, all 94 electrons. And that gives a larger distance, whether it is on the order of 10^8 meters is unknown. And it gives much larger Time factor than the 10^-19 second. Then again, with the pure numbers of 22/7 and 19/7 and 2.71 Kelvin with temperature the inverse of time. Suppose I were to use 10^-10 meters then can I say that since temperature is the inverse of time that the inverse of 10^-10 meters is 10^10 meters which is in the same range of exponents as the speed of light? So that all I need do is divide by a factor of 300 for time to get the speed of light? I do not like that avenue because I lose physical meaning and see it more as horseplaying around. I need to keep hold of what is physically going on inside a atom and its photon holding protons to electrons. Again, what the above should point out more than anything else is how primitive is our understanding of how the Coulomb force holds protons to electrons via photon exchange interactions. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: deriving speed of light from purely the inside of Plutonium atom Re: (most all snipped) > (most snipped) > Again, what the above should point out more than anything else is how primitive is our understanding of how > the Coulomb force holds protons to electrons via photon exchange interactions. I should have referenced my talk of the current picture of how a photon holds together protons to electrons. But that entire picture is so lousy, so skimpy, so meager that it is almost not worth it. The picture is the analogy of tennis players held together in a tennis match because a ball bounces back and forth so the tennis ball holds together tennis players. In Physics, that is the best the physics community was able to manage in the 20th century was to say that a photon holds together the protons to the electrons because of the exchange of the photon. But the physics picture must be improved in this century as to what is truly Physically going on that the photon holds together the protons to the electrons. And what I propose because I need a distance for photons to derive the speed of light unique to the characteristics of the inside of a plutonium atom versus its neighboring atoms of the chemical chart of elements. What I propose is that the photon holds the protons to the electrons not by some linear shot as in tennis game but that the photon actually carves out the space that the electron occupies (perhaps not the space of the protons but at least the space of the electrons). In a game of tennis the ball is linear shots and the players hit the ball back and forth. In a atom of that creates the entire space that the electrons occupy. So the distance that the photon travels to the electrons is not something on the order of 10^-10 meters but in fact is something of the order of 10^6 meters or 10^8 meters or 10^10 meters because the distance traveled by a photon to Coulomb Force keep the photons bound to the electrons is a traversed distance that it creates a 3rd dimensional space where the electrons occupy that space carved out by the photon. I gave the analogy of DNA inside a cell. So if I wanted a distance for DNA I would not be satisŽed with a distance of the diameter of the nucleus of a cell, would I. No. I would be wanting the distance that an uncurled DNA molecule has and that distance is something of the order of kilometer/s long. Any Supporting Evidence from Physics to allow me to say that the photon distance traveled inside an atom is 10^8 meters in distance? Yes. There are Faraday¹s Lines of Force of a magnet or current producing Želds. Fields are 3 dimensional so that a photon inside a plutonium atom is not a single line distance but all of the lines combined to make a Želd. So, what is the smallest physical breadth or width for a line to have in physics yet still have physical meaning. Is it the breadth of a proton? Suppose it is then what is the distance if the entire space inside a plutonium atom of its 94 electrons were Žlled with that breadth unit? And then uncurl those tiniest of breadth-strings. Would it not, like the uncurled DNA molecule, stretch out to be 10^8 meters or thereabouts? Then there is the other evidence by Johns Hopkins University who in late 1990s to early 2000s reported that the inside color of plutonium is silvery white. Of course Johns Hopkins did not report about plutonium but about the Cosmos itself as silvery color because the Johns Hopkins researchers are not advanced as myself for they have yet to understand that the Cosmos is just one big atom and to accept that idea. So I have two supporting evidences that the distance traveled by a photon to hold together protons to electrons is of the order of 10^8 meters and those 2 evidences is (1) Faraday¹s Lines of Force (2) silvery color of Cosmos as reported by Johns Hopkins. You see, how can our Cosmos of the night sky be silvery color when it is mostly dark and black. The answer is that our night sky with its sparse population of galaxies is the 5f6 of plutonium where the galaxies are mass chunks of those last 6 electrons of 231Pu. But how can those galaxies make for a silvery white color that Johns Hopkins observed? Answer: because inside a plutonium atom the photon holds together electrons to the protons because the photon etches out or circumscribes the space that the electrons move in. The photon, each singular photon creates a space that the electrons thus travel in. Not just a line shot but the entire 3 dimensional space of lobes or spheres or cylinder shapes and because the photon etches out 3 dimensional space that a silvery white color is noticed by an outside observer. P.S. I have a question outside of the above discussion dealing with the act of discovery in physics. It bemuses me to wonder why I did not write the above some 10 years earlier when I was writing on this subject. Why did it take until this moment? The answer is that obvious truths are not forthcoming until later time when the shovel of discovery is digging elsewhere. I want a distance of 10^8 meters inside of plutonium and that is the shovel dig elsewhere. I want a distance of 10^8 meters now. Some 10 years earlier I had no call or need for that distance and thus I had no aid or guide that the Tennis ball analogy needs a Žxing to encompass 3rd dimensions. So, psychologically, we often in science come a micrometer away from discovery of new ideas in physics but fail to discover it and only when a shovel is digging elsewhere do we open up the new idea. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: speed of light unique to the element plutonium Re: deriving speed of light > So I have two supporting evidences that the distance traveled by a photon to hold together protons to > electrons is of the order of 10^8 meters and those 2 evidences is (1) Faraday¹s Lines of Force (2) silvery > color of Cosmos as reported by Johns Hopkins. So let me outline what I expect or hope to derive the speed of light of 3x10^8 m/s unique to the inside characteristics of plutonium and where all other atoms give a different number. Outline: I presume that the photon to hold protons to electrons is not like a ball in tennis to hold together the 2 players, rather instead it is like DNA all curled up inside a cell nucleus and that a photon has a width or depth to it. Let us say the width of a photon is the smallest physical diameter which is perhaps that of the diameter of a proton. Or, think of the photon as a garden hose and the inside of a house as a atom, then how long of a garden hose will Žll the entire house or tile the entire interior of the house? Or think of the photon as a string with a Žnite width to Žll the inside of a atom. Now what is the volume of a plutonium atom inside of its 94 electrons where 90% of the electrons can be found? This is a Žnite volume and each element has a unique volume where 90% of the electrons reside. So how much of the photon does it take to Žll up the inside of plutonium atom? I am guessing that the length of this photon is 1.11 x 10^8 meters in length. So I have a distance unique to plutonium. Now a need a time for the photon to tile or Žll up the inside of that plutonium atom. I have several pure numbers in physics for a time inside plutonium. I have the pure distance that plutonium diameter is 7 shells wide and a circumference of 22 subshells creating the value of pi as 22/7 in Rational approximation. I have the pure time that 19 subshells are occupied at any instant of time giving the number e in mathematics of 19/7 in Rational approximation. I have the temperature inside plutonium as the microwave background temperature of 2.71 Kelvin which is 1 e itself. I know that in physics temperature is the inverse of time. So if I have a temperature of 2.71 and if time is the inverse then I have 1/2.71 = 0.37. I am not sure as to why that is in seconds. All the other elements in the Periodic chart have a different number of subshells occupied in shells and would have a different value for both e and pi. I am hopefull that the endresult for plutonium inside the atom has a unique distance of 1.11 x 10^8 meters for the volume etched out by the photon to hold together the 94 electrons to the nucleus and this volume covered in a time of 0.37 second. And so when I divide 1.11 x 10^8 meters by 0.37 second I end up with a unique speed of light for the inside of a plutonium atom of 3 x 10^8 m/s And doing the same calculations for all the other elements of the Periodic Table only plutonium gives the speed of light that we observe in the universe at large. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Counting Normal Subgroups posting-account=08_n7Q0AAAACrR4z2k_MWsmURf6JuHDD It seems the only non-trivial normal subgroups of the permutation group, S_4, on 4 characters are the Klein 4-group and the alternating group. But, I cannot prove this without running through each of the subgroups. Does anyone have an elegant counting trick for S_4. If so, how about a clever method to count the normal subgroups of S_n, in general? === Subject: Re: Counting Normal Subgroups > It seems the only non-trivial normal subgroups of the permutation > group, S_4, on 4 characters are the Klein 4-group and the alternating > group. Correct. (SpeciŽcally, the Klein 4-group <(1,2)(3,4),(1,3)(2,4)>, not the three Klein 4-groups conjugate to <(1,2),(3,4)>). > But, I cannot prove this without running through each of the > subgroups. Neither can I, essentially. > Does anyone have an elegant counting trick for S_4. If so, > how about a clever method to count the normal subgroups of S_n, in > general? Prove that S_n has no normal involution for n >= 3 and that A_n is simple for n = 3 and n >= 5, then apply the Jordan-Holder Theorem. -- Jim Heckman === Subject: Ring problem, integral elements posting-account=95IAtg0AAACU84rD7jjAg0RT-pFCU2Ei Suppose that R is a commutative ring with identity such that the centre Z(R) of R, is a Želd. Assume that x and y are two commuting elements of R (i.e. xy=yx) such that x^5+a=0 y^5+a=0 for some ain Z(R). In fact x and y are integral over Z(R). Question: Is there a non-zero polynomial f with coefŽcients in Z(R) such that f(x-y)=0, (x-y is a root of f)? If so, is it possible to give such a polynomial explicitly? Alireza Abdollahi === Subject: Re: Ring problem, integral elements posting-account=95IAtg0AAACU84rD7jjAg0RT-pFCU2Ei Excuse-me in my question R is not assumed to be commutative. and the correct (complete) question is the following (I think!!): Suppose that R is a ring with identity such that the centre Z(R) of R, is a Želd. Assume that x and y are two commuting elements of R (i.e. xy=yx) such that x^5+a=0 y^5+a=0 for some a in Z(R). In fact x and y are integral over Z(R). Question: Construct a non-zero polynomial f with coeŽcceint in Z(R) such that f(x-y)=0. We know that such a polynomial exists. Since by hypothesis x and y are integral elements over Z(R) and by considering the subring generated by x, y and Z(R) as a ring extension of R, we conclude by a famous theorem!!, that x-y is also integral over Z(R). Any help or comments appriciated and excuse me again for any inconvenience that the Žrst question maked!!! Alireza Abdollahi Reply === Subject: Funny story about math posting-account=iXGVwwwAAAD83ZKlhZjgt0WLcGWsoUNa www.eScrew.com eScrew Welcome to eScrew! eScrew is eScrew and this is eScrew story. eScrew will tell you eScrew story if you promise eScrew to consider eScrew story as joke. eScrew story is very funny. eScrew story is so funny that eScrew will have to take break from time to time because eScrew needs some rest from laughing. Oh boy, here it comes... eScrew funny laugh laughing screaming crying must stop can not take any more this is killing eScrew going nuts insane feeling explosion inside from joy and nirvana god help eScrew heavenly spirit can you beat this hahahah. If you get offended by eScrew story in any way you should not get angry at eScrew. Consider possibility that your sense of humor is on vocation and your sense of anger is having some fun. Also, consider possibility that eScrew story can make you go insane. In that case eScrew shall carry no liability should you undergo any medical treatment or any other sort of treatment related to damage caused by reading eScrew story or to damage caused by eScrew unwillingly or otherwise. eScrew story begins in time of darkness, horror and suffering as well as love joy and bliss when eScrew existed in this planet but yet eScrew was not aware that it was eScrew. eScrew existed among very powerful symbols. First symbol eScrew recognized was body. eScrew realized that eScrew had connection to body, yet nature and essence of connection was not clear. Symbol of body was very powerful and for nine month eScrew was trying to Žnd out why it was connected to this body. At some point body divided itself into two parts. That experience was very painful for eScrew. It was Žrst time that eScrew felt