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From: Will Twentyman on 13 Apr 2005 15:32 Eckard Blumschein wrote: > On 4/12/2005 11:29 PM, Will Twentyman wrote: > > >>>All this just reflects Cantor's claim that one can attribute different >>>quantities to infinity. Size means about the same as quantity, >>>Maechtigkeit or cardinality. It would be meaningful if it was correctly >>>founded. >> >>No, it is meaningful because it is precisely *defined*. > > I disagree. For me, a precise definition is sufficient to give a meaningful notion of different "infinite sizes". If that is not sufficient for you, what would be? If nothing, we will have to agree to disagree. >>>Let me also ask and answer how it was introduced: >>>Is cardinality really a big useful mathematical truth? I cannot confirm >>>that. Anything started at December 7th, 1871 when Cantor presented his >>>proof for the reals to allegedly be more than just countable. After the >>>won war against France, this was a time of euphoria. What a miracle! >>>More than infinite, and the best: Even the most posh people failed to >>>refute Cantor's claim. Well, there was a lot of quarrel. Cantor himself >>>named about 30 opponents of his theory, some of them very famous ones. >>>When he got mentally ill this was taken an indication for the huge >>>effort he made in order to create something epochal. The soap opera >>>continued with Bertrand Russell, Zermelo, and many others who attached >>>to the glory and took the attention away from the fact that a serious >>>basis is missing. >> >>The usefulness of Cantors ideas appear in categorizing sets, as well as >>defining some subsets of the reals with interesting measures. Now >>whether there are applications for the other sciences, I don't know. > > > Maybe, these categories and subsets of the reals are likewise junk. > Any real scientific breakthrough has a record of very widespread benefits. In mathematics, breakthroughs are often not realized until years or decades later. Complex numbers existed for a few hundred years as a curiousity. It wasn't until recently that applications were found for them. >>>>How can you say it is not more precise? >>> >>>Please read yourself how Cantor tried to answer objections. Would you >>>call it precise if he was not even able to convincingly explain how he >>>imagines his infinite whole numbers: even, odd or what? Would you call >>>it precise if Cantor mentioned Aristotele and Spinoza and declared they >>>were wrong without to explain why? Would you call it precise when Cantor >>>admitted that an opponent was correct but then he veiled the difference. >> >>Yes, > > How do you imagine and explain infinite whole numbers? I imagine them as sets, in this context, just as I imagine finite whole numbers as sets. 0 = {} 1 = { {} } = {0} 2 = { {} , {{}} } = {0,1} 3 = { {} , {{}} , {{},{{}}} } = {0,1,2} .... N+0 = N = {0,1,2,3,...} N+1 = {N,0,1,2,3,...} N+2 = {N,N+1,0,1,2,3,...} .... The above is also an explanation. >>no, no. However, the second no needs to be qualified, since his >>work is a precise explanation of why. I would not consider Aristotle a >>good choice for the foundation of modern mathematics. > > sometimes, I would rather trust in ancient than in certain modern > mathematics. If you were talking about philosophy, I'd be inclined to agree with you. In mathematics, however, the precision in thinking and standards in terminology have changed over the years. For example: Euclid's Elements is a wonderful work, but does not correspond at all to how geometry is viewed today, such as attempting to define things which cannot be precisely defined, or assuming a particular model. >>>An infinite number is by no means more precise than infinity or any >>>number. It is simply self-contradictory. >> >>There is a difference between cardinality and infinite numbers. Which >>are we talking about? Which do you object to? Neither requires the other. > > The close link between both is best to be seen from Cantor's original > work and also from David Hilbert's speech in honor of Weierstrass in > 1925. Transfinite cardinalities were thought like transfinite numbers, > numbers in excess of infinity. I understand that they are related concepts, but you seem to view them as a single whole. One can exist without the other. I also think you should examine ordinals as a different way of viewing "infinite numbers". >>>>You can use cardinality to >>>>compare the sizes of N, P(N), and P(P(N)). >>> >>>I accept that one has the freedom to define card(N). >>>However, the power set of N is not qualitatively related to it. >>>If it is non-countable infinite, then it has the quality oo and also the >>>quality