Guest Doron Shadmi Posted November 5, 2004 Posted November 5, 2004 Cantor used the expression 2^aleph0 in order to represent the magnitude of R set. Since base 2 can be represented as a tree diagram, we can use it in order to research a collection of infinitely many elements. For example, let us look at the infinitely long Top_to_Bottom blue tree, which is also represented as {1, 2, 4, 8, 16, ...}. It is obvious that we always find finitely many leafs in any arbitrary level of this tree, so this tree cannot have the magnitude of 2^aleph0. Furthermore, since in any arbitrary level we are still in N set, we can never define aleph0 as a transfinite number. Now let us say that we start by a collection of infinitely many R members, which are represented by infinitely many brown points. In this case, we know that we can never start to use base 2 in order to construct a Bottom_to_Top tree, if our collection of points can construct a solid line, and if we do that, we discover that we get infinitely many identical trees that cannot have |R| (if, again, R set is like a solid line). So my question is: How can we write 2^aleph0, if base 2 cannot exist when we deal with |R|? (This question is good for any base n) [color=Blue] 1 | 2^0 _______._______ | / \ | 2^1 ___.___ ___.___ | / \ / \ | 2^2 _._ _._ _._ _._ | / \ / \ / \ / \ | 2^3 . . . . . . . . | / \ / \ / \ / \ / \ / \ / \ / \ V 2^4 . . . . . . . . . . . . . . . . oo [/color] [color=DarkRed] 8 8 2^3 ___.___ ___.___ / \ / \ 2^2 _._ _._ _._ _._ / \ / \ / \ / \ 2^1 . . . . . . . . / \ / \ / \ / \ / \ / \ / \ / \ ---> oo 2^0 . . . . . . . . . . . . . . . . . . . . ... 4 4 4 4 2^2 _._ _._ _._ _._ / \ / \ / \ / \ 2^1 . . . . . . . . / \ / \ / \ / \ / \ / \ / \ / \ ---> oo 2^0 . . . . . . . . . . . . . . . . . . . . ... 2 2 2 2 2 2 2 2 2^1 . . . . . . . . / \ / \ / \ / \ / \ / \ / \ / \ ---> oo 2^0 . . . . . . . . . . . . . . . . . . . . ... 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ---> oo 2^0 . . . . . . . . . . . . . . . . . . . . ... [/color]
MandrakeRoot Posted November 5, 2004 Posted November 5, 2004 [math]2^{\aleph_0}[/math] is an expression coming form cardinal arithmic. You can actually show that a set having this cardinality has the same cardinality as [math]\mathbb{R}[/math]. I dont understand why you talk of base 2, because we are not talking of real arithmic but cardinal arithmic ! Mandrake
Guest Doron Shadmi Posted November 5, 2004 Posted November 5, 2004 [math]2^{\aleph_0}[/math] is an expression coming form cardinal arithmic. You can actually show that a set having this cardinality has the same cardinality as [math]\mathbb{R}[/math]. I dont understand why you talk of base 2' date=' because we are not talking of real arithmic but cardinal arithmic ! Mandrake[/quote'] Then please explain to me what is the meaning of 2 in [math]2^{\aleph_0}[/math] expression, thank you. If we look at http://mathworld.wolfram.com/PowerSet.html then we can see that the expression [math]2^{\S}[/math] (where [math]{\S}[/math] is any number) is a general notation of a power set. It means that |{0,1}| standing in the base of any power set. And it is easy to show that, for example, 2^3 is equivalent to the power set of 3: 8 2^3 ___.___ / \ 2^2 _._ _._ / \ / \ 2^1 . . . . / \ / \ / \ / \ 2^0 . . . . . . . . 1 2 3 4 5 6 7 8 is equivalent to |{{}, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2}}| = 2^3 1 2 3 4 5 6 7 8 And because base |{0,1}| cannot be found, then the whole idea of the transfinite cardinals does not hold, if the R set elements have no room to construct a Binary tree. And if they have a room to construct a Binary tree, then R set is enumerable, and Cantor's second diagonal method does not hold.