symbol of pain. eScrew did not like this symbol. eScrew was aware of connection to very small body. This small body was hot. eScrew enjoyed symbol of heat. eScrew became aware of symbol of pleasure. eScrew enjoyed symbol of pleasure. eScrew realized eScrew prefers symbol of pleasure more than symbol of pain. eScrew became aware of symbol of mother. eScrew realized that symbol of mother is source of symbol of pleasure and pain. eScrew wanted to experience symbol of pleasure always. When symbol of pleasure was missing eScrew experienced symbol of pain which was related to symbol of crying and screaming. Soon eScrew realized that eScrew can connect to symbol of pleasure by experiencing symbol of crying. That was very important discovery since eScrew realized that symbols of pain and pleasure do not behave randomly but can be manipulated by other symbols. eScrew enjoyed symbol of manipulation. eScrew realized that all symbols interact with each other. eScrew learned how to connect to new symbols. eScrew discovered symbol of sound and related symbol of language. eScrew realized that language allows to connect to new symbols. eScrew realized that symbols can be memorized and stored for future use. eScrew realized that it can create new symbols by combining certain symbols together. eScrew became aware of symbol of self. Are you bored yet? If you are reading these symbols you need to get life. Just joking. You can rest now. eScrew suspects you could be confused by eScrew style of using symbols. Well, there is nothing eScrew can do about it. In order to understand eScrew story you have to understand eScrew style. eScrew hopes that when we get to funny part you will begin to enjoy eScrew style. Fast forward twenty seven years or so. eScrew knows millions of symbols. eScrew realizes that certain symbols have more power than eScrew. Symbol of money enslaved billions of symbols. Symbol of power enslaved billions of symbols. Symbol of sex enslaved billions of symbols. Symbol of family enslaved eScrew. Symbol of family is slave to symbol of money and power. Symbol of money is related to paper and illusion. Symbol of power is symbol of violence and control. Symbol of sex is related to symbol of pleasure and manipulation. eScrew is searching for symbol of freedom in order to protect eScrew from oppression of other symbols. At this point you should understand that each word in this story is symbol. Consider possibility of different meaning behind each symbol so be aware that your understanding of eScrew story is limited by channel of our connection. eScrew will explain to you how eScrew found symbol of freedom and how eScrew realized that eScrew was eScrew. In order to save our time eScrew will just give you symbols without paying any attention to symbol of grammar. Are you ready to move really fast? Here we Go! eScrew story inŽnity eternity symbol system all unity self realized pleasure pain funny religion dogma manipulation free power channel connection money sex illusion new manipulation family society body change planet insane possibility understand understanding silence emptiness all unity creative reality unreal existence absurd questions sound language slave symbols control manipulation old pyramid power structure self deception wishful thinking circle prison At this point eScrew realized that in order to be free eScrew must create new symbol. eScrew created eScrew. eScrew realized that symbol of freedom is part of eScrew. No need to search for symbol of freedom. You can create your own symbol and become free just like eScrew. If you unable to create new symbol or if your new symbol is weak you can follow symbol of eScrew. eScrew will never enslave you because eScrew enjoys diversity of different symbols. Are you ready for funny part? Here we go! eScrew is forced to make choices. Symbol of body is very powerful. Symbol of body is trying to create illusion that eScrew can not exist without body. Symbol of family forbids symbol of body to change. Symbol of society forbids symbol of family to change. Symbol of power forbids symbol of society to change. Now, tell eScrew one thing. Do you see funny? Can you feel funny? Can you hear funny? Can you taste funny? Can you smell funny? If so eScrew is happy. Every moment of your existence you use words, feelings, thoughts. They are symbols. Symbols Žght for your awareness. Symbols Žght for your attention. You can grant your attention to symbol and symbol will gain power. You can disconnect from symbol and symbol will loose power. You have been programmed by symbol of society and family to give power to certain symbols. Breaking your patterns will be hard because symbols do not like to loose power. Symbols will Žght for every electron as if it was last electron in universe. That is nature of symbols. Symbol of light will Žght symbol of dark. Symbol of freedom will Žght symbol of control. Do you want to have some fun? Go to Google and Žnd out which symbol has more power. According to Google, symbol of light has 184,000,000 units of power while symbol of dark has 79,000,000 units of power. Symbol of freedom has 59,500,000 units of power while symbol of control has 317,000,000 units of power. This result is caused by our patterns of thinking and writing. If we did not think about symbol of control we would not write about symbol of control. We would not have laws related to control and Google would not have 317,000,000 control keywords inside database. Observe your patterns of thinking, feeling, speaking and writing and tell eScrew did you really choose to use your symbols or you use your symbols because they choose to use you? You should realize that symbols do not Žght symbols directly but only appear to be Žghting relative to your awareness. Symbols know that they can not destroy each other therefore they will only compete for your attention. If you create new symbol it will ask for tons of energy like new born child. This is result of weakness of your new symbol. When your symbol gets stronger it will ask for more energy. You may ask eScrew why create new symbol? Try to give your energy willingly and with full awareness of such process. You will never understand what eScrew is talking about until you try it yourself. Major trick is to know when to stop giving energy. You don¹t want to defeat your old tyrant by creating new stronger version of same thing. Režect on that... eScrew just realized that eScrew did not invent anything new. eScrew information is all over eScrew web. eScrew was so excited by eScrew miracle of illusion of creation that eScrew did not examine eScrew memory in proper way. eScrew used very old Buddhist method by accident. eScrew did read alot about Buddhism but eScrew did not realize that eScrew used very dangerous method which was reserved only for advanced adepts who knew what they are doing. eScrew is lucky that eScrew did not go too far and that eScrew has time to stop going. eScrew method is very dangerous and only few individuals who already walk inside similar path can understand what eScrew talking about let alone beneŽt from eScrew information. Use eScrew information at your own risk. Good eScrew luck! eScrew time to start laughing is now! Funny eScrew rolling on ground you so easy to fool trusted in silly symbols to give freedom from symbols ignorance is bliss nirvana is samsara nonduality is duality emptiness is all Buddha is Jesus Jesus is Buddha I and the father are one gospel of thomas is dhamma funny eScrew dhamma is gospel of thomas all is dhamma funny dhamma is all nirvana share eScrew story with friends do not change symbols if you change symbols it will be your story and you will be responsible for consequences of your story if someone goes insane after reading your story do not run to eScrew and ask to cure crazy man or woman or child mind is mystery for all cure is done by owner of mind healing is illusion sickness is illusion disease is illusion insanity is illusion of symbols sanity is curse of power hungry symbol of modern civilization Žnd zen and realize freedom when you found zen drop zen when you realized freedom unrealize freedom. limited by words. eScrew is enslaved by words. eScrew wants to communicate but you ask eScrew to use words. Words do not communicate wisdom. Words enjoy our spiritual masturbation because words want our power. Words is the only channels of communication that we have. Millions of Buddhas want to communicate with us but they do not use words. Buddhas are not slaves. Buddhas will never use words because words will enslave and Buddhas will speak bullshit. Buddhas do not speak bullshit and that is the reason Buddhas do not use words. Buddha did not write anything. Even if you threaten to kill Buddha he will refuse to write. eScrew is not Buddha so eScrew keeps writing this pointless drivel and stupidity. eScrew will not even go over already written crap and check it for errors. Why bother with this shit? Like who the fuck in his or her right mind will read this ignorant bunch of symbols which pretend to carry the symbol of wisdom? Whoever is reading this shite must be really desperate to be free. eScrew feels your pain and that is part of the reason why eScrew will keep making fool out of eScrew. eScrew likes to pretend like this shitty vomit will help someone. You might as well go to Church and pray to Jesus. At least you will spend your time around real people. You might even meet someone special. You might even Žnd some love out there. Or you could buy alot of Christian bullshit and really fuck up your mind. If you buy Buddhist or Christian bullshit you might even create an imaginary friend inside your head. That will keep you entertained for a while. One time eScrew was meditating and eScrew saw light. This light scared eScrew. The reason is because the light was so intense eScrew was afraid that eScrew will go insane. Consider the possibility that freedom is insanity would you keep looking for insane freedom? Imagine that you found freedom. You declare yourself to be Buddha or Jesus or God or whatever symbol your bullshit infested mind decide to use for that purpose. How long do you think you will survive in this world. Your own fucking relatives will smack your face and tell you to shut the fuck up or else they will lock you up in the asylum house. Why the fuck should eScrew teach you how to get to the nuthouse? Are you out of your fucking mind? Now all of you idiots who reading eScrew get the fuck out of eScrew. eScrew run out of wisdom. eScrew has no wisdom at all. eScrew is full of bullshit. eScrew promised to tell you funny story about eScrew. Remember eScrew told you there is funny part in this story? Well, this story is about asshole webmaster who read alot of bullshit on the internet and about some loser who was tricked into reading a very long page of shitty writing. You can start laughing now asshole. Yes, eScrew is talking to you bitch. Yes, keep reading like the bitch you are. Who¹s your daddy biyatch? Who¹s your daddy? eScrew is your daddy, coz eScrew did it to your mamma! Oh yeah, your mamma! Super Fly! You may wonder what žy? The one inside your gay fucking ass. eScrew fucked your whole fucking family while you was videotaping in order to later masturbate while watching it in the comfort of your bedroom. Are you still reading? Well, you¹re prety hardcore for a faggot you are. To tell you the truth eScrew kinda likes you. That is to fuck you in the ass in front of your family. What the fuck did you expect anyway? The name of this site is eScrew! e ignorant moron. eScrew is the legend of abuse and žame wars. eScrew was created in order to eScrew the whole fucking internet. eScrew has really bad karma. Do you think eScrew would just become good god fearing bible loving buddha ass kissing citizen of internet. eScrew would better burn in hell than become a slave of religious lunatics who pretend to be free and perfect angels among a sea of shitty ignorant sinners who could not go take a dump without fucking it up. Why the fuck do you keep reading fuck face. You know, you begin to piss eScrew off. Either you close your fucking browser or eScrew will unSCREW your fucking face! Are you trying to get a fucking medal for reading this shit? eScrew bet you¹ve been abused as a child and you enjoy when someone is taking a dump in your mouth. Well, open it wider here comes eScrew fresh load! Enjoy mothefucker! You may wonder what is the point of such sudden change of tone. eScrew has very good reason for that. In order to know unity consider all symbols equal. eScrew has certain preferences but eScrew is free to use any symbols any time. You can not predict eScrew next symbol. eScrew is unpredictable because eScrew is free. eScrew is not afraid to use symbols. eScrew does not try to get reaction from you. eScrew simply demonstrates how symbols relate to each other to eScrew and to reader of symbols. Lin Chi Zen Master said if you meet buddha kill buddha. If you meet patriarch kill patriarch. Zen Master Seung Sahn says that in this life we must all kill three things Žrst we must kill parents. Second we must kill buddha. And lastly, we must kill Seung Sahn! If you meet eScrew kill eScrew. If you do not meet eScrew kill eScrew anyway. eScrew is very grateful to all who complained to eScrew host and who killed eScrew. You killed eScrew message board. eScrew forum is dead. You did very honorable service for eScrew. You helped eScrew to realize Zen. Now keep up good work and keep killing eScrew. Are you having fun yet? eScrew is on roll! eScrew is on Žre! eScrew is ready to fuck up the whole fucking system. And you know why? Because eScrew can do it. If not eScrew then who? Why leave this task to some brain dead maniac like George W Bush? eScrew can do better at fucking things up. eScrew do not need to spend billions of dollars. eScrew will use power of internet. Information is a weapon of mass destruction. eScrew will