to be uncountable, as also has P(P(N)). >> >>However, there are no bijections between any of those three sets, and >>any mapping from N to P(N) or P(N) to P(P(N)) are limited to injections. >> This can certainly be defined to be a qualitative ranking of the three >>sets. Certain types of mappings either do or do not exist. > > I agree that it is impossible to map an uncountable set. As stated, that doesn't make sense. You agree that it is impossible to map an uncountable set *to what*? >>>I know that Cantor handled his cardinalities like numbers. However, this >>>is neither justified nor advantageous in any sense. >> >>It is convenient for ranking them. If I have several classes of sets, > > I am only aware of two different infinite sets: countable ones and > non-countable ones. I understand how Cantor fabricated even more than > more than infinite numbers. > > >>I >>would like to have a notation that is strongly suggestive of the >>existence or non-existence of a surjective (possibly bijective) map >>between any two sets in the classes to be compared. > > Why? If uncountable numbers cannot be mapped, then I do not see any > reason to conjecture anything. I have a feeling you are very confused about what it means to map a set to another set. Whether it's a set of numbers or something else is irrelevant. I get the sense that you don't know what I mean when I talk about maps between sets. Another word for "map" would be "function". >>The numbers give an >>intuitively understood labeling system that will allow someone to >>understand at a glance what types of maps can exist. Where do you not >>see the advantage of such a labeling system? > > In the case of uncountable numbers, because they cannot be mapped in > general. Again, this makes no sense. >>>>You can use cardinality to compare the sizes of N, P(N), and P(P(N)). >>>>oo does not distinguish >>>>between them at all, and countable/uncountable does not distinguish >>>>between the last two. >>> >>>That is true. The reason why there is no justification and no reason for >>>this distinction is in principle the same as expressed by Hilbert's >>>hotel. In so far the whole Cantorian concept is not even consequent. >> >>Hilbert's hotel only explains why the union of two countable sets is >>countable. How is it relevent to the rest of this discussion? > > If the union of two countable sets is countable then this is > repetitiously valid. P(P(N)) is the repetitious application of the power > operation. If something is uncountable then this quality cannot be > compared in grammatical sense. Aleph_2 sounds at least as amusing as > double death penalty. I guess, it has no useful application if it is > justifyable at all. IR+ is as uncountble as is IR no matter that IR* is > a subset of IR. All numbers constituting IR+ can be mirrored into IR and > vice versa. The countable union of countable sets is countable. P(N) is just the set of ALL subsets of N. No unions at all. Hilbert's hotel simply does not apply. >>>While I know these expressions, I wonder if aleph_2 has found any use in >>>application. >>>The countable infinite (IN, (Q ) makes sense to me, and the >>>non-countable infinite (IR) too. Anything else has to provide evidence >>>against the suspition that it is pure phantasmagora. >> >>RxR is aleph_1, so P(RxR) is aleph_2. P(RxR) is also the set of all >>relations on R. This has potential impact on various topics in >>analysis. I don't know off hand if it does, but it could. > > Hm. > > >>>Please tell me what operation or whatever it makes more convenient to >>>you. I only know tremendous trouble with it. >> >>Comparison of sets. > > Of infinite sets. No, comparison of all sets, finite or infinite. Cardinality is not a concept limited to infinite sets. > In that case there is one reasonable comparison: > either countable or not. Anything else seems to be speculative and of > questionable use. I'm starting to feel like we're going around in circles. Perhaps you just don't care about higher cardinalities. That doesn't mean they are things that can't be talked about. Also, many mathematicians don't care about whether there is ANY use to the work we do. The uses may appear later. >>>I would not have any reason to complain if Cantorian set theory was >>>satisfactory to me. I hope, overdue abandoning of Cantor's fallacious >>>infinite numbers will enforce a more reasonable rebuild of set theory. >> >>It is obvious that you don't care for it from top to bottom. Are there >>any particular definitions or axioms you object to? If it's just the >>"infinite numbers" you object to, that will not remove cardinality. >>Note: when you say oo+a=oo, you are talking about