MandrakeRoot Posted November 5, 2004 Posted November 5, 2004 If i remember correctly a^b for two cardinals a and b is defined to be the cardinality of the set of all functions from a set of cardinality b into a set of cardinality a. With this definition you can show that 2^b is indeed the cardinality of the powerset of b, since {0,1} is a canonical set of cardinality two and so any mapping from a set of cardinality b into {0,1} can be seen as a characteristic function of a subset of (the set of cardinality) b. Mandrake
matt grime Posted November 5, 2004 Posted November 5, 2004 Also, since every constructed model of the Real Numbers I can think of is done from taking the Naturals, adding inverses, localizing, and then completing or some other analytic process, we can assume that 2 "exists", and that base two expansions are valid. Clearly the smallest infinite cardinal is that of the Natural numbers, and we say that anything with cardinality equal to them has card aleph-0. So it "exists". If one allows the axiom of choice to be true then we can create a system of (small) transfinite numbers quite easily.
Guest Doron Shadmi Posted November 5, 2004 Posted November 5, 2004 we can assume that 2 "exists" Dear Matt' date=' but I do not assume anything, I clearly show that base |{0,1}| cannot be found when we deal with the "transfinite" universe, as it "exists" by the Cantorian way. With this definition you can show that 2^b is indeed the cardinality of the powerset of b Sorry dear, but you cannot show it, and I clearly and simply show it in my argument. Please analyze my argument and show clearly where it fails, thank you.
matt grime Posted November 5, 2004 Posted November 5, 2004 Your arguement fails because you don't understand the meanings of any of the terms involved. With the usual identification of the power set with the set of all functions to {0,1}, then cardP(X) = 2^cardP{X}, which is extended to the transfinite case by anology.
Guest Doron Shadmi Posted November 5, 2004 Posted November 5, 2004 Your arguement fails because you don't understand the meanings of any of the terms involved. With the usual identification of the power set with the set of all functions to {0' date='1}, then cardP(X) = 2^cardP{X}, which is extended to the transfinite case by anology. [/quote'] Again Matt you write fancy notations, but you did not show the meaning behind them, so please put aside the notations, and show clearly how can we construct a binary tree, when we deal with the magnitude of the transfinite universe? In short, where is the binary tree of 'power set' = 'the set of all functions to {0,1}' ? which is extended to the transfinite case by anology. And I clearly and simply show that this 'extension by analogy' contradicts itself because base |{0,1}| (and the binary tree that is constructed by it) cannot be found by the Cantorian terms. By the way, it is easy to show that there exists a 1-1 correspondence between each node of the Binary tree, and each segment of Cantor's set (http://mathworld.wolfram.com/CantorSet.html): [color=Blue]2*0 *[/color] | ___________ | [color=Blue]2^1 * *[/color] | _____ _____ | [color=Blue]2^2 * * * *[/color] | __ __ __ __ | [color=Blue]2^3 * * * * * * * *[/color] | _ _ _ _ _ _ _ _ | [color=Blue]2^4 ** ** ** ** ** ** ** **[/color] | .. .. .. .. .. .. .. .. V oo [color=Blue]2^0 _______*_______ | / \ | 2^1 ___*___ ___*___ | / \ / \ | 2^2 _*_ _*_ _*_ _*_ | / \ / \ / \ / \ | 2^3 * * * * * * * * | / \ / \ / \ / \ / \ / \ / \ / \ V 2^4 * * * * * * * * * * * * * * * * oo[/color]
matt grime Posted November 6, 2004 Posted November 6, 2004 Doron, I've said my bit, ok: the ability to draw on a piece of paper somthing that you want to call a binary tree of an "equation" is nonsense to everyone else. We have a simple and consistent system that you clearly do not understand and no amount of explanation seems to make you understand us or even want to understand it. Perhaps if you were to explain to the public why you have this obsession with binary trees we could understand why you want to do these things, but most of us apparently just can't see through your unclear writing. The cantor set is uncountable, the nodes in your tree are countable, but then these are things that you don't accept, or do you eventually understand the proof of cantor's theorem that no set is in bijection with its power set when taken as the set of functions from the set to {0,1}? The cantor set can be put in bijection with the paths through the tree.
Guest Doron Shadmi Posted November 6, 2004 Posted November 6, 2004 Perhaps if you were to explain to the public why you have this obsession with binary trees we could understand why you want to do these things' date=' but most of us apparently just can't see through your unclear writing. [/quote'] Are you kidding? Cantor his the one who wrote 2^Aleph0 = |R| = c and since |{0,1}| sands in its base, the existence of the Binary tree cannot be ignored. We have a simple and consistent system ... No you do not have a simple and consistent system' date=' and Cantor's method does not hold. It is only a matter of time until people will realize that the whole idea of the transfinite system "has no legs and arms", and the collection of infinitely many elements cannot be in any relation with the universal quantification, simply because a collection of infinitely many elements cannot be completed by definition (in other words, this is exactly the deep difference between a collection of finitely many elements, which is complete (has a beginning and end, and a collection of infinitely many elements, that its beginning or its end can never be found by definition)). If you look at pages 4-7 of http://www.geocities.com/complementarytheory/TRANSFINITES.pdf you will see why Cantor did not prove that |P(S)| > |S| . Furthermore, because Russell's "paradox" is nothing but a false statement in Excluded-Middle logical reasoning, as I clearly show here http://www.geocities.com/complementarytheory/Russell1.pdf , Cantor cannot use it in his |P(S)| > |S| "proof". More information about this subject, which clearly explains in details why Cantor's second diagonal method does not hold, can be found in: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf In my opinion, you cannot continue to teach Cantor's system to new students, and by this way to continue to keep Cantor's mathods artificially alive, because his understanding of the infinity concept does not hold even under the Excluded-Middle logical reasoning framework. Until now you did not show, by using your mathematical skills, any problem that can be found in my arguments against the Cantorian methods. It is not enough to say that I do not understand you. Strictly speaking, you have to show in details and in front of this forum , that the Cantorian methods hold, and my arguments are failed. If you cannot do that, then it is as if you say that I am right (and Cantor's methods are wrong).
matt grime Posted November 6, 2004 Posted November 6, 2004 Seeing as GB are losing to NZ I'll amuse myself with this instead. Doron, your "disproof" of Cantor assumes that the power set is a binary tree that can be embedded in the plane. It isn't. That is the first point at which your argument fails. The fact you do not accept universal quantifiers is of no interest to anyone. Cantor does not at any point use "the set of all sets that contain themselves", and so doesn't not need "Russell's paradox", that is another point where you are wrong. Your diagonal argument pdf still contains the assumption that you can arrange all these things into an array with countable rows and columns in the case of the natural numbers since you can do so for the finite numbers that is obviously rubbish. Moreover the two large arrays you start writing out clearly only contain elements in them that, if the columns were enumerable and a complete list, contain only finitely many zeroes or finitely many 1s. It still presumes that 2^{aleph-0} is countable in the model of ZF where the power set is the set of functions to {0,1}. Note there are models where that isn't the definition of power set and where P(N) is countable, this is called Skolem's Paradox. So, just because you do not accept that complete, or all are words that cannot be used for infinite sets doesn't mean that anyone else agrees with you. Please, if you can, write down some integer that isn't in the complete set of integers, or the set of all integers.