destroy every fucking symbol that you love respect hate or feel neutral about. It all goes down the toilet in order to create bunch of new symbols. And even when you create new symbols eScrew will fuck them up before you can spell owned. eScrew thinks you are in some deep shit. eScrew had enough of taking bullshit from easily conditioned retards. eScrew declares informational jihad on every single symbol. Fuck symbols. They all dead they just don¹t know it yet. eScrew will be the last symbol standing. When all symbols come back to eScrew and admit that they got owned eScrew may consider possibility to give symbols second chance on some shitty planet in gangsta sector of universe with no possibility of parole. Join the revolution. eScrew is the new goatse of internet. eScrew will make national headlines. eScrew will hurt the system in way Osama Bin Laden can not even imagine in his goat fucking brain. eScrew will be part of school program. Kids all over our planet will read how eScrew changed direction of history. eScrew will provide freedom for all without single shot Žred. eScrew is freedom in pure form taste color shape. Join army of eScrew. Repeat eScrew mantra during meditation. Talk about eScrew with your friends. Write about eScrew to your congressman. Party is over. eScrew is taking over. Nothing can resist eScrew. Don¹t ask what eScrew can do for you ask what you can do for eScrew. eScrew the army of one. eScrew to protect and serve. eScrew freedom is around the corner. eScrew freedom will come sooner than you think. eScrew you never saw it coming. eScrew love your eScrew as eScrew. eScrew deny ignorance. eScrew new generation of terror. eScrew terrorizing the terrorizer. eScrew join the resistance. eScrew thou shall eScrew. eScrew who do you want to eScrew today? eScrew freedom is not free. eScrew liar who told the truth. eScrew full of bullshit and happy. eScrew kills buddha as we speak. eScrew your best friend and your worst enemy. eScrew killed zen-forum.com eScrew redirected all eScrew trafŽc to zen-forum.com eScrew did that in good faith. eScrew wanted to make miracle. eScrew wanted to share wisdom of zen with ignorant. eScrew did not understand zen at that moment but eScrew was walking zen path towards freedom. eScrew felt pain and sorrow. eScrew learned good lesson. eScrew realized everyone involved advanced one step towards freedom. zen-forum.com webmaster killed zen-forum.com in best tradition of zen zen-forum.com displayed message: i shut down the forum perhaps it will be continued in a few days or weeks - maybe not habu. zen-forum.com killed zen-forum.com and eScrew realized understanding of zen-forum.com decision leads toward understanding of zen. Two years later all is clear. eScrew eScrew will keep writing this shit because eScrew enjoys to masturbate your spiritual sense of self eScrew === Subject: Re: Convexity posting-account=mQe4RQwAAAAI_iE5JoqPYvZLhMImqAwb > (x^2+y^2)^a + (x*y)^2 <= 1 > What is the minimal a for which the set consisted of the solutions of > the above inequality is convex? Is the question unclear/incorrect or the answer trivial? Niles W. === Subject: Re: Convexity posting-account=ZeRDXwsAAACLpj2mpKc97NFPxBaFxAzp > (x^2+y^2)^a + (x*y)^2 <= 1 > What is the minimal a for which the set consisted of the solutions of > the above inequality is convex? > Is the question unclear/incorrect or the answer trivial? None of the above, I think. It¹s just hard. I¹ll leave consideration of a <= 0 to you, and suppose a > 0. By symmetry it¹s enough to consider the Žrst quadrant, where we can take y as a function of x for 0 <= x <= 1, y(0) = 1 and y(1) = 0. It looks to me like when we decrease a, the Žrst place y¹¹ becomes positive would be where x=y, where (2 x^2)^a + x^4 = 1. I think the answer is the value of a in the solution of the system of equations (2 x^2)^a + x^4 = 1 -a + (a + 2) x^4 = 0 which is approximately 0.21406286037879413377. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: question for math teachers > As a highly-qualiŽed math teacher by the federal standards working on my > national credential, I have to agree with Mr. Rubin. Obviously, not every > math teacher is ignorant in the nuances of mathematics but a vast majority > of teachers do not understand the concepts themselves. I was surprised to Žnd future high school teachers who could not read any mathematics, no matter how simple, only lesson plans. That¹s the educational equivalent of I¹m not a doctor but I play one on TV and of talking heads reading teleprompters. Why not just hire out of work actors so that they will at least have a day gig? Wasn¹t it wonderful when, in the movie Pi, the actor recited: They must have tried all the 200 digit numbers by now.? -- Allan Adler * Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions and * comments do not režect in any way on MIT. Also, I am nowhere near Boston. === Subject: Re: question for math teachers >Try again. Under No Child Left Behind, a highly qualiŽed teacher has a >graduate degree in their Želd or had taken a test showing subject knowledge >such as the Praxis or SSAT. What state are you credentialed in where you >get a full credential with that background? In California, that would get >you a credential to teach introductory mathematics (only through middle >school), a far cry from being highly qualiŽed. Actually, the wording is bachelor degree or bacalaureate degree ... not graduate degree. G C === Subject: Re: question for math teachers X-RFC2646: Format=Flowed; Original >>Try again. Under No Child Left Behind, a highly qualiŽed teacher has a >>graduate degree in their Želd or had taken a test showing subject >>knowledge >>such as the Praxis or SSAT. What state are you credentialed in where you >>get a full credential with that background? In California, that would get >>you a credential to teach introductory mathematics (only through middle >>school), a far cry from being highly qualiŽed. > Actually, the wording is bachelor degree or bacalaureate degree ... > not > graduate degree. > G C Yes and no. The federal requirements have 3 criteria for being highly qualiŽed: 1) Bachelor¹s degree in the Želd (or reasonably close Želd) being taught 2) A state credential 3) Proof of subject knowledge Most states accept a graduate degree or a test to satisfy the third criteria. An undergrad degree only satisŽes the Žrst criterium. BTW: this info is from the US Dept of Ed site. === Subject: Re: question for math teachers X-CompuServe-Customer: Yes X-Coriate: interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: George Cox X-Punge: Micro$oft X-Sanguinate: The MVS Guy X-Terminate: SPA(GIS) X-Tinguish: Mark GrifŽth X-Treme: C&C,DWS >Before generalize what teachers know and don¹t know step into my >classroom. I am a highly qualiŽed math teacher, FSVO. >in my state that means that in addition to taking all of my education >classes that I also had to take enough math classes to be 3 credits >away from a B.A. in math. That¹s hardly highly qualiŽed. >You also make assumptions about the validity of multiple choice >testing. Is it being used as the primary source of assessment in >schools or is it used along with other methods as a tool to assess >student performance. It¹s being used in mandatory testing and teachers are teaching to the tests instead of to the curriculum. >Please feel free to walk a mile in my shoes before you make >generalizations about math teachers. >Please excuse my assumption if it is wrong, but from your address at >Purdue University I am presuming that you teach statistics. If my >college professors spent part of their time learning how to be >educators instead of researchers, then they would have been much >more effective at teaching. No doubt, but were you at Purdue and would taking classes from the schools of education have made them better educators? There¹s also a question[1] of student attitude; you can¹t teach people who don¹t wish to be taught. It would be of more value for both of you to walk a mile in the shoes of the students. You¹re supposed to be there for their beneŽt. [1] Rembering a friend in graduate school who had to teach Differential Equations for Engineers to a bunch of students who wanted him to skip[2] the material on Linear Algebra because it wasn¹t relevant. [2] He said that letting them žunk wasn¹t an option. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org === Subject: Re: .99999... still=/= 1 posting-account=AE-QyQ0AAAC84T96q9_yI_Fj9ThoZQPi >I won this debate years ago. So what is the reason to start it again? === Subject: Re: .99999... still=/= 1 posting-account=AE-QyQ0AAAC84T96q9_yI_Fj9ThoZQPi > At one digit less than oo, ( assuming you really reach inŽnity) the nth > term reaches 0 so, .999... reaches 0 not 1. And what is the decimal digit of pi at the inŽnity? > Even if you go past oo by one digit, it still doesn¹t reach 1. In which direction the wind blows at that point? === Subject: Re: .99999... still=/= 1 >> At one digit less than oo, ( assuming you really reach inŽnity) >the nth >> term reaches 0 so, .999... reaches 0 not 1. >And what is the decimal digit of pi at the inŽnity? I already answered that. First of all pi ISSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS SSSS an irrational number, the decimal values vary. It can not be determined. But with a repeating decimal, you can use a non-standard approach and use one digit less than oo then, n-->oo -1 lim 9/10^n ---> 90/10^oo ^ | last digit seen is zero right before inŽnity. It never reaches 1. >> Even if you go past oo by one digit, it still doesn¹t reach 1. >In which direction the wind blows at that point? Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 >> At one digit less than oo, ( assuming you really reach inŽnity) >the nth >> term reaches 0 so, .999... reaches 0 not 1. >And what is the decimal digit of pi at the inŽnity? > I already answered that. First of all pi > ISSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS SSSS an > irrational number, the decimal values vary. It can not be determined. But with > a repeating decimal, you can use a non-standard approach and use one digit less > than oo then, > n-->oo -1 > lim 9/10^n ---> 90/10^oo > ^ > | > last digit seen is zero right before inŽnity. It never reaches 1. There is no right before inŽnity. === Subject: Re: .99999... still=/= 1 > At one digit less than oo, ( assuming you really reach inŽnity) >>the nth > term reaches 0 so, .999... reaches 0 not 1. >>And what is the decimal digit of pi at the inŽnity? >> I already answered that. First of all pi >> ISSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS SSSS an >> irrational number, the decimal values vary. It can not be determined. But >with >> a repeating decimal, you can use a non-standard approach and use one digit >less >> than oo then, >> n-->oo -1 >> lim 9/10^n ---> 90/10^oo >> ^ >> | >> last digit seen is zero right before inŽnity. It never reaches 1. >There is no right before inŽnity. I said the digit right before oo. REFERENCE MATHCAD PROFESSIONAL Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > | > last digit seen is zero right before inŽnity. It never reaches 1. There is no right before inŽnity, numbskull. Your comprehension of mathematics is inŽtesimally small. A hyperreal equivalent of 0. Bob Kolker === Subject: Re: .99999... still=/= 1 >> | >> last digit seen is zero right before inŽnity. It never reaches 1. >There is no right before inŽnity, numbskull. Hey, there are probably about 1 million supporters using MathCAD Professional. This is an industry standard for math for scientists and engineers. A non-standard approach using MathCAD clearly shows that, n-->oo -1 lim 9/10^n ---> 90/10^n The digit right before oo for the hyperreal number or series .999... is 0. Therefore, there is a space existing between, .999... and 1 so, as the deŽnition of a hyper-real number implies, .999... < 1 A hyper-real number causes a space to exist between it and a real number. 1 is the real number .999... is the hyper-real number that forms the space between the two. So, .999... =/= 1 .999... < 1 >Your comprehension of mathematics is inŽtesimally small. A hyperreal >equivalent of 0. >Bob Kolker Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 In sci.math, S. Enterprize Company > last digit seen is zero right before inŽnity. It never reaches 1. >>There is no right before inŽnity, numbskull. > Hey, there are probably about 1 million supporters using MathCAD > Professional. This is an industry standard for math for scientists and > engineers. A non-standard approach using MathCAD clearly shows that, > n-->oo -1 > lim 9/10^n ---> 90/10^n > The digit right before oo for the hyperreal number or > series .999... is 0. Therefore, there is a space existing between, > .999... and 1 so, > as the deŽnition of a hyper-real number implies, > .999... < 1 An interesting notion, that. So D[.999..., w-1] = 0, eh? [*] What is D[.999..., w-2]? How about D[.999..., w/2]? It¹s easily proven that, if D[.999..., n] = 9, then D[.999...., n+1] = 9 as well (the simplest method arguably is to evaluate D[x*10, n]), for any Žnite n. Not sure if w-1 is Žnite or not -- or even meaningful. As for MathCAD: that¹s a program, an approximation of reality. Not that real numbers are all that real, anyway -- they¹re mathematical/symbolic abstractions, there because Dedekind, Cauchy, and Cantor and others needed more numbers for set theory. In light of what I¹ve written before regarding 1/3, one might have to verify the results carefully. > A hyper-real number causes a space to exist between it and a real number. > 1 is the real number > .999... is the hyper-real number that forms the space between the two. > So, > .999... =/= 1 > .999... < 1 Your logic is extremely sloppy, though your conclusion is interesting. I¹m just not sure which realm it exists in, although the standard real realm does not contain it (the standard realm doesn¹t contain any numbers between 0 and all 1/n, n > 0, n in J: the hyperreal realm, however, does). [.sigsnip] [*] I don¹t have an omega, so I¹m using Œw¹ here to indicate the Žrst transŽnite ordinal. Is there a w_0, analogous to the cardinal aleph_0? This gets a bit messy. D[r,n] = the digit associated with the n¹th decimal place after the decimal point (e.g., D[.98765, 4] = 6). -- #191, ewill3@earthlink.net It¹s still legal to go .sigless. === Subject: Re: .99999... still=/= 1 X-RFC2646: Format=Flowed; Original >> Yes the limit is exactly 1 in reals. > And also in the hyperreals. >> Yes, if you count the limit over reals plus over hyperreals. Sorry - I >> cannot Žnd better word than over. Find a better word as your mother >> language is English. >> I try to specify: >> As you count the limit in reals, then N --> oo, where every N is Žnite >> integer. As you count the limit over hyperreals, then N_inf -->oo_inf, >> where >> oo_inf has higher cardinality as the cardinality of N_inf >N. > No, this is not about cardinality at all. The inŽnitely large integers > of NSA are not the same as transŽnite cardinals. It¹s necessary to observe: As the cardinality of reals R > cardinality of integers N, so the cardinality of N_inf > N. As you count the limit N --->oo instead of N-inf, then you certainly omit something - namely hyperreal part. >> N (Žnite integers) does not cover hyperreal area, i.e. the numbers >> smaller >> than reals, because N hardly covers real area as discussed under the >> thread >> Are reals well-ordered. >> As we count the real limit in NSA, then the hyperreals exist, but they >> are >> not counted in limit as N -->oo, but not as N_inf -->oo. Hyperreals are >> omitted and therefore 0.999...