lama

Bloodhound and Matt, Please read very carfully post #46. Thank you.

"All" mean one and only one thing which is, we can find our first and last elements. We can say it only about finite things. We cannot say it about collections of infinitely many elements. This understanding is simple and clear, and any other approach is to define it by force. Edit: A new definition to "all": "All" means one and only one thing which is: No infinitely many elements can be found.

There is no objective thing called Math, and also it is not your privet property, so please hold your horses and be more modest, thank you.

N is a collection of infinitely many elements that cannot be completed, so you cannot use the term all in this case and also you cannot define a bijection (1-1 and onto) in an incomplete collection.

Do you understand that? In a collection of infinitely many elements' date=' an Epsilon is an invariant NEXT state (which is an inherent property of any collection of infinitely many alamants) that cannot give us the ability to force a universal quantification on this kind of a collection. So the cardinality of N is no more then |N| - Epsilon, which means that (the exact) w is undefined.

Yes' date=' because there are infinitely many [i']n[/i] in N but because what a clearly show in post #1 |N| cannot be defined and all we can get is |N| - epsilon.

Dear Bloodhound, Let us take for example the real line. If we say that our collection is the closed interval [0,1], then 0 AND 1 are members in our collection. If we use a function between [0 and x, than this function cannot be but a discontinuous function, and there are no members in the domain of the discontinuous function. The same holds between y and 1] so we get: Let cf be a continuous function. Let df be a discontinuous function. [0 <-df-> x1 <-df-> x2 ,... , Epsilon>0 ... <-cf-> ... Epsilon>0 , ..., y2 <-df-> y1 <-df-> 1] As you can see, we always has two separated collections: 1) A collection which is based on a continuous function that has infinitely many elements. 2) A collection of infinitely many elements which is based on infinitely many discontinuous functions (where each member in this set is the result of a quantum leap between the continuous function to a discontinuous function. Since we get two disjoined sets (one is based on cf and the other is based on df), we cannot use a universal quantification on both of them, and also not on each one of them, because in the case of infinitely many elements in both of them, there is always an Epsilon>0 between them. In other words, because we have a permanent Epsilon>0, no Universal Quantification can be used on a collection of infinitely many elements. By the way there is also the case that the collection of discontinuous functions is a finite collection (as I wrote in my previous posts), but then it is understood that this set has a universal quantification, where the set that is based on a continuous function, has no universal quantification (because Epsilon > 0 ). This is, by the way, the reason why Dedekind's Cut is problematic, because also in this case we have Epsilon>0, for example: L Epsilon>0 c Epsilon>0 R and we cannot find all of Q members in L and R sets. Also we have to understand that if Epsilon=0 then L member = c = R member.

Dear Matt, but in order to define that w is a limit ordinal (has no predecessor) you first have to show that the transfinite universe exists. And it cannot exist, because no set of infinitely many elements is a complete collection (include also its maximal AND minimal elements). So, your example is circular because you are using a transfinite element in order to show the existence of a transfinite element, but you do not show how this transfinite element (w in this case) can exist, by using a universal quantification on a collection of infinitely many elements and also to save it as a magnitude of infinitely many elements. I clearly and simply show why it can be done, but you did not show how it can be done. All you show is that some how out of the blue, there exists w. If you cannot explain how w exists, then you cannot use it as a model. In a continuous function a small input gives a small output, and this is not the case if we define a function between one of the memebers, which is not maximal or minimal, and one of the members which is maximal or minimal (as I clearly show in post #42).

Matt wrote: "But what about infinte sets that have maximal and minimal elements, Doron?" So dear bloodhound, please show us such a well-ordered set where both its minimanl and maximal members are included in it, and it is also a set with infinitely many members. If you read again posts #1 , #19 and #23 you will find out why this set cannot be found.

Matt, We are talking here about ideas and not about accurate formal nonations, so please do not use your energy on less important things like corrected formal notations, and open your mind to the ideas. The Ideas are not any formal notations of them.

ok, corrected to [math]s\mapsto s[/math] , thank you.

Well, in this case I used bloodhound notations for identity map, so if they are not correct then please correct them, thank you. As for your question about ends. Take any non-empty finite collection and then you can find its finite cardinal and also (if this collection is well ordered) its first AND lest elements.

Dear Bloodhound, I'll do my best to explain again post #25 in a more formal way, but I need your help in order to do it right, so here it is again: If [math]S[/math] has is well-defined, then the definition is [math]S[/math] and not any finite or infinitely many products of this definition. So mapping between infinitely many elements cannot return the Identity map of [math]S[/math] (please look at post #23 in order to understand this). If you want to find the Identity map of [math]S[/math], look at its definition (which does not depend on 'Quantity', 'Size', 'Magnitude', 'ALL' s in S, etc...). But if we say that one of the properties of [math]S[/math] is to be infinite, then [math]s\mapsto s[/math] (identity map) is exactly this unbounded state, and not a bijection of it. Please read again post #23 in order to understan my idea here, and also ask me about any non-convetional expressions of me, so we can together write it in its formal way. Thank you. Yours, Doron

Dear Bloodhound, Thank you for you questions: "Accurate Definition" is Well-defined. "endless thing" is unbounded. "accurate result" is one and only one result. "endless state" is infinitely many ... Since [math]\mathbb{R}[/math] is a collection of infinitely many elements' date=' then we need a universal quantification in order to prove the bijection in this case. But then we see that is we use a universal quntification, then our set does not exist or our set is a finite set. Since [math']\mathbb{R}[/math] has infinitely members, and since we need a universal quantification in order to prove the bijection, we cannot prove that there is a bijection.

If [math]S[/math] has an accurate definition, then the accurate definition is [math]S[/math] and not any finite or infinitely many products of this definition. So mapping between infinitely many elements is an endless thing that cannot return any accurate result, which in this case is the Identity of [math]S[/math]. If you want to find the Identity of [math]S[/math], look at its definition (which does not depend on 'Quantity', 'Size', 'Magnitude', 'ALL' s in S, etc...). But if we say that one of the properties of [math]S[/math] is to be infinite, then [math]s\mapsto s[/math] (identity map) is exactly this endless state, and not a bijection of it.

Then you use A AND B and get nothing XOR a collection of finitely many elements. In other words' date=' there is no connection between the definition of the Prime numbers, and how many of them can be found in some collection, which means: There can be finite or infinitely many prime numbers, but exactly a one and only one definition of them. So the definition is actually the prime number(s) and not their collection. [b']N[/b] is not the definition of the Natural numbers, but only a collection of them, and the Cntorian |N| is based on an A AND B approach, and therefore it does not hold.

No' date=' it has to be: For [b']each[/b] x in R x^2 is positive, because I clearly show that if R is a collection of infinitely many elements, then we cannot use a universal quantification on such a collection, because: In a collection of infinitely many elements, an Epsilon is an invariant NEXT state (which is an inherent property of any collection of infinitely many elements) that cannot give us the ability to force a universal quantification on this kind of a collection. Do you really can't see this beautiful state? An inherent and invariant NEXT state prevents the use of a universal quantification, it is simple, and it is right in front of your mind, don't you see it? Actually, a collection of infinitely many elements cannot exist without the permanent existence of this Epsilon, and this is an amazing thing, because it gives us the deep insight that the existence of infinitely many elements actually depends on the existence of a single element as its inherent permanent NEXT state. Also please pay attention that I use the word 'state' and not 'process', in order to clarify that I am talking about a timeless simultaneous NEXT, which is an inherent signature of any collection of infinitely many elements... We can use ALL only when we have a collection of finitely many elements. We do not have to think twice in order to understand that Cantor's approach is a complete rubbish that cannot be exist even as an abstract universe.

Matt, No one can conclude anything by using mapping between collections with infinitely many elements, because these collections are un-bounded and therefore not completed by their very own nature. All what Cantor did is to show a bijection between a collection of finitely many elements, and then he forced a universal quantification on these collections, in order to get by force his required result. But this result is clearly ill-defined because we cannot conclude something about a collection of infinitely many elements by trying to eliminate by force its unbounded nature in order to define our requested results. Cantor’s extension-by-force of cardinality and ordiality simply does not hold water, so Matt you are the one that continue to force your ill-defined conditions on collections of infinitely many elements by forcing on them a universal quantification. By this forcing method you lose your ability to distinguish between a collection of finitely many elements (where a universal quantification can be related to them) and a collection of infinitely many elements (where a universal quantification cannot be related to them) and this is exactly the deep difference between these kinds of collections, that actually gives us the ability to perfectly distinguish between them. If you ignore this deep and simple difference, then your system is based on ill-defined terms. Simple as that, so do not sell us stories about your "well-defined" universe. Also please pay attention to the very important fact, which is: I am not a constructivist that says that collections of infinitely many elements do not exist. I clearly say that collections of infinitely many elements do exist and they are deeply and totally different from finite collections, and the first and the most important difference is: We cannot force on them a universal quantification, as I clearly and simply show in my abstract model at post #1. I totally agree with you' date=' but I add that they are ill-defined by Cantor, therefore must not be a part of the Language of Mathematics and its logical reasoning. Really?? In Post #1 I clearly and simply show that I am right!!! If you want to show that I am wrong you have to introduce to all of us your model that clearly explain, by using only infinitely many elements, why you are right and I am wrong. You did not do it, you are just talking about it, and just talking is exactly nothing in this case. Also I have noticed that you look at yourself as some kind of a duke or a king, that shares his wisdom with his worshiped people. So, King Matt: I clime that King Cantor is naked … , please prove me wrong!!! In a collection of infinitely many elements, an Epsilon is an invariant NEXT state (which is an inherent property of any collection of infinitely many alamants) that cannot give us the ability to force a universal quantification on this kind of a collection. You can conclude accurate things by using mapping, only between collections with finitely many elements. If you disagree with me, then you have to show how a universal quantification can be related to a collection of infinitely many elements. Can you do this? ------------------------------------------------------------------------------------ Some general point of view: There is more than one school of thought in the world of Mathematics. Please look at: http://en.wikipedia.org/wiki/Philosophy_of_mathematics