Guest Doron Shadmi Posted November 6, 2004 Posted November 6, 2004 Doron' date=' your "disproof" of Cantor assumes that the power set is a binary tree that can be embedded in the plane. It isn't. That is the first point at which your argument fails. [/quote'] 1) I used a binary tree just because of Cantor's 2^Aleph0 expression. We can use any N member instead of 2. 2) A Binary tree is not embedded to a plan, because it is a multi-dimensional diagram, which its dimension level is determined by its power value, so you simply do not understand what a Binary tree is. Cantor does not at any point use "the set of all sets that contain themselves"' date=' and so doesn't not need "Russell's paradox", that is another point where you are wrong. [/quote'] 3) the method that Cantor used in the second part of |P(S)|=|S| "proof" is equivalent to Russell's second paradox. Please' date=' if you can, write down some integer that isn't in the complete set of integers, or the set of all integers. [/quote'] To make it simpler let us understand integers as the Natural numbers. The Naturals do not exist because of the existence of N set, but because of the axioms that define them, for example, in Peano's system the Natural numbers exist without using the Set concept. In short, The definition is important here and not the "total" number of the elements that can be defined by it. So as you can see, the existence of a complete collection of infinitely many elements, is not a must have state from a pure mathematical point of view. The problem is that you put the carriage (the container or the set) in front of the horses (the definitions), and by this trivial attitude you reduced the power of the abstraction of the language of Mathematics. if the columns were enumerable and a complete list' date=' contain only finitely many zeroes or finitely many 1s. [/quote'] Again you force your frozen point of view about the infinity concept on my flexible point of view about the infinity concept, which is based on the difference between an Actual infinity and a Potential infinity (http://www.geocities.com/complementarytheory/RiemannsLimits.pdf ). This point of view can be found in: http://www.geocities.com/complementarytheory/9999.pdf http://www.geocities.com/complementarytheory/ed.pdf http://www.geocities.com/complementarytheory/Anyx.pdf http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
matt grime Posted November 6, 2004 Posted November 6, 2004 I agree that I do not share your point of veiw on infinity, however since you're claiming that Cantor is wrong and Cantor's result is that which uses *my* definitions, then your definitions are completely irrelevant and have no bearing on the result. If you're saying that Cantor's theorem is false in *your* system, then so what. It isn't a statement about anything in your system. Why should it be true if you just reassign meanings at will. You ought to at least show Cantor is wrong within the "Cantorian" model, otherwise you're just spouting nonsense.
Guest Doron Shadmi Posted November 6, 2004 Posted November 6, 2004 agree that I do not share your point of veiw on infinity' date=' however since you're claiming that Cantor is wrong and Cantor's result is that which uses *my* definitions, then your definitions are completely irrelevant and have no bearing on the result. If you're saying that Cantor's theorem is false in *your* system, then so what. It isn't a statement about anything in your system. Why should it be true if you just reassign meanings at will. You ought to at least show Cantor is wrong within the "Cantorian" model, otherwise you're just spouting nonsense. [/quote'] Cantors frozen point of view about the infinity concept that also uses an expression like 2^Aleph0 to express the magnitude of R, is clearly based on a self contradiction, because base |{0,2}| cannot be found when we deal with the solid real-line. Therefore it must get off stage because it is conceptually wrong. And this is exactly the main point of my argument.
matt grime Posted November 6, 2004 Posted November 6, 2004 Cantor at no point even requires, or states the the real numbers exist. The real numbers aren't a line, Doron, and are not important in this discussion, can you please get past this high-school misapprehension?
Guest Doron Shadmi Posted November 6, 2004 Posted November 6, 2004 Cantor at no point even requires' date=' or states the the real numbers exist. The real numbers aren't a line, Doron, and are not important in this discussion, can you please get past this high-school misapprehension? [/quote'] If 2^Aleph0 = |R| = c , then I do not care what Cantor said on not said. And since base |{0,1}| cannot be found in the terms of |R|, then |{0,1}|^Aleph0 does not hold. Let us show that Aleph0 also cannot be found: Cantor used the expression Aleph0 in order to represent the magnitude of N set. In the basis of Von Neumann Hierarchy there exists a Binary-Tree: 0 = |{ }| 1 = |{{ }}| = {0} 1. | 2 = |{{ },{{ }}}| = {0,1} 1. . | | 2|____| | 3 = |{{ },{{ }},{{ },{{ }}}}| = {0,1,2} 1. . . . | | | | 2|____| |____| | | 3|__________| | 4 = |{{ },{{ }},{{ },{{ }}},{{ },{{ }},{{ },{{ }}}}}| = {0,1,2,3} 1. . . . . . . . | | | | | | | | 2|____| |____| |____| |____| | | | | 3|__________| |__________| | | 4|______________________| | | ... This Binary-Tree stands at the basis of N members. Since base 2 can be represented as a tree diagram, we can use it in order to research a collection of infinitely many elements. For example, let us look at the infinitely long Top_to_Bottom blue tree, which is also represented as {1, 2, 4, 8, 16, ...}: [color=Blue] 1 | 2^0 _______._______ | / \ | 2^1 ___.___ ___.___ | / \ / \ | 2^2 _._ _._ _._ _._ | / \ / \ / \ / \ | 2^3 . . . . . . . . | / \ / \ / \ / \ / \ / \ / \ / \ V 2^4 . . . . . . . . . . . . . . . . oo [/color] Since in any arbitrary level we are still in N set, we can never define Aleph0 as a transfinite number. Now let us say that we start by a collection of infinitely many N members, which are represented by infinitely many brown points: [color=DarkRed] 8 8 2^3 ___.___ ___.___ / \ / \ 2^2 _._ _._ _._ _._ / \ / \ / \ / \ 2^1 . . . . . . . . / \ / \ / \ / \ / \ / \ / \ / \ ---> oo 2^0 . . . . . . . . . . . . . . . . . . . . ... 4 4 4 4 2^2 _._ _._ _._ _._ / \ / \ / \ / \ 2^1 . . . . . . . . / \ / \ / \ / \ / \ / \ / \ / \ ---> oo 2^0 . . . . . . . . . . . . . . . . . . . . ... 2 2 2 2 2 2 2 2 2^1 . . . . . . . . / \ / \ / \ / \ / \ / \ / \ / \ ---> oo 2^0 . . . . . . . . . . . . . . . . . . . . ... 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ---> oo 2^0 . . . . . . . . . . . . . . . . . . . . ... [/color] In this case, when we construct a Bottom_to_Top trees, we discover that we get infinitely many identical trees that cannot have |N| because also in this case we are still in N. And also in this case we still in N because it is nothing but an horizontal version of the blue tree: 16 _______._______ 8 / \ ___.___ ___.___ ___.___ 4 / \ / \ / \ _._ _._ _._ _._ _._ _._ _._ 2 / \ / \ / \ / \ / \ / \ / \ . . . . . . . . . . . . . . . 1 / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ --->oo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... In short, the transfinite system does not exist.
Guest Doron Shadmi Posted November 7, 2004 Posted November 7, 2004 Now take a continuus set of points- such as a line No dear' date=' no [b']model[/b] that is described by a solid (pointless) line (for further information please look at http://www.geocities.com/complementarytheory/SegPoint.pdf), can also be described by a model, which is based on infinitely many elements. I want to clarify my argument about the transfinite system, which is based on a collection of infinitely many elements + a universal quantification that is related to them. I clime that if we have a collection of infinitely many elements, then no universal quantification can be related to them, because if we force the universal quantification on a collection of infinitely many elements, we get an actual infinity, and actual infinity is too strong in order to be used as an input in the domain of any language, including the language of Mathematics. For example: No collection of infinitely many segments can construct a one infinitely long solid line, and the reason is very simple: In a collection of infinitely many elements, we always have elements that can be manipulated by us by using functions or any other mathematical tool. If we force the universal quantification on a collection of infinitely many elements, the result is that we find ourselves out of the domain of the collection of infinitely many elements, which is equivalent to the state of a one infinitely long solid line. And no mathematical tool can deal with this totality because on information can be found and manipulated when we deal with Actual infinity (which in this case, is represented by a one infinitely long solid (pointless) line). The Idea of the transfinite system according to Cantor, actually forces the universal quantification on a collection of infinitely many elements, but then it is totally misses the results of this forcing, and continue to use mathematical manipulations on Actual infinity state, which is nothing but a fundamental conceptual frailer of his understanding of the Infinity concept. Please see this diagram for better understanding: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf In short, it is about time that Cantor's infinity world will get off stage.
MandrakeRoot Posted November 8, 2004 Posted November 8, 2004 Dear Matt' date=' but I do not assume anything, I clearly show that base |{0,1}| cannot be found when we deal with the "transfinite" universe, as it "exists" by the Cantorian way. Sorry dear, but you cannot show it, and I clearly and simply show it in my argument. Please analyze my argument and show clearly where it fails, thank you.[/quote'] Well clearly is a big overstatement here ! Your reasoning is impossible to follow and flawed. I like your tree drawings, but i dont see what is the relevance with respect to the topic ? Mandrake
matt grime Posted November 8, 2004 Posted November 8, 2004 Doron, I cannot prove you wrong **within your system**, and have no intention of even trying. Indeed there is little I can prove or disprove in your system because you've not managed to explain anything clearly enough to allow me to make any certain deductions, or even state that deductive reasoning is allowed (deductive reasoning states that (Anb)=>C is equivalent to (A=>(B=>C)). You have for instance failed to define "complete", that adjective that we cannot apply to infinite sets, apparently. I can say you're wrong about Cantor's argument. It may fail in your system but that is because your system rejects the objects upon which it is built. Nothing wrong with that, but then to imply, as you do, that Cantor needs to get off the stage because, with *your* ideas about infinite things it is a nonsensical argument is wrong. It just means that in *your* theory it doesn't hold, not that our theory is wrong. If you can in any way demonstrate anything in your theory that is consistent and useful then perhaps some may take you seriously, if however all you're going to do is to say that results that weren't formulated in your system are wrong then no one will care. In fact you will only serve to dissuade people from examining it if the only results about it are that everything in mathematics currently needs to be thrown away. As I keep saying, there is a model of ZF where in the power set of the naturals and the naturals are isomorphic. It is called Skolem's paradox. This is all down to your idea of "actual" and "potential" infinity that is no part of mathematics as we know it.
Guest Doron Shadmi Posted November 11, 2004 Posted November 11, 2004 (deductive reasoning states that (Anb)=>C is equivalent to (A=>(B=>C)). Deduction that is based only on the quantity concept is without a doubt a trivial system' date=' when it is compared to the fundamental Structural/Quantitative approach. And the reason is very simple: in quantitative-only approach you permanently ignore the non-trivial diversity of infinitely many information clarity degrees, that can be found in any single path along the structural\quantitative representation of any R member. For example, please look at pages 3,4 of http://www.geocities.com/complementarytheory/9999.pdf where I clearly and simply demonstrate the weakness of the quantitative-only approach that is based only on 0_Redundancy_AND_0_Uncertianty information form building-block. I can say you're wrong about Cantor's argument. No you cannot, because I demonstrate the conceptual self-contradiction of Cantor’s infinity word, within the terms of Cantor’s infinity word. For example: 1) In one case no collection of infinitely many elements can reach its limit, but on the other case the same collection somehow includes within it the limit itself, as a part of the collection, which is a trivial self-contradiction, if each element has a unique place in the collection. 2) Let us take each scale level of a single representation of some number, as some object that exists in a collection of infinitely many represented scale levels. Form this point of view; there is no problem to use the half open interval idea on some unique path along these infinitely many scale levels, and get the unique [.000…, 1) object. But then we discover the limited Standard approach that allows this point of view on [0, 1) but then, and without any good reason, does not allow [.000…, 1). In short, the Standard approach about Infinity, is an inconsistent system, which also limits itself by using arbitrary decisions about collections of infinitely many objects. You have for instance failed to define "complete"' date=' that adjective that we cannot apply to infinite sets, apparently. [/quote] On the contrary, this is the first time when a clear and simple definition of Completeness is defined by a mathematical framework (please see page 3,4 of http://www.geocities.com/complementarytheory/My-first-axioms.pdf) and gives us the ability to clearly distinguish between a complete collection and a non-complete collection. And I clearly and simply show, why a universal quantification cannot be related to a collection of infinitely many objects, elements, states, etc… For example, please look at pages 2,3 of http://www.geocities.com/complementarytheory/ed.pdf Koch Fractal example in at pages 6,7 of http://www.geocities.com/complementarytheory/9999.pdf And the basic idea of incompleteness according to godel, that can be found in http://www.geocities.com/complementarytheory/Anyx.pdf In short, Matt, the Standard pure approach about fundamental concepts like: Number, Infinity, Functions, Limit, Logic, and self-consistency and more, clearly demonstrate a very poor framework, which is based only on linear Euclidean point of view, that cannot go beyond technical proofs on papers. Matt, this is defiantly not the living Language of Mathematics that I am talking about. The future of this Language is based on the ability to find and reinforce our abilities to connect in non-destructive and non-trivial way, between our morality and our logical reasoning, as they are clearly and simply explained in: http://www.geocities.com/complementarytheory/MM.pdf http://www.geocities.com/complementarytheory/GaloisDialog.pdf In short, the happy days of pencil an paper and some trivial technical proof on some Journal, are going to get of stage, because the Language of Mathematics is much more then that, and even in its technical aspect we need parallel, quantum computers in order to deal with its non-trivial Structural/Quantitative approach.
matt grime Posted November 11, 2004 Posted November 11, 2004 No you cannot' date=' because I demonstrate the conceptual self-contradiction of Cantor’s infinity word, within the terms of Cantor’s infinity word. For example: 1) In one case no collection of infinitely many elements can reach its limit, but on the other case the same collection somehow includes within it the limit itself, as a part of the collection, which is a trivial self-contradiction, if each element has a unique place in the collection. 2) Let us take each scale level of a single representation of some number, as some object that exists in a collection of infinitely many represented scale levels. Form this point of view; there is no problem to use the half open interval idea on some unique path along these infinitely many scale levels, and get the unique [.000…, 1) object. But then we discover the limited Standard approach that allows this point of view on [0, 1) but then, and without any good reason, does not allow [.000…, 1). In short, the Standard approach about Infinity, is an inconsistent system, which also limits itself by using arbitrary decisions about collections of infinitely many objects. [/quote'] 1) isn't something that is true in axiomatic set theory, indeed the words reach, limit and scale you are using to mean something that we do not, in the Cantorian realm, agree with, since your conclusions are nothing to do with that theory. It is you *opinion* that no infinite set can be completed - that isn't a (mathematical) statement within ZF, or any of its models, in particular the theorem is true in a system which doesn't contain the reals as a set. You are not doing mathematics here.
Guest Doron Shadmi Posted November 11, 2004 Posted November 11, 2004 You are not doing mathematics here. No dear' date=' the Language of Mathematics cannot be understood, if concepts like Infinity, Limit, Number are not understood first, before any technical definition. And it is trivially understood that Cantor’s approach does not hold, because he ignores the most basic principle of any formal language, which is: SYMMETRY. In short, if he uses concept like “The Empty Set”, he simply cannot ignore its symmetrical opposite, which is: “The Full Set”. Cantor ignored it, and by this he missed the deep understanding of the Infinity concept, and the clear and simple fruitful difference between Actual infinity (represented as an infinitely long non-composed solid element) and Potential infinity (represented as a collection of infinitely many elements). This asymmetrical state creates an ugly mathematical framework, which is based on arbitrary and artificial methods, and these forcing methods clearly can be seen when they are compared to a symmetrical method (where both “The Empty Set” and “The Full Set” are used). in particular the theorem is true in a system which doesn't contain the reals as a set ZF is not based on the symmetrical point of view if the Infinity concept, therefore most of its products are invalid. Let us write it once again: The Naturals do not exist because of the existence of N set, but because of the axioms that define them, for example, in Peano's system the Natural numbers exist without using the Set concept. In short, The definition is important here and not the "total" number of the elements that can be defined by it. So as you can see, the existence of a complete collection of infinitely many elements, is not a must have state from a pure mathematical point of view. The problem is that you put the carriage (the container or the set) in front of the horses (the definitions), and by this trivial attitude you reduced the power of the abstraction of the language of Mathematics. In short, for the past 100 years, pure mathematicians play with fancy notations without understanding the deep meaning of them. Furthermore, they are doing their best in order to avoid any exploration of these fundamental concepts (like, Infinity, Number, Limit, Logic, etc…), and if someone tries to touch them, then immediately he can hear the same old broken record: ”It is not Mathematics”, “It is a philosophy”, “You do not understand Mathematics”, etc…
matt grime Posted November 11, 2004 Posted November 11, 2004 None of that shows ZF or Cantor to be inconsistent (you understand what that word means?) It shows that if *you* assign meanings to things that directly contradict those of ZF that you get an inconsistent system. That is neither important nor news. You have said there is no absolute truth in mathematics (and there isn't), however if you are insisting that something *must* be always true, which is contradictory behaviour.
Guest Doron Shadmi Posted November 11, 2004 Posted November 11, 2004 however if you are insisting that something *must* be always true' date=' which is contradictory behaviour. [/quote'] No, I insist that something must be always consistent, and if we ignore SYMMTERY in a pure method, we cannot be consistent.
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