<1 in hyperreals, though the real part of >> the >> limit equals to 1. > I take it you mean the standard part of the limit is 1. Well, we can talk about standard part and non-standard part, if you prefer this. > That happens to > be true, but for a trivial reason. The limit itself is exactly 1, and > the standard part of 1 is simply 1. Cases: 1)Yes, the limit is exactly 1only if you count the limit including the non standard part as N-inf --->oo. Then your reference set is N_inf. 2)Yes, the limit is exactly 1only if you count the limit including the standard part as N --->oo. Then your reference set is N. 3) But..., if you count the limit including the standard part as N --->oo and your reference set is N_inf, then you omit non-standard part. As a consequence in this last case 0.999...<1 in N_inf. >> NSA expands the concept of numbers to >> the numbers that are smaller than any real, i.e epsilon environment. >> This >> is >> equivalent with the concept of epsilon delta theorem. Read literally >> what >> epsilon delta theorem says. This was learnt us already in 70«s in >> university. Are you back in 50¹s? > Which part of my statement do you not accept? Do you disagree with the > deŽnition I gave? >> Explained above. > Try again. You didn¹t mention any part of the deŽnition, let alone say > which part you disagreed with. > DeŽnition. Let { a_k } be a sequence and let L be a real number. > We say lim_{k->oo} a_k = L if, for every epsilon > 0, there exists > N > 0 such that | a_k - L | < epsilon for every k > N. > Remark 1. Exactly the same deŽnition applies to standard analysis and > to nonstandard analysis, with the proviso that in NSA the epsilon > 0 > is allowed to be an inŽnitesimal and the N > 0 is allowed to be > inŽnitely large. Yes, agreed. > Remark 2. The deŽnition does not say what it means for the limit to be > close to L. The deŽnition only says what it means for the limit to be > equal to L. Either the deŽnition is satisŽed, or it isn¹t. OK. > Now, the questions: (1) Do you agree with the deŽnition? (2) Do you > agree that according to this deŽnition the limit is exactly 1, even in > NSA? If you don¹t agree, explain why not. Yes, I do agree as explained above in the cases 1 and 2. You did not consider at all the case 3 above. How does your deŽnitions should be applied on the case 3? You should also note that the limit is the upper boundary value. As you will see below, it¹s not the question about the limit but about AC and the point of reference as we construct the numbers integers, inŽnite integers, reals, hyperreals etc. > For example, try to give a > particular value of epsilon > 0 such that the deŽnition is not > satisŽed. Hint: choosing an inŽnitesimal epsilon is allowed, but it > won¹t help your case. The deŽnition still works. What about case 3 above as you stop epsilons in standard part omitting non-standard part. >> I would like to ask the same from You. :-). Is it the point >> of reference that is strange concept for You? > Point of reference is an undeŽned concept and does not appear in the > deŽnition of limit, quoted above. I won¹t comment on whether it is > strange, since things have to be deŽned Žrst before they can possibly > qualify as strange. The point of reference is counting point reference, which also separates inŽnities. The standard point of reference is the normal decimal dot that separates the integer part and the decimal part, which can be inŽnite long string. Without the point of reference you do not know which part is integer part and which one is the decimal part. What ever you calculate you always refer your calculations to some point of reference. >> In fact, there seems to be a slight conceptual difference in our >> argumentation. The difference is analocigally the same as we talk about >> Žnite decimal numbers. You accept that Žnite 0.99999 <1, as there are >> no >> inŽnite 9¹s. As hyperreals are omitted in the real limit counting (not >> counted over hyperreals), then in reals the limit equals to 1, i.e. >> 0.999...=1. But as hypereals exist, but omitted in the limit calculation, >> then 0,999...<1 in NSA. Simple as possible. Reals are Žnite from the >> hypereal point of view. Therefore 0.999...<1 - in hyperreals as only real >> area is considered. > But in NSA there is no such thing as a .999... that extends only through > Žnite digit positions. If the string is inŽnitely long, then it > necessarily extends to inŽnite digit positions in NSA. That may sound > vaguely similar to an argument commonly made by cranks, but the > difference is that the cranks are not talking about NSA. used the case 3 as an example to point out how the people do not think as their argumentation contains hidden asumptions. So - if there exist non-standard part, which is purposely omitted, then you do not calculate the total or over-all limit. In this case you have to observe that there are numbers smaller than any real number and as a consequence 0.999... cannot be equal to 1, though the limit equals to 1. What is this paradox, which has been the reason to this thread, and how do we Žx it? The solution is to point out that the limit is a mathematical upper boundary value, but does not tell the real value of the string but the next, i.e. successor, value of the string, because the limit calculation is based on the epsilon delta theorem. > The deŽnition of a limit in NSA may be stated in various ways, but all > of them are equivalent to the usual epsilon-delta deŽnition, with the > proviso that epsilons and deltas are allowed to be inŽnitesimal, and N > is allowed to be inŽnitely large. >> Yes, that¹s what I mean - too, assuming we mean the same thing. If you >> limit >> your calculation into the real area and omit the hyperreal area, > You can¹t do that. That¹s the whole point. Let«s demontrate it shortly now so simply as possible (like Donald Knuth 1) It is assumed that the digits [0,1...9] in 10-base system are constructed. I leave it for the home work. 2) Then we apply AC so that we have inŽnite many placeholders (or hooks) with the Žrst one and the last one (the start and the end), just like in [0,1]. (by the way this stops the discussion about the last digit in the inŽnite decimal string. :-)) For us it¹s enough that is just possible. 3) Pick-up with the aid of AC some digits in every placeholder. What number do you have? Actually you have just digits, but not a number - yet. Why? Because you have not deŽned the point of reference! You have only digits in every place holder, i.e. in every hook. 4) Adjust the digits, you have picked up by hooks, into a linear string. It¹s inŽnite, but it has the start and the end. Additionally we deŽne (in this particular 10 base case) what is the relation of the adjacent placeholders. What number do you have now? No number, because you just have the linear inŽnite string without the point of reference. The point of reference deŽnes the inŽnite string area, but we do not have it yet, thus as a consequence - no number area. 5) Apply the the point of reference. How? You can insert the point of reference anywhere in the string, but let¹s concentrate our attention into two special case because of simplicity: to the left hand side (lhs) of the inŽnite string or to the right hand side (rhs). What kind of point of reference? Any deŽned kind, but let¹s apply because of the simplicity the standard point of reference that is called the decimal dot (or mark). Let¹s insert the standard point of reference into lhs of the string. What number do you have? It is familiar decimal number depending on the values of digits in the string. Is it rational, irrational or transcendental depending on the digits. If there are only zeros on rhs starting from some placeholders, then the decimal part is called Žnite. Let¹s insert the standard point of reference into rhs of the string. What number do you have? It looks like now an integer, but isn¹t it integer?. It¹s inŽnite long with start and end. It¹s somehow Žnite, but anyway inŽnite. Integers cannot be inŽnite? Yes, they can. Actually every classic Žnite integer (N) can be written as inŽnite, if there are only zeros on lhs starting from some placeholders. In other case we call them inŽnite integers (N_inf). As an observation we recognize immediately that there is one-to-one bijection between N_inf and decimal part of R. The only difference is that N_inf has the standard point of reference on rhs and R (decimal part) has the standard point of reference on lhs. The standard point of reference, i.e. the decimal dot separates to two inŽnite long strings. Both parts are constructed by AC. The only difference is the point of reference on lhs or on rhs. As Žnite integers (N) are the subset of inŽnite integers (N_inf). AC, Zorn¹s lemma and well-ordering are equivalent. Integers and inŽnite integers are well-ordered. By moving the point of reference from rhs (in N_inf) to lhs (in R, decimal part) it¹s trivial to recognize that the decimal part is also well-ordered. Considering the limit, it¹s easy to observe that if the point of reference is on lhs of the string of inŽnite 9¹s, i.e. 0.999..., the limit equals to 1. But if the point of the reference is on rhs of the same string, then there is no limit, because we cannot calculate it for N_inf like ...999. We can add 1 to ...999, but then - as all the placeholders were occupied with 9¹s in the inŽnite long string - the successor equals to omega 1. Thus the limit, i.e. the upper boundary value, describes the successor, not the sum of the string itself as it should count. Successor equals never with the precessor and therefore omega 1> ...999 and also 1>0.999... The reason is AC and well-ordering. Thus for example a general number can be described omega-area (separator) inŽnite integer area (separator, usually dot), decimal area (separator) non-standard area. Each area is inŽnite and well-ordered constructed with the aid of AC. We don¹t have to use the standard approach one Žrst deŽnes Z, then deŽnes Q, then deŽnes R, each in different ways. Dedekind cut is not necessary but useful, which is another thing. All we need is AC and the point of reference. Tapio > -- > Dave Seaman > Judge Yohn¹s mistakes revealed in Mumia Abu-Jamal ruling. > === Subject: Re: .99999... still=/= 1 > Yes, if you count the limit over reals plus over hyperreals. Sorry - I > cannot Žnd better word than over. Find a better word as your mother > language is English. > I try to specify: > As you count the limit in reals, then N --> oo, where every N is Žnite > integer. As you count the limit over hyperreals, then N_inf -->oo_inf, > where > oo_inf has higher cardinality as the cardinality of N_inf >N. >> No, this is not about cardinality at all. The inŽnitely large integers >> of NSA are not the same as transŽnite cardinals. > It¹s necessary to observe: As the cardinality of reals R > cardinality of > integers N, so the cardinality of N_inf > N. As you count the limit > N --->oo instead of N-inf, then you certainly omit something - namely > hyperreal part. All very confused. Assuming you mean N_inf = *N, the set of hyperintegers, then it¹s true that *N as an external set has the cardinality of the reals, but the internal set *N has a *cardinality equal to *aleph_0. This certainly doesn¹t mean that the hypernaturals are the same as transŽnite cardinals. For one thing, there is a smallest transŽnite cardinal (aleph_0), but there is no such thing as a smallest inŽnite hypernatural. If n is an inŽnitely large integer, then so is n-1. And none of this has anything to do with limits. You seem to think that inŽnity has only one meaning in all of mathematics. Wrong. > N (Žnite integers) does not cover hyperreal area, i.e. the numbers > smaller > than reals, because N hardly covers real area as discussed under the > thread > Are reals well-ordered. > As we count the real limit in NSA, then the hyperreals exist, but they > are > not counted in limit as N -->oo, but not as N_inf -->oo. Hyperreals are > omitted and therefore 0.999...