cardinality. > > Transfinite cardinals are infinite numbers. This would perhaps not cause > much damage. I would like to put my finger squarely onto an other moot > point. Cantor's introduction of numbers beyond any sense distracted > attention from important peculiarities of the real numbers. All experts > will agree that the continuum is something special. Cantor ascribed this > enigma to the silly idea of more than infinitly many numbers while he > did not question the identity of the real numnbers. His set theory > treats real numbers as if they were just natural ones. I am not sure > whether or not it is really justified to believe that all axioms are > valid for the real numbers too. In particular, AC does not remedy the > problems of lacking identity and missing successor. I rechecked M280... I have no idea where you got this notion of the reals lacking identity. Similarly, your idea of a missing successor is unclear. What is clear is that you do not know the definition of "countable". >>>Would you regard someone correct who just followed his intuitive guess >>>and therefore performed operations that were and are still incorrect >>>except for the idea that one declines to decide whether the operand is >>>infinite or a number? >> >>Now I don't have any idea what you're referring to. I get the sense you >>feel there are no proofs associated with Cantors work. > > I dealt with most of his proofs in detail. The decisive one was his > second diagonal argument using a suggestion by Paul du Bois Reymond. I > already explained that this argument suffered from wrong interpretation. My summary: You say it doesn't do what it claims to do because the case is actually what it proves. The only response I can have to that is: so you agree it's right? >>>>>>>2) Why did he manage to find so much support? >>>>>> >>>>>>Because his results are consistent with the axioms and definitions he used. >>>>> >>>>>That is definitely not true. Read the original papers! >>>> >>>>See the note above regarding my skills with German. >>> >>>Well, you might deal with my arguments independently. >> >>I am attempting to do so. I suspect we disagree on the axioms or the >>validity of the proofs. That or you simply think the definitions or >>their popularizations are nonsense. > > No. We disagree because I found out that Cantor misinterpreted the fact > that the reals are not countable. He assumed that one of two sets must > have either a smaller or the same or a larger size than the other one. > While this is correct for finite sets, it is wrong for infinite ones. Translation: we disagree on the definition of "larger" as applied to infinite sets. >>>>>No. He just made the wrong assumption that the reals can be mapped. This >>>>>cannot work despite of AC. >>>> >>>>If you accept that the reals are numbers, then they can be mapped to >>>>*something*. >>> >>>Please check the basis for such confidence. Perhaps, you just reiterates >>>what was told to you. You can certainly just draw a line as to include >>>infinitely many reals. The unresolvable problem is: Nobody is able to >>>resolve this line into all single reals. We know how Cantor understood >>>mapping, cf. his diagonalizations. >> >>f(x)=x is a mapping from the reals to the reals. > > One already fails to map x itself in the sense of listening the reals > one by one onto a something like a matrix or the like. A mapping doesn't have to be something that can be represented in a list or matrix. A mapping between a set and N can be represented as a list, however. A mapping is just a function from one set (domain) to another (range). >>>In that I follow Cantor's definition, and additionaly I accept that e.g. >>>pi is a real number. However, the real numbers are something special in >>>that they lost the property of ordinary numbers to be numerically >>>identifiable. >> >>The second sentence didn't make sense to me. Are you talking about some >>reals being non-constructible? > > In order to understand the notion of infinity you have to abandon any > idea that you can reach it by means of counting/numbers. Infinity is a > different quality. What do you mean by "numerically identifiable"? Infinity is not mentioned in the confusing sentence. >>>Can you please send me a map of the reals? I guess: It does not work. >>>Forget cardinality together with AC. Forget this dark German >>>megalomaniac chapter in history of mathematics. >> >>A map is from a set to a set. For example: f(x)=x is a map from R to R. >> What do you think a map is? > > In general, a map is a plan to look at, usually on a sheet of paper. > Well maybe, I should call it a list in order to be understood correctly. > List has also different meaning. So try to guess what I might call map > or list. When discussing set theory, you are wrong. A map is a function from one set to another. Since f(x) is a function from R to R, it is also a map from R to R. It is a mathematical object. It has nothing to do with roadmaps, etc. -- Will Twentyman email: wtwentyman at copper dot net
