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From: David Kastrup on 13 Apr 2005 08:10 Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > On 4/12/2005 11:21 PM, David Kastrup wrote: > > >> And the core of Cantor's argument is sound, > > It is fallacious. He introduces non existing infinite numbers in order > to misinterpret the unquestionably different qualities of rational und > real numbers. Just the latter are uncountable. Cantor's thinking is at > best fallatious. He ignores that both the rational and the real numbers > do likewise have the quality to be infinite, and infinite cannot be > enlarged. You are babbling pseudophilosophical sophistic hogwash. That is simly irrelevant to the math. Cantor showed a strict ordering of surjectability between certain sets, and this property of surjectability can be expressed in the language of cardinalities. Whether you want to assign names to cardinalities or not, the sets _are_ ordered into classes that can, with regard to surjectability, be ordered. > Consequently, the missing possibility to represent the reals in a > list cannot be attributed to a larger size (Maechtigkkeit, > cardinality) of the reals but it relates to something else. It does not matter to what it "relates". You are trying to protest against people putting the cart before the horse, when the problem just is that you are facing backwards. The fundamental mathematical issue is the existence of surjections between sets, and an order established by that. Cardinalities are just a convenient means for describing that order. They are descriptive, not prescriptive. >>> So he tried to show that there are more reals as compared to the >>> "size" of the set of the rationals. >> >> Nope. He showed that no bijection can be established. > > His second diagonal argument was an evidence by contradiction. Proof by contradiction. > Cantor misinterpreted it by claiming that there are more real numbers > than his list contained. Nonsense. Cantor never made such a list. And he actually _proved_ that no such list can be created. > He overlooked the correct possiblity that his assumed list simply > did not represent the reals. This is too stupid for words. He _proved_ that the "assumed" list does not represent the reals. This, and nothing else, is what the proof is all about. How Cantor can "overlook the correct possibility" that his proof shows what it indends to prove is beyond me. You really have no clue whatsoever what you are talking around. >> And that means that there is an order of cardinalities, where >> cardinalities are considered as an indicator of surjectability of >> sets. > > Please indicate the pertaining pages where he claimed and proved > that. This is what cardinalities are about. That is their whole and single purpose. Ordering surjections of sets. You don't "prove" names for entities and properties. You define them. >>> I argue that such comparison lacks any basis. Infinity is a >>> quality, not a quantity. >> >> You are just waffling around with stupid terms. > > The difference between quality and quantity is quite fundamental. You are waffling. The relative orderings of sets do not depend on terminology. >> The existence and non-existence of surjections is a _hard_ fact >> that has nothing to do with any philosophy of "infinity". And it >> also is a hard fact that being surjectable is a transitive and >> reflexive property. > > What do you think about surjection between IR and IR+? exp(x) maps IR onto IR+, ln(x) maps IR+ onto IR. Equal cardinality, since a bijection exists. >> If the reals obeyed the laws of ordinary numbers, they would be >> structural equivalent to them and could be put into a one-on-one >> correspondence with them. > > Yes. I agree with you that the reals do differ from ordinary > numbers. > >> Cantor showed that this is impossible, and thus the reals fail to obey >> the laws of integers and can't be brought in correspondence with them. > > I agree. > >> This is what his second diagonal argument is about. Exactly that, >> and nothing else. > > So far I do not object. > And who added the whole cardinality story? Infinite numbers are insane. > Cantor got famous by means of a incredible misinterpretation. I am > suggesting a less spectacular explanation that does not contradict to > very fundamental rules. You are babbling. Cardinalities are just names for a relative ordering of sets based on surjections. There is nothing to interpret here. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Arthur Fischer on 13 Apr 2005 08:22 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. __ Arthur
From: Dik T. Winter on 13 Apr 2005 08:53 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? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Matt Gutting on 13 Apr 2005 09:11 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". You appear to agree, then, with the conclusion of his proof. > >>>Actually, infinity is not a quantity but a quality that cannot be >>>enlarged or exhausted. Whether or not an infinite set is countable >>>depends on its structure. The reals are obviously not countable because >>>one cannot even numerically approach/identify a single real number. >> >>No, that is NOT the reason the reals are not countable. > > > 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? 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. 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. He doesn't; he believes them to be uncountable, but disagrees with your statement about the reason why. Matt > > Eckard > >
From: Matt Gutting on 13 Apr 2005 09:33
Eckard Blumschein wrote: > On 4/12/2005 9:03 PM, Matt Gutting wrote: > >>>>I'm not sure what you mean by "loss of approachable identity". The >>>>difference between the rationals and the reals is that every convergent >>>>sequence of reals converges to a real, while not every convergent >>>>sequence of rationals converges to a rational. >>> >>> >>>In other words, numerical representations of rational numbers do not >>>require infinitely many numerals. Convergency invites to restrict to a >>>finite number of coefficients. In that case you do not reach a real >>>number but are satisfied by an rational approximation instead. Real >>>numbers are fictitious. >>> >> >>Numerical representations of reals need not require infinitely many >>numerals either. And whether even a rational number requires an infinite >>number of digits to be represented depends on the method of representation >>chosen. > > > Such objections were already cleared away in many discussions. Do you > really need resuming this issue? Please take into account that I (like > Cantor) refer to a uniform repesentation of all reals together e.g. the > decimal one. In this "system" the natural number 4 is embedded like > 4.000000000... =3.999999999999999... Infinitely many numerals are > essential even if we do not refer to an irrational number. > > > >>I'm not sure what you mean by "Convergency invites to restrict to a finite >>number of coefficients". Do you mean that to say "this sequence converges >>to the real number r" is to say that "r can be represented as a number >>which begins with the digits of one of the elements of this sequence"? >>That is true. However, what r *is* and what r is *approximated by* are two >>different things, and mathematicians keep that fact in mind. In this sense, >>the real numbers are not fictitious. > > > The real numbers exist mathematically in the sense they are fictions. > > >>>Even embedded natural numbers cannot be numericall identified without >>>all infinitely many numerals e.g. 3,99999999999999999999999999999... >> >>I thought you just said that "numerical representations of rational numbers >>do not require infinitely many numerals"? Since 4 = 3.999999... is an integer >>and therefore a rational number, you appear to be contradicting yourself here. > > > No. See above. > > > >>>Did you refer to infinite sets? >>> >>> >> >>No, but neither did you. You were speaking here, as I understood it, >>simply of how one compares one set to another ("Isn't a quantitative >>measure a quantity to compare with?"). > > > Yes. In that case I referred to finite sets. Okay, but I was intending to be able to refer both to finite and to infinite sets. The statements I made, although they didn't specifically refer to finite or to infinite sets, are applicable to both. > > >>My point is that one can describe a method of comparison between sets >>which (i) works for infinite sets exactly the way it does for finite sets, >>(ii) gives meaningful, well-defined comparisons for any two sets, (iii) >>does not refer to "quantity" in any way, and (iv) yields exactly the same >>results as a quantitative comparison does for those sets to which "quantitative >>comparison" applies. Having found such a comparison, > > > I wonder if you really found it. Concerning infinite sets I am only > aware of the possiblity do decide whether or not there is a bijection > being synonymous to countable. Very specifically, here is my method of comparison: Given two (for the moment, let's say non-empty) sets A and B, if one can find an injection, but not a surjection, from A to B, say that A "comes before" B or that A is "smaller than" B. Similarly, if one can find an injection, but not a surjection, from B to A, say that A "comes after" B or that A is "bigger than" B. Finally, if one can find both an injection and a surjection (that is, a bijection) between A and B, say that A "occupies the same position as" B or that A is "as big as" B. (I put "smaller than","bigger than", and "as big as" in quotes in the paragraph above to make it clear that these concepts are not to be literally interpreted in the usual sense. This is also why I provided equivalent phrasings not as laden with specifically mathematical meaning.) Since injections and surjections can be found between infinite sets as well as between finite ones, my method fulfills conditions (i) and (ii). Obviously it also fulfills condition (iii). It is a bit of work to show (iv), but it turns out to be true as well. So I would say, yes, I have indeed found such a comparison. To extend some of Will Twentyman's work here, the existence of a bijection between sets demonstrates that the two sets occupy the same position in an order of cardinalities. The lack of a bijection indicates that the two sets are in different positions in this order. One can find two infinite sets which occupy different positions in this order. > > > >>I don't >>see an immediate connection between "boundlessness" (something that cannot be >>exhausted or limited) and "something that cannot be enlarged". > > > Descriptions are indeed sometimes difficult. Try to describe the > principle of a sewing machine: Using a needle with the eye on top. > > > >>>A definition should not contain the defined expression. >> >>I believe your native language is German? > > > Yes. > > >>You may be confusing "infinite" (in this sense, perhaps "unbegrenzt")and "indefinite" ("unbestimmt"). > > > No. > > >>The definition does not, in this case, contain the defined expression. > > > It does. I refer to: > > >>>>"Infinity: >>>>2 an indefinitely great number or amount. > > > in English, "indefinitely" has two differnt meanings: > - without clear definition, vague e.g. a view > - without limit, e.g. a value > "2" relates to "without limit". I don't believe it does. In this case, the editors of the dictionary refer to the fairly common practice (especially in literature) of saying, for example, "an infinity of stars" when what they mean could also be expressed by "a lot of stars" or even "an unknown, but obviously very large, number of stars". In other words, this describes an English usage of the word "infinity" to describe a large finite number whose value is unknown. > > > >>Perhaps I misinterpreted what you meant by saying "A part of mathematics would >>go slippery..." Would you mind explaining that? > > > Just an example. Children at school must not be taught Cantor's nonsense > infinite whole numbers. There are no infinite numbers. No one is saying that there are numbers which are infinite. There are cardinalities that don't describe finite sets. And there are different cardinalities which describe different infinite sets. > > Eckard > |