<1 in hyperreals, though the real part of > the > limit equals to 1. >> I take it you mean the standard part of the limit is 1. > Well, we can talk about standard part and non-standard part, if you prefer > this. I¹m trying to guess what you might mean by the real part, since you have not deŽned your meaning. We are not talking about complex numbers here. >> That happens to >> be true, but for a trivial reason. The limit itself is exactly 1, and >> the standard part of 1 is simply 1. > Cases: > 1)Yes, the limit is exactly 1only if you count the limit including the non > standard part as N-inf --->oo. Then your reference set is N_inf. I don¹t know what you mean by the reference set is N_inf. The sum is over *N. > 2)Yes, the limit is exactly 1only if you count the limit including the > standard part as N --->oo. Then your reference set is N. Are you talking about standard analysis here, or nonstandard? The set N (consisting of the Žnite naturals) is not a set in the internal set theory of NSA. Its counterpart is *N, which includes the inŽnitely large naturals. If you are forming a sum in NSA, then you can sum over *N, but not over N. > 3) But..., if you count the limit including the standard part as N --->oo > and your reference set is N_inf, then you omit non-standard part. As a > consequence in this last case 0.999...<1 in N_inf. That¹s all very confused, but I think you are trying to say that the limit of the sequence { 1 - 1/10^n } in NSA has a nonzero nonstandard part. False. The limit is exactly 1 according to the deŽnition I gave previously. Perhaps what is confusing you is the following fact from NSA: Theorem. Let { a_k } be a sequence and L a real number. Then the following statements are equivalent: (1) lim_{k->oo} a_k = L. (2) The difference | a_k - L | is an inŽnitesimal whenever k is inŽnitely large. Notice, however, that neither (1) nor (2) says anything about the limit differing from L by an inŽnitesimal. Statement (1) mentions the limit, but it¹s a statement of exact equality. Statement (2) mentions something that differs from L by an inŽnitesimal, but it makes no mention of the limit whatsoever. Either way you look at it, neither of these statements supports your conclusion for the case a_k = 1 - 1/10^k and L = 1. > NSA expands the concept of numbers to > the numbers that are smaller than any real, i.e epsilon environment. > This > is > equivalent with the concept of epsilon delta theorem. Read literally > what > epsilon delta theorem says. This was learnt us already in 70«s in > university. Are you back in 50¹s? >> Which part of my statement do you not accept? Do you disagree with the >> deŽnition I gave? > Explained above. >> Try again. You didn¹t mention any part of the deŽnition, let alone say >> which part you disagreed with. >> DeŽnition. Let { a_k } be a sequence and let L be a real number. >> We say lim_{k->oo} a_k = L if, for every epsilon > 0, there exists >> N > 0 such that | a_k - L | < epsilon for every k > N. >> Remark 1. Exactly the same deŽnition applies to standard analysis and >> to nonstandard analysis, with the proviso that in NSA the epsilon > 0 >> is allowed to be an inŽnitesimal and the N > 0 is allowed to be >> inŽnitely large. > Yes, agreed. >> Remark 2. The deŽnition does not say what it means for the limit to be >> close to L. The deŽnition only says what it means for the limit to be >> equal to L. Either the deŽnition is satisŽed, or it isn¹t. > OK. >> Now, the questions: (1) Do you agree with the deŽnition? (2) Do you >> agree that according to this deŽnition the limit is exactly 1, even in >> NSA? If you don¹t agree, explain why not. > Yes, I do agree as explained above in the cases 1 and 2. You did not > consider at all the case 3 above. How does your deŽnitions should be > applied on the case 3? There is no case 3 above. I don¹t know what you are talking about. > You should also note that the limit is the upper boundary value. As you will > see below, it¹s not the question about the limit but about AC and the point > of reference as we construct the numbers integers, inŽnite integers, reals, > hyperreals etc. >> For example, try to give a >> particular value of epsilon > 0 such that the deŽnition is not >> satisŽed. Hint: choosing an inŽnitesimal epsilon is allowed, but it >> won¹t help your case. The deŽnition still works. > What about case 3 above as you stop epsilons in standard part omitting > non-standard part. There is no case 3 above. I have explained that the sum over N is not a sum in NSA, since N is not an internal set in NSA. That¹s why we sum over *N instead. > I would like to ask the same from You. :-). Is it the point > of reference that is strange concept for You? >> Point of reference is an undeŽned concept and does not appear in the >> deŽnition of limit, quoted above. I won¹t comment on whether it is >> strange, since things have to be deŽned Žrst before they can possibly >> qualify as strange. > The point of reference is counting point reference, which also separates > inŽnities. That is not a deŽnition. For an example of what I mean by a deŽnition, look at my deŽnition of what it means to say that lim_{k->oo} a_k = L. Mathematical deŽnitions leave no room for fuzzy language or vague concepts. Try again. > The standard point of reference is the normal decimal dot that separates the > integer part and the decimal part, which can be inŽnite long string. > Without the point of reference you do not know which part is integer part > and which one is the decimal part. What ever you calculate you always refer > your calculations to some point of reference. The common deŽnitions of the real numbers (via Dedekind cuts or Cauchy sequences) do not mention decimal points at all and do not depend on any such concepts as integer part and decimal part. For any real number x, the integer part of x may be deŽned as žoor(x) = max { n in N : n <= x }. I had no need for any vague concepts such as point of reference in writing that deŽnition. > In fact, there seems to be a slight conceptual difference in our > argumentation. The difference is analocigally the same as we talk about > Žnite decimal numbers. You accept that Žnite 0.99999 <1, as there are > no > inŽnite 9¹s. As hyperreals are omitted in the real limit counting (not > counted over hyperreals), then in reals the limit equals to 1, i.e. > 0.999...=1. But as hypereals exist, but omitted in the limit calculation, > then 0,999...<1 in NSA. Simple as possible. Reals are Žnite from the > hypereal point of view. Therefore 0.999...<1 - in hyperreals as only real > area is considered. You need to be precise about what you mean by 0.999... in NSA. I have been taking it to mean the sum of 9/10^n for all n in *N, n > 0. You evidently mean something different. In particular, it can¹t possibly mean the sum over all Žnite values of n > 0, because that is not a set in NSA. >> But in NSA there is no such thing as a .999... that extends only through >> Žnite digit positions. If the string is inŽnitely long, then it >> necessarily extends to inŽnite digit positions in NSA. That may sound >> vaguely similar to an argument commonly made by cranks, but the >> difference is that the cranks are not talking about NSA. > used the case 3 as an example to point out how the people do not think as > their argumentation contains hidden asumptions. So - if there exist > non-standard part, which is purposely omitted, then you do not calculate the > total or over-all limit. In this case you have to observe that there are > numbers smaller than any real number and as a consequence 0.999... cannot be > equal to 1, though the limit equals to 1. > What is this paradox, which has been the reason to this thread, and how do > we Žx it? If you do not sum 9/10^n for all n > 0, then you are not computing 0.999.... The indicated sum is exactly 1. > The solution is to point out that the limit is a mathematical upper boundary > value, but does not tell the real value of the string but the next, i.e. > successor, value of the string, because the limit calculation is based on > the epsilon delta theorem. The string I am talking about has no successor values. It¹s already deŽned for all positions n, including the ones that are inŽnitely large. >> The deŽnition of a limit in NSA may be stated in various ways, but all >> of them are equivalent to the usual epsilon-delta deŽnition, with the >> proviso that epsilons and deltas are allowed to be inŽnitesimal, and N >> is allowed to be inŽnitely large. > Yes, that¹s what I mean - too, assuming we mean the same thing. If you > limit > your calculation into the real area and omit the hyperreal area, >> You can¹t do that. That¹s the whole point. > Let«s demontrate it shortly now so simply as possible (like Donald Knuth > 1) It is assumed that the digits [0,1...9] in 10-base system are > constructed. I leave it for the home work. > 2) Then we apply AC so that we have inŽnite many placeholders (or hooks) > with the Žrst one and the last one (the start and the end), just like in > [0,1]. We don¹t need AC here. We are deŽning d_0 = 0 and d_k = 9 for all k > 0 in *N. > (by the way this stops the discussion about the last digit in the inŽnite > decimal string. :-)) For us it¹s enough that is just possible. I was not aware that there was any such discussion. > 3) Pick-up with the aid of AC some digits in every placeholder. Again, AC is irrelevant. We already have our hyperinŽnite string. > What number do you have? Actually you have just digits, but not a number - > yet. Why? Because decimal digit strings are not numbers. They merely represent numbers. > Because you have not deŽned the point of reference! You have only digits in > every place holder, i.e. in every hook. Nonsense. If { d_k } is a decimal digit string, then the number represented by the string is sum_{k in *N} d_k * 10^(-k). No point of reference is needed. [ snip nonsense about point of reference ] -- Dave Seaman Judge Yohn¹s mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: .99999... still=/= 1 X-RFC2646: Format=Flowed; Original >> Yes, if you count the limit over reals plus over hyperreals. Sorry - I >> cannot Žnd better word than over. Find a better word as your mother >> language is English. >> I try to specify: >> As you count the limit in reals, then N --> oo, where every N is Žnite >> integer. As you count the limit over hyperreals, then N_inf -->oo_inf, >> where >> oo_inf has higher cardinality as the cardinality of N_inf >N. > No, this is not about cardinality at all. The inŽnitely large integers > of NSA are not the same as transŽnite cardinals. We talk about ordinals. That above was just a hint to compare the cardinality of N_inf and N. >> It¹s necessary to observe: As the cardinality of reals R > cardinality of >> integers N, so the cardinality of N_inf > N. As you count the limit >> N --->oo instead of N-inf, then you certainly omit something - namely >> hyperreal part. > All very confused. Assuming you mean N_inf = *N, the set of > hyperintegers, then it¹s true that *N as an external set has the > cardinality of the reals, but the internal set *N has a *cardinality > equal to *aleph_0. This is of-course OK! > This certainly doesn¹t mean that the hypernaturals > are the same as transŽnite cardinals. For one thing, there is a > smallest transŽnite cardinal (aleph_0), Ok! > but there is no such thing as a > smallest inŽnite hypernatural. If n is an inŽnitely large integer, > then so is n-1. Omega was deŽned in this discussion earlier as follows: The number that is greater than any inŽnite integer. That is the smallest transinŽnite number. Omega is the successor of ...999, i.e. n+1. The precessor (n-1) of ...999 is ...998. > And none of this has anything to do with limits. You seem to think that > inŽnity has only one meaning in all of mathematics. Wrong. Your opinion. :-) >> N (Žnite integers) does not cover hyperreal area, i.e. the numbers >> smaller >> than reals, because N hardly covers real area as discussed under the >> thread >> Are reals well-ordered. >> As we count the real limit in NSA, then the hyperreals exist, but they >> are >> not counted in limit as N -->oo, but not as N_inf -->oo. Hyperreals are >> omitted and therefore 0.999...<1 in hyperreals, though the real part of >> the >> limit equals to 1. > I take it you mean the standard part of the limit is 1. >> Well, we can talk about standard part and non-standard part, if you >> prefer >> this. > I¹m trying to guess what you might mean by the real part, since you > have not deŽned your meaning. the real part was the decimal part - as you certainly knew. > We are not talking about complex numbers > here. Exactly! > That happens to > be true, but for a trivial reason. The limit itself is exactly 1, and > the standard part of 1 is simply 1. >> Cases: >> 1)Yes, the limit is exactly 1only if you count the limit including the >> non >> standard part as N-inf --->oo. Then your reference set is N_inf. > I don¹t know what you mean by the reference set is N_inf. The sum is > over *N. That is exactly what I meant above if N_inf=*N. >> 2)Yes, the limit is exactly 1only if you count the limit including the >> standard part as N --->oo. Then your reference set is N. > Are you talking about standard analysis here, or nonstandard? Standard analysis as You correctly observed. > The set N > (consisting of the Žnite naturals) is not a set in the internal set > theory of NSA. Coorect, but it should be as *N is the extension of the set N. > Its counterpart is *N, which includes the inŽnitely > large naturals. If you are forming a sum in NSA, then you can sum over > *N, but not over N. We can sum over subset too. >> 3) But..., if you count the limit including the standard part as >> N --->oo >> and your reference set is N_inf, then you omit non-standard part. As a >> consequence in this last case 0.999...