Cantor Confusion
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From: Virgil on 8 Dec 2006 15:01 In article <45796D33.1070606(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 9:32 PM, Virgil wrote: > > In article <4577FC31.9000700(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> > Who gave you the power to dictate what a "number" is, EB? > >> > >> I just realized how brutally Cantor raped the old and correct notion of > >> number and feel safe with Gauss all the others. > > > > Where does one find that "old and correct notion of number"? > > Already in ancient times. > Even Stifel, who lived at the same time with Martin Luther understood > that the irreals behave like fog. > Gauss utterance concerning infinity is known. Equally "old and "correct notions" put the earth at the center of the universe with everything else rotating around it. The age of a notion does not guarantee correctness. Even Euclid's axioms have had to be revised. > > > And why > > does "old" make it any more correct than "new". > > As a rule new things get ripe and also reliable as older ones. Only because the older things that are wrong eventually tend to be found wrong and discarded. > Unfortunately, some illusions and in perticular beliefs got pandemic and > are not yet overcome. After more than a hundred years, Cantor's daring > ideas did not prove futile. And, I predict, will not prove futile over the next 100, or even 1000. > > > > There are lots of new "numbers", that are just as correct as old ones. > > I do not like Dedekind's booklet "Was sind und was sollen die Zahlen?" > but I cannot imagine him using the expression "correct numbers". I can not imagine him using "fictitious numbers" either. > > Cardinal numbers are not incorrect but the notion cardinality lacks a > reasonable basis. Injective, surjective and bijective functions are unreasonable? > > > >> > A "number" in mathematics is what the majority of mathematicians agree > >> > it is, regardless of what anti-mathematicians like EB try to dictate. > >> > >> The majority of really important mathematics perhaps lived before Cantor > >> or did not take issue towards his at best somewhat strange and > >> absolutely unfounded violation of the notion number. > > > > The vast majority of extant mathematics was created since Cantor's death. > > No. I measure the relevance of a theory not by counting more or less > useless papers. Cantor's whole set theory did not manage to justify > itself by an exemplary application of aleph_2. Requiring everything to have a use or application is not a requirement in mathematics. As Churchill once asked, "of what use is a baby?" > > > No one right now can say with any certainty how important that new math > > will untimately prove to be. > > I am certain that unfounded fancy will never become important. At least > I am not aware of such case. Your ignorance of a possible use does not bind future generations. Mathematics is rife with developments made without immediate uses that later became highly useful. One of the latest being number theory. G.H. Hardy, who lived until 1947, prided himself that his work in number theory would never have any practical use, but almost lived long enough to see how wrong he was. It is now the basis for all secure electronic transmission and storage of information and currency. > > > > > >> If there was really general agreement among mathematicians, then there > >> would be an acceptable printed definition. Since Cantor's definition of > >> set has been declared untennable without substitute, > > > > By whom? > > Adolf F. And who made him God?
From: cbrown on 8 Dec 2006 15:01 Han de Bruijn wrote: > Lester Zick wrote: > > > On Thu, 7 Dec 2006 03:00:21 -0500, David Marcus > > <DavidMarcus(a)alumdotmit.edu> wrote: > > > >>If you use ZFC (or something similar) as your foundation for > >>mathematics, then everything is a set. Of course, while solid > >>foundations are good to have, if you are living on an upper floor, you > >>may prefer to ignore what is going on in the basement. > > > > So you're saying that set "theory" is all of mathematics? Of course > > since what you say isn't necessarily true that's not exactly a ringing > > endorsement of set "theory". > > It's quite simple. Set Theory can not be the foundation for mathematics, > because NOT EVERYTHING IS A SET. E.g. a calculation is mathematics, but > it's not a set. But calculations performed on the real numbers can, in principle, be translated to operations on sets and back again to obtain identical results. Of course, we might disagree whether something is mathematics, just as we might disagree over whether something is an example of Italian cooking. Can you give an example of something you consider mathematical that you feel cannot be modeled using sets as described by ZFC? > Set theory may be of limited use, but it's supremacy is > complete nonsense, and will be overruled in time. "Supremacy"? It's not somehow "better" than other branches of mathematics. The fact that in principle we can perform calculations equivalently by performing certain set theoretic operations doesn't mean that the latter is a "good" way of performing calculations. Set theory is primarily a tool used to unify different branches of mathematics by providing these different branches with a common language. Cheers - Chas
From: Virgil on 8 Dec 2006 15:29 In article <4579771C.1060402(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 9:15 PM, Virgil wrote: > > >> > The countability of a set requires no more than it have countably MANY > >> > members, but is totally independent of the nature of those members. > >> > >> No. The numbers also have to have an approachable address. > > > > Which axiom says that? > > The customer is king. > That may be a commercial axiom, but is not a mathematical one. > > > >> BTW: The meaning of "many" already includes countably. > > > > Where is that written in stone? > > German: how many = wieviele (considered a number) > how much = wieviel (considered a single entity) Mathematics is not tied to the German language. The peculiarities of one language are irrelevant. Anything that cannot be said in other languages as well is not mathematics. > > >> > >> > >> > > >> > For example: > >> > > >> > The set of square roots of prime naturals is as countable as the set of > >> > prime naturals itself by an obvious bijection, > >> > >> In this case you are using addresses approachable via bijection. You are > >> not really counting the square roots but primarily the prime naturals. > > > > In set theory, counting is done by bijection. So those square roots are > > /really/ counted as much as the primes are counted. > > Bijection is not yet counting. If one bijects with a standard set of objects, such as an ordinal "number" as a set, one has counted. And if one bijects with a set which has been so bijected, one also counts. So the square roots of the primes have, by that standard, been counted. > Bijection is not yet counting. You can also biject {Virgil, Cantor, > Dedekind} with {nice, naive, clever}. This is no counting, not even a > ranking. But two or more sets once having been pairwise bijected, counting one set counts all of them. And even without actually counting them one can then say they have the same "count".
From: Virgil on 8 Dec 2006 15:31 In article <bed62$45797cde$82a1e228$28318(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > stephen(a)nomail.com wrote: > > > But everything can be modelled as a set. > > Define "everything" and prove that claim. > > Han de Bruijn Within ZF or ZFC everything that can be modeled must be modeled as a set as that is all one has to model with.
From: Virgil on 8 Dec 2006 15:38
In article <4579836F.4010806(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 8:51 PM, Virgil wrote: > > In article <4577F17F.7000005(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > >> Dedekind himself let the question open whether his cut belongs to the > >> left or the right side. He was also lacking insight. > > > > Actually, Dedekind saw that it made no difference which way it went, as > > long as one one was consistent about it. Those, like EB, who fail to see > > what Dedekind saw are the ones lacking insight. > > One one? Perhaps a typo. > > Dedekind's lack of insight is obvious: He imagined exemplary rationals > like distinct points on the continuous line. Euclid did the same with "commeasurability". But Dedekind did not need any "line" to do this, as he could construct each rational as a set and take the set of all of them with a set as order relation on the set of all of them, etc. > On this basis he claimed to > be able to separate all points and all reals, respectively, into left > and right ones. While he may claim having defined a cut somewhere, he > cannot properly and directly attribute a belonging address. Those poor homeless cuts still exist as sets. And the set of all of them is a set with a definable order and arithmetic. Which set can be shown to have all the properties one wants for "the real ordered field". |