Courage?
in [Logic]
|
Prev: arithmetic in ZF
Next: Derivations
From: David Kastrup on 12 Apr 2005 17:03 rusin(a)vesuvius.math.niu.edu (Dave Rusin) writes: > In article <vcbu0mbx6es.fsf(a)beta19.sm.ltu.se>, > Torkel Franzen <torkel(a)sm.luth.se> wrote: >>Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: >> >>> You did nor by chance refer to Bertrand Russel who was, according to >>> Lavine, responsible for applying Cantor's basic idea to the reals? >> >> No, no! The furry evil creatures! Look out for them. > > That would be the Trolls, surely? Trolls are anything but furry. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: David Kastrup on 12 Apr 2005 17:21 Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > On 4/12/2005 1:59 PM, David Kastrup wrote: > >> You consider Cantor at fault for the reals not being in >> one-to-one correspondence with the natural numbers > > No. I consider him intending and deliberately drawing the wrong > conclusion from his second diagonal argument. To me it is clear that ^^^^^^^^^^^^^^^^^ > there is no one-to-one correspondence between reals and rationals. > Cantor was mislead by his intuition. ^^^^^^^^^^^^^^^^ If I have to choose between the intuitions of those two persons, the choice is pretty easy to make. But I don't have to choose based on the persons. I can choose based on the argument. And the core of Cantor's argument is sound, while there is no rhyme, reason or coherency to yours. > 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. And that means that there is an order of cardinalities, where cardinalities are considered as an indicator of surjectability of sets. > I argue that such comparison lacks any basis. Infinity is a quality, > not a quantity. You are just waffling around with stupid terms. 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. Associating names with certain equivalence classes of sets is not a philosophical problem, as you make it out to be. Those names require no physical properties or existence. They are a fast way to check for the relative size of sets as defined by the existence of surjections. That's all. >> You are also missing a clue. > > If you did not yet get the point you might look into M280. I have already looked into rubbish of yours, and it was distasteful to the extreme to see how clumsy a charlatan can nowadays earn academic degrees. >>> Platonian thinking is perhaps more appropriate. One cannot force >>> the reals to have the same properties as exhibited by ordinary >>> numbers. >> >> Congratulations. Exactly this, and nothing else, is what Cantor's >> diagonal proof was all about. > > You are joking. I am dead serious. 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. 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. This is what his second diagonal argument is about. Exactly that, and nothing else. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Will Twentyman on 12 Apr 2005 17:29 Eckard Blumschein wrote: > On 4/12/2005 12:02 AM, Will Twentyman wrote: > : > >>>>Eckard Blumschein wrote: >>> >>>>>>>http://iesk.et.uni-magdeburg.de/~blumsche/M280.html > > >>>You are correct in that meanwhile nobody asks for why and how >>>cardinality was introduced. >> >>This seems unlikely at best. Without looking at any notes, it seems >>like cardinality gives a meaningful method of comparing the "size" of >>sets and defining what we mean by "size". > > > 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*. > Let me ask and answer the question why it was introduced: > Cantor's thinking was strongly guided by his intuition. That was one > reason why his unbelievable claims were so appealing. Cantor himself > wondered after he got evidence for what already Albert von Sachsen > (1316-1390) found out. A more intelligent man would have expected that > the whole universe does not contain more points than any linear > interval. Obviously, Cantor was unable to grasp the notion infinity. > Maybe his teacher Weierstrass is to blame for that. As known, W. spoke > of infinite numbers. So he failed to make quite clear that infinity and > numbers strictly exclude each other. It seems odd to accuse someone of not understanding infinity when he worked with it so much. [snip] > 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. >>It is a natural extension of >>the cardinality of a finite set. > > I wonder if cardinality of finite sets ever played any role except for > embellishmet to transfinite cardinality. Perhaps not. >> > The underlying notion of infinite whole >> >>>numbers is undecided between infinite and numbers mutually excluding >>>each other. Cantor's thinking was correspondingly split. He did not >>>decide between the meaning oo of what he called Maechtigkeit and later >>>cardinality and his intention and pracice to use it as and like just a >>>number that served as an infinite or even more than infinite measure of >>>quantity. So it is different from infinity but definitely not more >>>precise. It is just nonsense. >> >>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, 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. His reasoning was good, but even that has potential flaws. > 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. >>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 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 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. 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? >>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? >>>As far as I know, the only decisive question is whether or not an >>>infinite set is bijective to the set of natural numbers. In that case it >>>is calles countable infinite, else non-countable. >> >>P(N) and P(P(N)) are also standards. This is where aleph_0, aleph_1, >>aleph_2 get started. Both aleph_1 and aleph_2 are uncountable, but they >>are different cardinalities. > > > 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. >>>Cantor got (in)famous just because he claimed to have revealed different >>>levels of infinity. He was a bluffer. >> >>Most people > > Is this not possibly exaggerated. Perhaps. >>seem to find it convenient as a measure of levels of >>infinity. I do. > > Please tell me what operation or whatever it makes more convenient to > you. I only know tremendous trouble with it. Comparison of sets. >>If you don't care to think of it that way, just focus >>on the definition as it applies to sets. > > 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. >>>>I don't think he was planning >>>>on the results. >>> >>>He was keen to create the most unusual. >> >>Most unusual or most correct? I've seen several odd ways of looking at >>things that are useful for easily getting a correct result. > > > Would you regard someone correct who deliberately ignored Aristotele, > Cauchy, Galilei, Gauss, Kronecke, Lebniz, and many others without any > convincing argument for that. Perhaps not convincing to you, anyway. > 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. >>>>>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. 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. f(x)=x^2 is a mapping from the reals to the non-negative reals. A mapping is between sets. The characterization of the mappings that was of concern was the domain and range of the mapping, not the individual elements. >>If you don't accept that the reals are numbers (which I >>think is your position), then the entire topic of conversation is moot. > > In that I follow Cantor's definition, and additionaly I accept that e.g. > pi is an 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? >>>Do not deny compelling arguments. Neither my feeling nor yours matters. >> >>Which is why I work with the ZF version of set theory and accept the >>various infinite cardinalities. > > You might feel that ZF or ZFC is appropriate. There is no tenable basis > for that. True. However it preserves the spirit of Cantor's set theory to the best of my knowledge and is the version I am most familiar with. >>>Such map does not exist. There is no approachable well-ordered map of >>>the reals. >> >>AC asserts (without proof and non-constructively) a well-ordering of the >>reals. There is no bijective map between the reals and rationals >>precisely because they have different cardinality. > > I do not like arbitrary definitions with the only purpose to conceal > mistakes. > 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? -- Will Twentyman email: wtwentyman at copper dot net
From: Will Twentyman on 12 Apr 2005 17:46 Eckard Blumschein wrote: > On 4/12/2005 2:00 PM, Barb Knox wrote: > > >>I don't understand: here you appear to accept the distinction between >>countably and not-countably infinite, yet your main point in this thread >>seems to have been that there is only a single "oo" that cannot be added to >>or otherwise extended. How do you reconcile those 2 views? > > > 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? > 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. -- Will Twentyman email: wtwentyman at copper dot net
From: Gerry Myerson on 12 Apr 2005 19:30
In article <d3gaqq$bca$1(a)panix2.panix.com>, lrudolph(a)panix.com (Lee Rudolph) wrote: > Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> writes: > > >In article <vcb4qec7718.fsf(a)beta19.sm.ltu.se>, > > Torkel Franzen <torkel(a)sm.luth.se> wrote: > > > >> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > >> > >> > Since it is > >> > impossible to completely write down all infinitely many numerals of just > >> > one single real number, it is also impossible to name its successor. > >> > >> The impossibility of naming the successor of a real number is indeed > >> the central flaw in today's mathematics. Little can be done about it, > >> I'm afraid. > > > >Fortunately, no real number has ever died, so the problem > >of naming a successor has not arisen. > > Then what are all those cardinals doing in a conclave, eh? Eh? Offhand, sir, I'd say the backstroke. (Do you think we'll ever see cardinals in a convlex?) -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email) |