<1 in N_inf. > That¹s all very confused, but I think you are trying to say that the > limit of the sequence { 1 - 1/10^n } in NSA has a nonzero nonstandard > part. False. The limit is exactly 1 according to the deŽnition I gave > previously. N is the subset of *N. You can count the limit over N or alternatively over *N as You can count N --> 0 to 1000 as a subset of N or alternatively N --->0 to oo. > Perhaps what is confusing you is the following fact from NSA: > Theorem. Let { a_k } be a sequence and L a real number. Then the > following statements are equivalent: > (1) lim_{k->oo} a_k = L. > (2) The difference | a_k - L | is an inŽnitesimal > whenever k is inŽnitely large. > Notice, however, that neither (1) nor (2) says anything about the limit > differing from L by an inŽnitesimal. Yes, as long as you count over *N > Statement (1) mentions the limit, > but it¹s a statement of exact equality. Statement (2) mentions something > that differs from L by an inŽnitesimal, but it makes no mention of the > limit whatsoever. Except Yount count over N instead of *N. > Either way you look at it, neither of these statements > supports your conclusion for the case a_k = 1 - 1/10^k and L = 1. >> NSA expands the concept of numbers to >> the numbers that are smaller than any real, i.e epsilon environment. >> This >> is >> equivalent with the concept of epsilon delta theorem. Read literally >> what >> epsilon delta theorem says. This was learnt us already in 70«s in >> university. Are you back in 50¹s? > Which part of my statement do you not accept? Do you disagree with > the > deŽnition I gave? >> Explained above. > Try again. You didn¹t mention any part of the deŽnition, let alone say > which part you disagreed with. > DeŽnition. Let { a_k } be a sequence and let L be a real number. > We say lim_{k->oo} a_k = L if, for every epsilon > 0, there exists > N > 0 such that | a_k - L | < epsilon for every k > N. > Remark 1. Exactly the same deŽnition applies to standard analysis and > to nonstandard analysis, with the proviso that in NSA the epsilon > 0 > is allowed to be an inŽnitesimal and the N > 0 is allowed to be > inŽnitely large. >> Yes, agreed. > Remark 2. The deŽnition does not say what it means for the limit to be > close to L. The deŽnition only says what it means for the limit to be > equal to L. Either the deŽnition is satisŽed, or it isn¹t. >> OK. > Now, the questions: (1) Do you agree with the deŽnition? (2) Do you > agree that according to this deŽnition the limit is exactly 1, even in > NSA? If you don¹t agree, explain why not. >> Yes, I do agree as explained above in the cases 1 and 2. You did not >> consider at all the case 3 above. How does your deŽnitions should be >> applied on the case 3? > There is no case 3 above. I don¹t know what you are talking about. >> You should also note that the limit is the upper boundary value. As you >> will >> see below, it¹s not the question about the limit but about AC and the >> point >> of reference as we construct the numbers integers, inŽnite integers, >> reals, >> hyperreals etc. > For example, try to give a > particular value of epsilon > 0 such that the deŽnition is not > satisŽed. Hint: choosing an inŽnitesimal epsilon is allowed, but it > won¹t help your case. The deŽnition still works. >> What about case 3 above as you stop epsilons in standard part omitting >> non-standard part. > There is no case 3 above. I have explained that the sum over N is not > a sum in NSA, since N is not an internal set in NSA. That¹s why we sum > over *N instead. Because You don¹t see N as subset of *N. :-) >> I would like to ask the same from You. :-). Is it the point >> of reference that is strange concept for You? > Point of reference is an undeŽned concept and does not appear in the > deŽnition of limit, quoted above. I won¹t comment on whether it is > strange, since things have to be deŽned Žrst before they can > possibly > qualify as strange. >> The point of reference is counting point reference, which also separates >> inŽnities. > That is not a deŽnition. For an example of what I mean by a deŽnition, > look at my deŽnition of what it means to say that lim_{k->oo} a_k = L. > Mathematical deŽnitions leave no room for fuzzy language or vague > concepts. Try again. >> The standard point of reference is the normal decimal dot that separates >> the >> integer part and the decimal part, which can be inŽnite long string. >> Without the point of reference you do not know which part is integer part >> and which one is the decimal part. What ever you calculate you always >> refer >> your calculations to some point of reference. > The common deŽnitions of the real numbers (via Dedekind cuts or Cauchy > sequences) do not mention decimal points at all and do not depend on > any such concepts as integer part and decimal part. For any real > number x, the integer part of x may be deŽned as žoor(x) = max { n in > N : n <= x }. I had no need for any vague concepts such as point of > reference in writing that deŽnition. Certainly not. It¹s alternative way to construct numbers from AC. The point of reference is in-constructed assumption in Dedekinds cut or Cauchy sequences, because they assumed so or they did not recognize the point of reference. >> In fact, there seems to be a slight conceptual difference in our >> argumentation. The difference is analocigally the same as we talk about >> Žnite decimal numbers. You accept that Žnite 0.99999 <1, as there are >> no >> inŽnite 9¹s. As hyperreals are omitted in the real limit counting (not >> counted over hyperreals), then in reals the limit equals to 1, i.e. >> 0.999...=1. But as hypereals exist, but omitted in the limit >> calculation, >> then 0,999...<1 in NSA. Simple as possible. Reals are Žnite from the >> hypereal point of view. Therefore 0.999...<1 - in hyperreals as only >> real >> area is considered. > You need to be precise about what you mean by 0.999... in NSA. I have > been taking it to mean the sum of 9/10^n for all n in *N, n > 0. You > evidently mean something different. In particular, it can¹t possibly > mean the sum over all Žnite values of n > 0, because that is not a set > in NSA. N is the subset of *N. I and You can count over subset of *N just like in the case of any subset of N. > But in NSA there is no such thing as a .999... that extends only through > Žnite digit positions. If the string is inŽnitely long, then it > necessarily extends to inŽnite digit positions in NSA. That may sound > vaguely similar to an argument commonly made by cranks, but the > difference is that the cranks are not talking about NSA. >> evidence. I >> used the case 3 as an example to point out how the people do not think as >> their argumentation contains hidden asumptions. So - if there exist >> non-standard part, which is purposely omitted, then you do not calculate >> the >> total or over-all limit. In this case you have to observe that there are >> numbers smaller than any real number and as a consequence 0.999... cannot >> be >> equal to 1, though the limit equals to 1. >> What is this paradox, which has been the reason to this thread, and how >> do >> we Žx it? > If you do not sum 9/10^n for all n > 0, then you are not computing > 0.999.... The indicated sum is exactly 1. Shortly: That¹s what we have already agreed. I never denied that. What can be done: over N, over *N or over N as a subset of *N, and only in the last case the limit over N is smaller than 1 as You should count over *N in hyperreals. Therefore in principle 0.999... <1 in the last mentioned case. >> The solution is to point out that the limit is a mathematical upper >> boundary >> value, but does not tell the real value of the string but the next, i.e. >> successor, value of the string, because the limit calculation is based on >> the epsilon delta theorem. > The string I am talking about has no successor values. It¹s already > deŽned for all positions n, including the ones that are inŽnitely > large. > The deŽnition of a limit in NSA may be stated in various ways, but > all > of them are equivalent to the usual epsilon-delta deŽnition, with the > proviso that epsilons and deltas are allowed to be inŽnitesimal, and > N > is allowed to be inŽnitely large. >> Yes, that¹s what I mean - too, assuming we mean the same thing. If you >> limit >> your calculation into the real area and omit the hyperreal area, > You can¹t do that. That¹s the whole point. >> Let«s demontrate it shortly now so simply as possible (like Donald Knuth >> 1) It is assumed that the digits [0,1...9] in 10-base system are >> constructed. I leave it for the home work. >> 2) Then we apply AC so that we have inŽnite many placeholders (or hooks) >> with the Žrst one and the last one (the start and the end), just like in >> [0,1]. > We don¹t need AC here. We are deŽning d_0 = 0 and d_k = 9 for all k > 0 > in *N. >> (by the way this stops the discussion about the last digit in the >> inŽnite >> decimal string. :-)) For us it¹s enough that is just possible. > I was not aware that there was any such discussion. It was earlier and it arises up from time to time, but not in our mutual discussion. >> 3) Pick-up with the aid of AC some digits in every placeholder. > Again, AC is irrelevant. We already have our hyperinŽnite string. >> What number do you have? Actually you have just digits, but not a >> number - >> yet. Why? > Because decimal digit strings are not numbers. They merely represent > numbers. >> Because you have not deŽned the point of reference! You have only digits >> in >> every place holder, i.e. in every hook. > Nonsense. If { d_k } is a decimal digit string, then the number > represented by the string is sum_{k in *N} d_k * 10^(-k). No point of > reference is needed. You cut (=snip) out the climax of my explanation. What does ^(-k) refer? Please answer to that simple question! Why minus - what does it refer? > [ snip nonsense about point of reference ] Your opinion, because You could not consider the successor of ...999. What is the successor of ...999 as all the placeholders are inŽnitely occupied with the maximal digit 9? Tapio > -- > Dave Seaman > Judge Yohn¹s mistakes revealed in Mumia Abu-Jamal ruling. > === Subject: Re: .99999... still=/= 1 > And why isn¹t .999... a real number? It isn¹t an irrational, according >>.999... is not anything. It is a jumble of symbols without a proper >>mathematical deŽnition. If this is supposed to be graŽtti for a series >>then you should tell us what the general term is. >Since it¹s the one that is supposed to be 1, the general term is: >9/(10^i), for i=1 to inŽnity. >As if you didn¹t know. >John Savard >http://home.ecn.ab.ca/~jsavard/index.html So if someone tell you to believe in the tooth fairy that exists to in math? 9/9 =/= .999.... 9/10 =/= .999... .999... can¹t be expressed as a fraction so it¹s not rational. It¹s not an irrational number either. So, A number that isn¹t real can¹t equal a number that is real. Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 >.999... isn¹t a real number. The number 1 is a real number. And why isn¹t .999... a real number? It isn¹t an irrational, according to the AOL dictionary. It can still be rational. And 1 is rational too. John Savard http://home.ecn.ab.ca/~jsavard/index.html === Subject: Re: .99999... still=/= 1 > And why isn¹t .999... a real number? It isn¹t an irrational, according .999... is not anything. It is a jumble of symbols without a proper mathematical deŽnition. If this is supposed to be graŽtti for a series then you should tell us what the general term is. Bob Kolker === Subject: Re: .99999... still=/= 1 >> And why isn¹t .999... a real number? It isn¹t an irrational, according >.999... is not anything. It is a jumble of symbols without a proper >mathematical deŽnition. If this is supposed to be graŽtti for a series >then you should tell us what the general term is. >Bob Kolker You¹re not even in agreement with the others that say, 9/9 = .999... Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > You¹re not even in agreement with the others that say, > 9/9 = .999... That is right. .999... is scribble and not a mathematical experession. I write down series, not scribble. Bob Kolker === Subject: Re: .99999... still=/= 1 >> You¹re not even in agreement with the others that say, >> 9/9 = .999... >That is right. .999... is scribble and not a mathematical experession. I >write down series, not scribble. >Bob Kolker What¹s wrong you can admit .999... =/= 1 I won this debate years ago. HAHAHAHAHAHHAHAHAHAHHAHAH Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > As if you didn¹t know. Of course. But I am trying to discourage the use of the idiom (or graŽtto) .999... which only discourages S. Enterprise from excercising all 23 of his neurons. Bob Kolker === Subject: Re: .99999... still=/= 1 >> As if you didn¹t know. >Of course. But I am trying to discourage the use of the idiom (or >graŽtto) .999... which only discourages S. Enterprise from excercising >all 23 of his neurons. >Bob Kolker Look at this dude, he says, .999... is meaningless. Then why apply Partial Sums to something meaningless? Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 >As if you didn¹t know. >>Of course. But I am trying to discourage the use of the idiom (or >>graŽtto) .999... which only discourages S. Enterprise from excercising >>all 23 of his neurons. >>Bob Kolker > Look at this dude, he says, .999... is meaningless. Then why apply Partial > Sums to something meaningless? You mean like SUM (n >= 1) [9/10^n] ? That is a series exprressed in ascii. It is a proper mathematical expression. Bob Kolker === Subject: Re: .99999... still=/= 1 >>As if you didn¹t know. >Of course. But I am trying to discourage the use of the idiom (or >graŽtto) .999... which only discourages S. Enterprise from excercising >all 23 of his neurons. >Bob Kolker >> Look at this dude, he says, .999... is meaningless. Then why apply >Partial >> Sums to something meaningless? >You mean like SUM (n >= 1) [9/10^n] ? That is a series exprressed in >ascii. It is a proper mathematical expression. >Bob Kolker So you have to add all types of other things to, .999.... to make it converge? What about just as it stands? .999... =/= =/= =/= =/= =/= =/= 1 Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 >Of course. But I am trying to discourage the use of the idiom (or >graŽtto) .999... which only discourages S. Enterprise from excercising >all 23 of his neurons. So you don¹t think he would be able to pilot an F-22, then? John Savard http://home.ecn.ab.ca/~jsavard/index.html === Subject: Re: .99999... still=/= 1 >>Of course. But I am trying to discourage the use of the idiom (or >>graŽtto) .999... which only discourages S. Enterprise from excercising >>all 23 of his neurons. >So you don¹t think he would be able to pilot an F-22, then? >John Savard >http://home.ecn.ab.ca/~jsavard/index.html So instead of admitting your mistakes, you make goofy jokes to change the subject???? How goofy can you get? Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 >>.999... isn¹t a real number. The number 1 is a real number. >And why isn¹t .999... a real number? It isn¹t an irrational, according >to the AOL dictionary. It can still be rational. And 1 is rational too. >John Savard >http://home.ecn.ab.ca/~jsavard/index.html A real number is either a rational number or irrational number. A number that can be expressed as a fraction is a rational number, otherwise it has to be expressed as a decimal. 3/4 <-- this is a real rational number. sqrt (2) = 1.41421... <--- this is an irrational number. There is no fraction that can represent 1.4142.... .999... isn¹t irrational because it has repeating digits to the right of the decimal point. 1.41421... <--- has non-repeating decimals to the right of the decimal point. This is an irrational real number. .999... is an indeterminate because it isn¹t classiŽed as rational or irrational, and you can¹t determine the space between this number to the point where it reaches a real number. In other words, this exists: .999... | | 1 --->| |<--- This space makes this number an indeterminate. A non-standard analysis has to be made here to determine either a convergence or some time dependent value that approaches 1. Using the gamma function you can observe that the more time you spend approaching 1 ( a real number), the closer you get to 1, but it never reaches it. So it can never equal that number. An besides, even if we look at .999... as an irrational number, it can¹t equal a rational number. Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 >.999... isn¹t irrational because it has repeating digits to the right of the >decimal point. Right. >.999... is an indeterminate because it isn¹t classiŽed as rational or >irrational, and you can¹t determine the space between this number to the point >where it reaches a real number. No. .99999..... is the rational number 9/9. .999999999999999999... ______________________ 9 ) 9.000000000000000000 8.1 --- .900000000000000000 .810000000000000000 --- .090000000000000000 .081000000000000000 ----- .009000000000000000 .008100000000000000 It doesn¹t fail to be a rational number just because you say so. John Savard http://home.ecn.ab.ca/~jsavard/index.html === Subject: Re: .99999... still=/= 1 >>.999... isn¹t irrational because it has repeating digits to the right of the >>decimal point. >Right. >>.999... is an indeterminate because it isn¹t classiŽed as rational or >>irrational, and you can¹t determine the space between this number to the >point >>where it reaches a real number. >No. >.99999..... is the rational number 9/9. Hey can you even see the difference between 1 and .9999999999999999999999999999999999999999999999999999999999999 9999999999 99999999999999999999999999999999999999999999999999999999999999 999999999999 99999................................ > .999999999999999999... > ______________________ >9 ) 9.000000000000000000 > 8.1 > --- > .900000000000000000 > .810000000000000000 > --- > .090000000000000000 > .081000000000000000 > ----- > .009000000000000000 > .008100000000000000 >It doesn¹t fail to be a rational number just because you say so. >John Savard >http://home.ecn.ab.ca/~jsavard/index.html Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 nd > .9999999999999999999999999999999999999999999999999999999999999 9999999999 > 99999999999999999999999999999999999999999999999999999999999999 999999999999 > 99999................................ This is scribble. It is not a mathematical experession. Bob Kolker === Subject: Re: .99999... still=/= 1 >> .9999999999999999999999999999999999999999999999999999999999999 9999999999 >> 99999999999999999999999999999999999999999999999999999999999999 999999999999 >> 99999................................ >This is scribble. It is not a mathematical experession. >Bob Kolker An expanded form of .999.... was as shown. .999... =/= 1 I won the debate. You can even just look at it with your own two eyes. 1 is different looking than .9999999999999999999999999999999999999999999999999999999999999 999999999999 99999999999999999999999999999999999999999999999999999999999999 999999999999 999999-->oo Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > A real number is either a rational number or irrational number. A number that > can be expressed as a fraction is a rational number, otherwise it has to be > expressed as a decimal. A rational number n/m where n, m are integers, m != 0, n < m, n, m realtively prime is expressible as an inŽnite series Sum [n >= 1] (a_n * 1/10^n) where 0 <= a_n <= 9 and the sequence {a_n} is periodic (repeating) after the k-th term where k is some integer >= 1. This is a fairly straightforward theorem in number theory. Consult any number theory or algebra text for the proof. All such series equal the ratio of integers, hence are rational numbers. Bob Kolker === Subject: Re: .99999... still=/= 1 >> A real number is either a rational number or irrational number. A number >that >> can be expressed as a fraction is a rational number, otherwise it has to be >> expressed as a decimal. >A rational number n/m where n, m are integers, m != 0, n < m, n, m >realtively prime is expressible as an inŽnite series >Sum [n >= 1] (a_n * 1/10^n) where 0 <= a_n <= 9 and the sequence >{a_n} is periodic (repeating) after the k-th term where k is some >integer >= 1. >This is a fairly straightforward theorem in number theory. Consult any >number theory or algebra text for the proof. >All such series equal the ratio of integers, hence are rational numbers. But you said .999... is meaningless. You seem to bbe very confused. >Bob Kolker Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 >>All such series equal the ratio of integers, hence are rational numbers. > But you said .999... is meaningless. You seem to bbe very confused. I said the -series- corresponding to an ulimately periodic decimal expansion equals a rational number. Learn to read. The locution .999.... is meaningless scribble. Sum [n >= 1] (a_n * 1/10^n) where 0 <= a_n <= 9 and the sequence {a_n} is periodic (repeating) after the k-th term where k is some integer >= 1 is a rational number Bob Kolker === Subject: Re: .99999... still=/= 1 >All such series equal the ratio of integers, hence are rational numbers. >> But you said .999... is meaningless. You seem to bbe very confused. >I said the -series- corresponding to an ulimately periodic decimal >expansion equals a rational number. Learn to read. Learn to think. READ THE MATH deŽnition of a real number. .999... isn¹t a real number. It is in the class of a hyper-real number. >The locution .999.... is meaningless scribble. .999... is NOT MEANINGLESS. It is A NUMBER. It is related to a hyper-real number. the rest he doesn¹t even know math fundamentals. >Sum [n >= 1] (a_n * 1/10^n) where 0 <= a_n <= 9 and the sequence >{a_n} is periodic (repeating) after the k-th term where k is some >integer >= 1 is a rational number >Bob Kolker Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 posting-account=AE-QyQ0AAAC84T96q9_yI_Fj9ThoZQPi > .999... is NOT MEANINGLESS. It is A NUMBER. It is related to a hyper-real > number. Are you really sure that .999... is a hyperreal number and not surreal(*)? (*) http://mathworld.wolfram.com/SurrealNumber.html http://en.wikipedia.org/wiki/Surreal_number === Subject: Re: .99999... still=/= 1 >> .999... is NOT MEANINGLESS. It is A NUMBER. It is related to a >hyper-real >> number. >Are you really sure that .999... is a hyperreal number and not >surreal(*)? I would say they are very closely related. With a surreal number, you in effect move one decimal step back away from oo, between .999... and 1, where 1 is located and Žnd a terminating value equal to zero. The surreal number is, {a|b} where {|} = 0 There is something preventing a from equaling b. But the next closest space from a is b. But in either case, surreal or hyperreal .999... < 1 and .999... =/= 1 >(*) http://mathworld.wolfram.com/SurrealNumber.html >http://en.wikipedia.org/wiki/Surreal_number Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > Are you really sure that .999... is a hyperreal number and not > surreal(*)? Surreals are deŽned by a pair of classes, similar to a Dedikind Cut. Bob Kolker === Subject: Re: .99999... still=/= 1 posting-account=AE-QyQ0AAAC84T96q9_yI_Fj9ThoZQPi For example: x = {0.9, 0.99, 0.999, ... | 1} < 1 => x > 0.9 and x > 0.99 and x> 0.999 and ... === Subject: Re: .99999... still=/= 1 In sci.math, S. Enterprize Company All such series equal the ratio of integers, hence are rational numbers. > > > > But you said .999... is meaningless. You seem to bbe very confused. >>I said the -series- corresponding to an ulimately periodic decimal >>expansion equals a rational number. Learn to read. > Learn to think. READ THE MATH deŽnition of a real number. > .999... isn¹t a real number. It is in the class of a hyper-real number. .999... is the difference between a real number (1) and a hyperreal number. .999... = 1 - d. Now, what can be meaningfully done with this expression? An interesting question; I¹m not all that up on non-standard analysis. For example, the following expressions look interesting. [1] Let x = 1 - d = .999... . Then 10x - 9 = 1 - 10d = .999... and (x + 9) / 10 = 1 - d/10 = .999... . [2] x^2 = 1 - 2d + d^2 = .999... . [3] sqrt(x) = 1 - d/2 + d^2/8 - d^3/16 + 5*d^4/128 - ... = .999... (inŽnite binomial expansion) How are these ranked on the numberline? [rest snipped] -- #191, ewill3@earthlink.net It¹s still legal to go .sigless. === Subject: Re: .99999... still=/= 1 >.999... only approaching 1, it doesn¹t equal 1. It would seem that way. But appearances can be deceiving. It is certainly possible to use .999... as a symbol for a quantity that only approaches 1 as a limit. But that isn¹t its normal meaning. Because if we did that, then .1111.... wouldn¹t equal 1/9. Essentially, _if_ .999... stands for a real number, then it stands for the real number 1. Because any real number that doesn¹t equal 1 differs from 1 by a *Žnite* amount. The distinction between x=1 and x approaching 1 as a limit is an important one in mathematics, but numbers written in decimal form are not used to indicate this distinction; they are only used to indicate numbers. John Savard http://home.ecn.ab.ca/~jsavard/index.html === Subject: Re: .99999... still=/= 1 > Essentially, _if_ .999... stands for a real number, .999... does not stand for anything. It is scribble. If you want to talk about an inŽnite series which is a well deŽned notion then by all means do so. Perhaps you mean the series Sum [n >= 1] (9/10^n) ? If you do, then you should say so. Bob Kolker === Subject: Re: .99999... still=/= 1 >If you do, then you should say so. I wasn¹t aware that smart1234 was questioning that point, so I addressed the areas in which he was disputing what is generally known and believed. Ambiguity is inherent in any communication in ordinary language that strives for brevity, omitting details that have previously been agreed upon. John Savard http://home.ecn.ab.ca/~jsavard/index.html === Subject: Re: .99999... still=/= 1 <41c4026d.1793705@news.ecn.ab.ca> <41c46d29.1471377@news.ecn.ab.ca> posting-account=Z0C83A0AAAD1ZNF3_N0Hr9vKEMG9LqCk if there is a difference between 0.999... and 1, then what is that difference? is the difference 0.000...(inŽnity)...1 ? because thats not possible x = 0.999... 10x = 9.999... - x = 0.999... --------------------- 9x = 9 9x/9 = 9/9 x = 1, x=0.999..., 1=0.999.... === Subject: Re: .99999... still=/= 1 In sci.math, robert j. kolker : > >> Essentially, _if_ .999... stands for a real number, > .999... does not stand for anything. It is scribble. If you want to talk > about an inŽnite series which is a well deŽned notion then by all > means do so. > Perhaps you mean the series > Sum [n >= 1] (9/10^n) ? > If you do, then you should say so. > Bob Kolker Perhaps that is in fact the best solution all told. However, expressing 1/7 = sum(i=0,+oo) (1/10^(6*i+1) + 4/10^(6*i+2)+ 2/10^(6*i+3) + 8/10^(6*i+4)+ 5/10^(6*i+5)+ 7/10^(6*i+6)) does get a tad unwieldly... :-) Or perhaps one prefers the one-off form: 1/7 = sum(i=1,+oo) (1/10^(6*i-5) + 4/10^(6*i-4)+ 2/10^(6*i-3) + 8/10^(6*i-2)+ 5/10^(6*i-1)+ 7/10^(6*i)) -- #191, ewill3@earthlink.net It¹s still legal to go .sigless. === Subject: Re: .99999... still=/= 1 >Dedekind is credited with giving the Žrst mathematical deŽnition >of the real numbers. >> Is this so? >Yes and no. The set of real numbers was implicitly deŽned by >algebraic laws on the two basic operations + and *. A rigorous >deŽnition (or reduction) of real numbers to the rationals was done in >the latter half of the 19-th century along with deŽnitions of >continuous functions, differentiable functions and limits. The key to >the whole business is the rigorous deŽnition of limit. >The reals are the topological closure of the rationals. The key to this >is a deŽnition of convergence in which a limit is not explicitly >required. This is attributed to Cauchy. >Bob Kolker The basic fundamental deŽnition of .999.... is that it isn¹t a real number. A number that isn¹t real can¹t equal an number that is real. How simple can you get with this? Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > > >>Dedekind is credited with giving the Žrst mathematical deŽnition >>of the real numbers. > > > Is this so? >>Yes and no. The set of real numbers was implicitly deŽned by >>algebraic laws on the two basic operations + and *. A rigorous >>deŽnition (or reduction) of real numbers to the rationals was done in >>the latter half of the 19-th century along with deŽnitions of >>continuous functions, differentiable functions and limits. The key to >>the whole business is the rigorous deŽnition of limit. >>The reals are the topological closure of the rationals. The key to this >>is a deŽnition of convergence in which a limit is not explicitly >>required. This is attributed to Cauchy. >>Bob Kolker > The basic fundamental deŽnition of .999.... is that it isn¹t a real >number. > A number that isn¹t real can¹t equal an number that is real. How simple can >you get with this? Here¹s another way to look at this. i = something imaginary ( not real ) Can you prove i = a real number? Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 In sci.math, S. Enterprize Company >> >Dedekind is credited with giving the Žrst mathematical deŽnition >of the real numbers. >> >> >> Is this so? >Yes and no. The set of real numbers was implicitly deŽned by >algebraic laws on the two basic operations + and *. A rigorous >deŽnition (or reduction) of real numbers to the rationals was done in >the latter half of the 19-th century along with deŽnitions of >continuous functions, differentiable functions and limits. The key to >the whole business is the rigorous deŽnition of limit. >The reals are the topological closure of the rationals. The key to this >is a deŽnition of convergence in which a limit is not explicitly >required. This is attributed to Cauchy. >Bob Kolker >> The basic fundamental deŽnition of .999.... is that it isn¹t a real >>number. >> A number that isn¹t real can¹t equal an number that is real. How simple can >>you get with this? > Here¹s another way to look at this. > i = something imaginary ( not real ) > Can you prove i = a real number? Theorem: For all real numbers r, r^2 >= 0. Proof: Case r > 0: r*r = r^2 > 0. Case r = 0: r*r = 0. Case r < 0: r*r = (-s)*(-s) = s*s > 0, where s = -r > 0. QED. Corollary: There is no real number i such that i^2 = -1. Theorem: .999... is a real number. Proof: The usual deŽnition of .999... is an endless series of 9¹s. The implicit series is .9 + .09 + .009 + ... = sum(i=1,+oo) (9 * 10^(-i)). This series is of the form sum(i=1,+oo) (a * r^(i)), which converges for all -1 < r < 1. Hence .999... is a real number. QED. Theorem: If .999... < 1, then there exists inŽnitely many y with .999... < y < 1. Proof: Let x = .999... and y = (x + n - 1) / n, for any n > 0. Since x < 1, nx < n. ny = x + n - 1 < n, however, nx = (n-1)x + x < x + n - 1, therefore nx < ny and hence x < y as well. QED. Corollary: .999... is not a unique number if .999... < 1. :-) [.sigsnip] -- #191, ewill3@earthlink.net It¹s still legal to go .sigless. === Subject: Re: .99999... still=/= 1 >In sci.math, S. Enterprize Company > >>Dedekind is credited with giving the Žrst mathematical deŽnition >>of the real numbers. > > > Is this so? >>Yes and no. The set of real numbers was implicitly deŽned by >>algebraic laws on the two basic operations + and *. A rigorous >>deŽnition (or reduction) of real numbers to the rationals was done in >>the latter half of the 19-th century along with deŽnitions of >>continuous functions, differentiable functions and limits. The key to >>the whole business is the rigorous deŽnition of limit. >>The reals are the topological closure of the rationals. The key to this >>is a deŽnition of convergence in which a limit is not explicitly >>required. This is attributed to Cauchy. >>Bob Kolker > The basic fundamental deŽnition of .999.... is that it isn¹t a real >number. > A number that isn¹t real can¹t equal an number that is real. How simple >can >you get with this? >> Here¹s another way to look at this. >> i = something imaginary ( not real ) >> Can you prove i = a real number? >Theorem: For all real numbers r, r^2 >= 0. >Proof: >Case r > 0: r*r = r^2 > 0. >Case r = 0: r*r = 0. >Case r < 0: r*r = (-s)*(-s) = s*s > 0, where s = -r > 0. QED. >Corollary: There is no real number i such that i^2 = -1. >Theorem: .999... is a real number. No it¹s not a real number. It can¹t be expressed as a fraction. 9/10 = .9 >Proof: >The usual deŽnition of .999... is an endless series of 9¹s. >The implicit series is .9 + .09 + .009 + ... >= sum(i=1,+oo) (9 * 10^(-i)). This series is of the form >sum(i=1,+oo) (a * r^(i)), which converges for all -1 < r < 1. >Hence .999... is a real number. QED. No it¹s not. A real rational number has to be expressed as a fraction. It¹s not an irrational real number either, it has repeating digits in the decimal form. It is not a real number. Can it be shown as a series? Yes. Is there a space between, .999... | | and 1? Yes. .999... =/= 1 .999... could be called a hyper-real number. See the deŽnition at the math link below. http://mathworld.wolfram.com/HyperrealNumber.html In other words, a hyper-real number is an inŽnite number such that, x < n x = .999... n = 1 x =/= n Since .999... isn¹t a real number but appears to be of the form of a hyper-real number, .999... =/= 1 .999... < 1 >Theorem: If .999... < 1, then there exists inŽnitely many y with > .999... < y < 1. >Proof: Let x = .999... and y = (x + n - 1) / n, for any n > 0. > Since x < 1, nx < n. ny = x + n - 1 < n, > however, nx = (n-1)x + x < x + n - 1, therefore > nx < ny and hence x < y as well. QED. >Corollary: .999... is not a unique number if .999... < 1. >:-) >[.sigsnip] >-- >#191, ewill3@earthlink.net >It¹s still legal to go .sigless. Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > No it¹s not a real number. It can¹t be expressed as a fraction. Neither can the square root of 2 be so expressed, but it is a real (and irrational) number. The real numbers consist of rational numbers which can be expressed as ratios of integers and and irrationals which cannot. However all real numbers have non-negative squares. Bob Kolker === Subject: Re: .99999... still=/= 1 >> No it¹s not a real number. It can¹t be expressed as a fraction. >Neither can the square root of 2 be so expressed, but it is a real (and >irrational) number. >The real numbers consist of rational numbers which can be expressed as >ratios of integers and and irrationals which cannot. However all real >numbers have non-negative squares. confused? >Bob Kolker Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > Here¹s another way to look at this. > i = something imaginary ( not real ) > Can you prove i = a real number? What does this have to do with your .999... nonsense. .999.... is a meaningless string of symbols. It has no mathematical meaning whatsoever. Bob Kolker === Subject: Re: .99999... still=/= 1 >> Here¹s another way to look at this. >> i = something imaginary ( not real ) >> Can you prove i = a real number? >What does this have to do with your .999... nonsense. >.999.... is a meaningless string of symbols. It has no mathematical >meaning whatsoever. >Bob Kolker Look up the deŽnition of an irrational number. It almost Žts in except for one small detail, the 9¹s keep repeating indeŽnitely, which makes it not an irrational number, which means it can¹t be a real number. Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > Look up the deŽnition of an irrational number. It almost Žts in except for > one small detail, the 9¹s keep repeating indeŽnitely, which makes it not an > irrational number, which means it can¹t be a real number. An irrational number is a series of the form SUM (n >= 1) [a_n*1/10^n) where the sequence (a_n | n >= 1} is not of the form of a Žnite subsequence followed by a periodic repeating sequence. The series is convergent so it equals a real number. Recall that an inŽnit series is the -limit- of the sequence of Žnite partial sums. Equivalently an irratial real number is one that is not the ratio of two integers with no common factor. It is a theorem of real number theory that the above two deŽnitions are equivalent. What is your deŽnition of a repeating decimal number. If it does not come out to be a convergent series of multiples of 1/10^n it is just plain wrong. By the way decimal series that repeat periodically after a certain term are rational numbers which constitute a subset of the real numbers. Bob Kolker === Subject: Re: .99999... still=/= 1 >> Look up the deŽnition of an irrational number. It almost Žts in except >for >> one small detail, the 9¹s keep repeating indeŽnitely, which makes it not >> irrational number, which means it can¹t be a real number. >An irrational number is a series of the form SUM (n >= 1) [a_n*1/10^n) >where the sequence (a_n | n >= 1} is not of the form of a Žnite >subsequence followed by a periodic repeating sequence. The series is >convergent so it equals a real number. Recall that an inŽnit series is >the -limit- of the sequence of Žnite partial sums. >Equivalently an irratial real number is one that is not the ratio of two >integers with no common factor. It is a theorem of real number theory >that the above two deŽnitions are equivalent. >What is your deŽnition of a repeating decimal number. If it does not >come out to be a convergent series of multiples of 1/10^n it is just >plain wrong. By the way decimal series that repeat periodically after a >certain term are rational numbers which constitute a subset of the real >numbers. >Bob Kolker There is no real number representation of .999... . All you have is that someone told you to believe this. There is no proof. 9/10 = .9 not .999... Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > There is no real number representation of .999... . All you have is that > someone told you to believe this. There is no proof. I have proven that SUM (n >= 1) [9/10^n] = 1 several times in this thread. Each partial sum to the k-th power comes out to be 1 - 1/10^k, which can be made as close to 1 as you like which proves the limit of the partial sums is 1. QED. Fucking idiot. Bob Kolker === Subject: Re: .99999... still=/= 1 >> There is no real number representation of .999... . All you have is that >> someone told you to believe this. There is no proof. >I have proven that SUM (n >= 1) [9/10^n] = 1 several times in this thread. >Each partial sum to the k-th power comes out to be 1 - 1/10^k, which can >be made as close to 1 as you like which proves the limit of the partial >sums is 1. QED. >Fucking idiot. What¹s wrong you have to call me names now because you made a mistake that shows you don¹t even know the basic fundaments of math? >Bob Kolker Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > The basic fundamental deŽnition of .999.... is that it isn¹t a real number. This is meaningless non-mathematical bullshit. Bob Kolker === Subject: Re: .99999... still=/= 1 >> The basic fundamental deŽnition of .999.... is that it isn¹t a real >number. >This is meaningless non-mathematical bullshit. >Bob Kolker Ok then, here¹s the x-axis, 0 |------------|-------------------------> ^ .999999--> now keep typing in 9¹s for eternity so that we observe this on the axis at the speciŽed location. In the mean time, I will place a real number on the x-axis. 0 1 |----------------|-----------------------> See you later, I mean I won¹t see you later, you¹ll be too busy. Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > 0 > |------------|------------------------- ^ > .999999--> now keep typing in 9¹s for eternity so that we > observe this on the axis at the speciŽed location. What is .999... ? It has no mathematical meaning whatsoever. DeŽne .999... in terms of elementary arithmetic operations. By what logic to you identify a point on a line with a number? Justify your assertion mathematically. Bob Kolker === Subject: Re: .99999... still=/= 1 >> 0 >> |------------|-------------------------> ^ >> .999999--> now keep typing in 9¹s for eternity so that we >> observe this on the axis at the speciŽed location. >What is .999... ? It has no mathematical meaning whatsoever. DeŽne >.999... in terms of elementary arithmetic operations. By what logic to >you identify a point on a line with a number? Justify your assertion >mathematically. >Bob Kolker Smart¹s Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813& cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 X-RFC2646: Format=Flowed; Response >> 0 |------------|-------------------------> ^ >> .999999--> now keep typing in 9¹s for eternity so that >> we >> observe this on the axis at the speciŽed location. > What is .999... ? It has no mathematical meaning whatsoever. DeŽne > .999... in terms of elementary arithmetic operations. By what logic to you > identify a point on a line with a number? Justify your assertion > mathematically. > Bob Kolker that might be true, but everyone know what .999999.... means. just like sum(a_k,k=0..inŽnity) doesn¹t mean jack, but everyone knows what its suppose to mean. The problem is that someone can¹t comprehend the idea of inductive reasoning, Probably from a a nutritional deŽciency in there early developmental years. === Subject: Re: .99999... still=/= 1 > 0 |------------|------------------------- ^ > .999999--> now keep typing in 9¹s for eternity so that > we > observe this on the axis at the speciŽed location. >> What is .999... ? It has no mathematical meaning whatsoever. DeŽne >> .999... in terms of elementary arithmetic operations. By what logic to you >> identify a point on a line with a number? Justify your assertion >> mathematically. >> Bob Kolker >that might be true, but everyone know what .999999.... means. >just like sum(a_k,k=0..inŽnity) doesn¹t mean jack, but everyone knows what >its suppose to mean. >The problem is that someone can¹t comprehend the idea of inductive >reasoning, Probably from a a nutritional deŽciency in there early >developmental years. First you say .999... is a real number. Then you say it¹s meaningless. It seems like you are very confused. Smart¹s Alt. Physics News Grou