From: Will Twentyman on 13 Apr 2005 16:11 Eckard Blumschein wrote: > On 4/13/2005 2:22 PM, Arthur Fischer wrote: > >>Eckard: >> >>Just out of curiosity, could you provide your definitions for the >>following concepts: >> >>- finite set >>- infinite set >>- countable set >>- uncountable set >>- non-countable set >>- enlarging a set >> >> >>Of course, mathematically precise definitions would be preferable, and >>dictionaries do not, in general, provide for such definitions. > > > Regrettably my time is limited. The pertaining definitions are easily > available except for the last one. I would just like to try and comment > on "enlarging a set" in case of infinite sets: This simply does not work. It is very difficult to comment on your assertion without a definition or some sort of explanation as to what "enlarging a set" is supposed to mean. -- Will Twentyman email: wtwentyman at copper dot net
From: Will Twentyman on 13 Apr 2005 16:35 Eckard Blumschein wrote: > On 4/13/2005 3:11 PM, Matt Gutting wrote: > >>Eckard Blumschein wrote: >> >>>On 4/12/2005 11:46 PM, Will Twentyman wrote: >>> >>>>>Cantor was mislead by his intuition. >>>>>I do not attribute the difference between countable and non-countable to >>>>>the size of the both infinite sets. >>>> >>>>What do you view the difference between them to be? >>> >>>The difference resides in the property of each single real number >>>itself. Cantor assumed his list represents all real numbers. Actually, >>>nobody can provide any list of real numbers, not even two subsequent of >>>them can be named. >> >>Cantor assumed such a list could be made, in order to prove that "nobody >>can provide any list of real numbers". > > I could not find this sentence in his original papers. Look for the statement of the theorem, along with a definition of "uncountable". >>You appear to agree, then, with the conclusion of his proof. > > I disagree with the conclusion that there are real numbers in excess of > his allegedly complete list. Actually, his list was not complete because > nobody can really write such list. When I read this my first thought was the following: "I disagree with his conclusion. Actually, I agree with his conclusion." The only way I can make sense of your response is to assume you don't know what he was trying to say. >>>This was your statement. Where is your evidence for it or at least some >>>justification? Simply tell me the successor of pi. >> >>What do countability and successorship have to do with each other? > > Any countable set is bijective to the natural numbers. Tell me any > natural number, and I will tell you its successor. > Please tell me the successor of sqrt(2). However, bijections are not unique. There is a bijection between the natural numbers and the integers. One bijections would cause the successor of -1 to be 1. A different bijection would cause the successor of -1 to be 100. Without knowing which bijection I have in mind, you cannot predict what the successor of a particular integer will be. Asking for the successor of sqrt(2) is meaningless in that context. >>The rationals >>can be ordered in such a way that each one has a defined successor. > > That's it. But there are many different such orderings, none of which need be consistent with each other. There is no canonical "successor" for a rational number. >>They can >>also be ordered in such a way that none has a defined successor. This ordering >>has nothing to do with whether the set is countable or not. > > This is a typical fallacy. The standard < relation does an excellent job of this. If you claim the successor is the "next largest" rational, you cannot find it because the rationals are dense under the < ordering. >>Further, you are asking Will to tell you the successor of pi. This request seems >>to indicate that you think he believes the reals to be countable. > > No. I just argue that reals deserve to be treated differently from > ordinary numbers. Well, since I consider reals, rationals, and complex numbers all to be "ordinary numbers", you may want to be a little more precise. Depending on what you mean by ordinary numbers, I can probably find all kinds of differences between them. -- Will Twentyman email: wtwentyman at copper dot net