If ALL n is in N then we cannot avoid the idea of a SUM when we think about |N|. Wrong' date=' I improved the arithmetic between infinitely many elements, for example: By Cantor Aleph0 = |[b']N[/b]|, which is the cardinality of N set. By Cantor: aleph0+1=aleph0, aleph0-2^aleph0 has no meaning, aleph0 < 2^aleph0, 3^aleph0=2^aleph0, etc... My solution to Aleph0 concept My concept of aleph0 is based on "cloud-like" magnitude of any collection of infinitely many elements. For example: aleph0+1 > aleph0 If A = aleph0 and B = aleph0 - 2^aleph0, then A > B by 2^aleph0, where both A and B are collections of infinitely many elements. Also 3^aleph0 > 2^aleph0 > aleph0 > aleph0 - 1, etc... Fore more details please look at: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf Strictly speaking, Actual infinity (infinitely long non-composed element) is too strong to be used as an input. Potential infinity (infinitely many elements, which never reaches Actual infinity, and therefore cannot be completed) is the name of the game. For further information please look at: http://www.geocities.com/complementarytheory/ed.pdf http://www.geocities.com/complementarytheory/9999.pdf http://www.geocities.com/complementarytheory/Anyx.pdf 5. All you have is to look at my representation' date=' and by using its abstract model in your mind you immediately define the invariant proportion, unless you have no ability to think abstract thoughts. 6. Yes (http://mathworld.wolfram.com/CardinalNumber.html) and their extension to transfinite cardinals can be done only if we force a universal quantification on a collection of infinitely many elements, which is something that cannot be done, and I clearly and very simply show it in my abstract model. 7. This bijection cannot be found between collections of infinitely many elements' date=' because endless mapping is a meaningless thing. More to the point: a collection of infinitely many elements cannot be completed, as I clearly show in post #1. 8. Thank you for your advice Matt, but I already did it, and the results of what I discovered are clearly and simply shown in my abstract model at post #1 9. Ho, dear Matt, no one of us really Knows the true, and therefore why do you think that everything must behave as you wish?

Dear Daymare17, There is more than one school of thought in the world of Mathematics. Please looka at: http://en.wikipedia.org/wiki/Philosophy_of_mathematics The main schools of the modern time are Formalism and Logicism. But as you say, things can be changed, and we hope that the changes are for the better. If you ask me, then after all, all of us including our abstract thoughts belong to the changing reality. One of the important things in the language of Mathematics is the ability to share ideas in a very accurate way between different persons around our planet. In order to do this, we have to ignore the changing reality and stick to unchanged models, which can be easily understood and developed by the people which share them. So, first you find that the time concept does not exist in this way of thinking, which is called logical reasoning. There are advantages in this timeless way of thinking, because they can help us to take some perspective point of view from changing reality, and maybe discover some deep connections that maybe or maybe not can be translated to the changing reality, but these deep connections can help us to organize our abstract/non-abstract world in cleverer, efficient, beautiful and unexpected ways, which can change our (at least) near reality. I agree with you, that if we go too far by totally ignore the non-trivial complexity of the changing reality, by using mostly a False_XOR_True (0 either 1) way of thinking, then there is a real danger that we become a closed community of scholars that all they do is to develop methods to share their scholastic ideas, where the ideas themselves are not developed beyond the 0 xor 1 way of thinking, which is defiantly a trivial model, when it is compared to the non-trivial changing reality. The keyword, in my opinion, is to find balanced gateways between the abstract worlds of our trivial models, and the non-trivial changing reality. It means that from time to time our most fundamental abstract concepts like (for example) the Number concept must be re-searched, re-examined, re-discussed, re-thought and understood from new points of view, which are based on new insights that are created/discovered in both abstract and non-abstract word. For example, concepts like redundancy and uncertainty that are coming form quantum phenomena, information theory, chaos, etc... can be used as first-order properties of the Number concept, and as the result, we can get a much more comprehensive framework, that can help us to develop abstract/non-abstract worlds, which are less trivial than the standard Black_XOR_White approach. In order to see some example of this approach please look at: http://www.geocities.com/complementarytheory/TheBestOf.pdf Yours, lama

Please think about circumference/diameter ratio, which does not depend on a circle's size. The same holds for the infinitely many triangles in post #1.

I pick lama.

Dear Matt, After your warm wellcome, please reply a detailed answer about post #1. Thank you.