From: Matt Gutting on 13 Apr 2005 20:15 David Kastrup wrote: > Matt Gutting <tchrmatt(a)yahoo.com> writes: > > >>Eckard Blumschein wrote: >> >> >>>This was your statement. Where is your evidence for it or at least >>>some justification? Simply tell me the successor of pi. >> >>What do countability and successorship have to do with each other? > > > Uh, everything? > > >>The rationals can be ordered in such a way that each one has a >>defined successor. They can also be ordered in such a way that none >>has a defined successor. This ordering has nothing to do with >>whether the set is countable or not. > > > It pretty much has everything to do with it. If you can make a subset > of the rationals obey the Peano axioms, then this subset is in > one-to-one correspondence with the natural numbers. If you can make > the entire rationals obey the Peano axioms by choosing a different > successor relation), then the rationals are in one-to-one > correspondence with the naturals. > I'm sorry, this was phrased badly. I realize that one can rearrange the rationals in such a way that one can define the successor of a number, and I realize that it is this fact that allows us to speak of the rationals (or any set) as being countable. There are many ways to do so for a given set, so that it makes no sense to talk about *the* successor of a number unless the ordering is either given or obvious. Because Eckard asked for "the" successor of pi, but did not specify an ordering, I assumed he meant "the successor of pi under the natural ordering of the reals". What I wanted to point out is that the lack of a successor under that ordering, or under any given ordering, did not *in itself* imply the lack of a successor under any possible ordering. (The same is true for rationals - no rational has a successor under the natural ordering on Q, but that does not hinder Q from being countable.) I probably should have said "What do countability and successorship IN A GIVEN ORDERING have to do with each other?" Matt
From: Dik T. Winter on 13 Apr 2005 21:54
In article <425D4F63.2010500(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > On 4/13/2005 2:53 PM, Dik T. Winter wrote: > > In article <425D0A7B.50309(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > > ... > > > The difference resides in the property of each single real number > > > itself. Cantor assumed his list represents all real numbers. Actually, > > > nobody can provide any list of real numbers, not even two subsequent of > > > them can be named. > > > > Isn't sqrt(1), sqrt(2), sqrt(3), sqrt(4), ... a list of real numbers? > > Of course I meant a complete list of real numbers regardless on what > base: e.g. decimal or hexadecimal Eh? The base has nothing to do with it. You can not give a complete list of real numbers, period. This only means that there is no bijection between the natural numbers and the reals. And that is all. The reason is *not* that you can not name the successor of a real. You can also not name the successor of a rational number, but there is a bijection between the rational numbers and the natural numbers. Or do you know the successor of 1/3 in the rational numbers? Ordening has nothing to do with the bijection. Moreover you wrote (and I insert the word "all"): > > > itself. Cantor assumed his list represents all real numbers. Actually, > > > nobody can provide any list of "all" real numbers, Yes, and that is what Cantor showed. I.e. he gave a proof that a list of "all" real numbers does not exist, whatever way you order them. And so his primary assumption was false. You do not give proof that his assumption is false; he gave that proof. You only base yourself on not being able to giving a list when you retain order, but you do not show that a list is not possible when you impose a different order on the reals. And it is exactly *that* what was shown by Cantor. But let's try the following thought experiment. Consider the reals greater than 0, but less than or equal to 1. Start with 0.1. Define as the successor of a real number the number where the first digit is increased by 1, if that first digit was 9, make it 0 and increase the second digit by 1, etc. (I.e. add 0.1 with backwards carry.) So we get the ordering: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.01, .... 0.91, 0.02, .... 0.92, .... Now this successor function works perfectly well on *all* reals (%), so for each real we can find a successor. On the other hand, there are pairs of reals of which we do not know which comes first. We do not even know whether 1/3 comes before 1/7 in that sequence or not. So we now have a clear successor function, but we still can not do the mapping. -- (%) as always we have to adjust for infinitely many 9's or 0's... -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |