Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Eckard Blumschein on 8 Dec 2006 07:04 On 12/7/2006 10:04 PM, Virgil wrote: > In article <45780B92.4000304(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/7/2006 4:10 AM, Virgil wrote: >> > In article <4576D4C7.4070207(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > >> >> On 12/5/2006 10:41 PM, Virgil wrote: >> >> > In article <457580FA.1030700(a)et.uni-magdeburg.de>, >> >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> >> >> >> >> >> >> That there are more rationals than reals? >> >> > >> >> > WE do not make any such assumption. >> >> >> >> I perhaps meant more reals than rationals. I apologize. >> >> >> >> Dedekind as well as Cantor started from this idea. Cantor even >> >> fabricated the notion cardinality in order to quantify the putative >> >> difference in size. Considering the rationals a subset of the reals, >> >> dull people conclude that there must be more reals than rationals. >> > >> > There certainly cannot be fewer, or even just as many, so what other >> > option is open/ >> >> the 4th one > > I think that to find that 4th one must use most of a 5th. Sorry, an insider has to know: There are exactily 4 options, cf. e.g. Fraenkel 1923 or a good book on logic.
From: Eckard Blumschein on 8 Dec 2006 08:02 On 12/7/2006 9:58 PM, Virgil wrote: > In article <45780ABE.4070108(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/7/2006 4:32 AM, Virgil wrote: >> > In article <4576DCFC.6040901(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> >> >> >> Every single integer is a countable element. >> >> > >> >> > Since the set of integers is a subset of the set of reals, every integer >> >> > is a real in any real set theory, whatever EB may claim. >> >> >> >> Just this is fallacious. The reals are continuous like glue. >> > >> > Being like glue is not a mathematically relevant quality. The Dedekind >> > cut for a real, e.g., for sqrt(2), is as actual as is the rational for >> > 1/2. >> >> >> Dedekind dealt with numbers ordered like points along a line. He assumed >> to consider _all_ points and _all_ rationals. This was the elusive basis >> of several fallacies. See my reply this morning. > > The only fallacies here are by those who proclaim that ZF or NBG must > have internal contradictions because they do not conform to their > critic's intuitions. Forget ZF, NBG, and intuitions altogether. The original topic is Cantor and confusion, your confusion. >> >> >> >> Embedded genuine numbers lost their numerical address >> >> >> >> > Countability >> >> >> > /Uncountability are properties of -sets-, not individuals. >> >> >> >> >> >> Do not reiterate what I know but deny. >> >> > >> >> > Then do not reiterate your falsehoods >> >> >> >> Cantor's proof (DA2) of uncountability required numbers of perfectly >> >> infinite length. >> > >> > Cantor's first proof >> >> Would you please clarify what proof or paper you are referring to! > > The one before what you mislabel "DA2" If you refer to Cantor's so called 1st diagonal argument stolen from Cauchy, then you may reiterate what you intended to say. DA1 just illustrates that discrete points can be arranged along a line. So rational numbers are countable. > >> > You understand: Such numbers contradict common sense. >> > >> > Common sense is irrelevant in mathematics. >> >> It is only irrelevant if it contradicts to well-founded theory and has >> proven wrong. In case of Cantor's transfinite numbers all putative >> proofs turned out to be not correctly founded. > > According to EB, who confesses that he is not a mathematician. Nonetheless I can read and reason. >> >> Writing very fasr and without proofreading, I cannot guarantee correct >> >> command of my English. Sometimes words or letters may be missing or >> >> confused. Nevertheless, all these arguments of mine are most likely >> >> flawless. >> > >> > Your flaws in English are excusable, your flaws in logic are not. >> >> I will hopefully understand at least one flaw of mine in logic, provided >> you are correct. Do not hesitate pointing me to what you consider wrong >> and why. > > Been there! Done that! Have the t-shirt! A dirty kind of surrender.
From: stephen on 8 Dec 2006 08:07 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> 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. Set theory may be of limited use, but it's supremacy is > complete nonsense, and will be overruled in time. > Han de Bruijn But everything can be modelled as a set. You simply do not understand what "foundation" means in this context. Any calculation can be rewritten as a set theory problem. It would be long, cumbersome, and impractical, but it could be done. Just as an computer program can be transformed into a Turing machine. I do not know what you mean by "supremacy". Do you think 486 assembly language is the "supreme" programming language? It currently is sort of a de facto candidate for a foundational programming language. Stephen
From: stephen on 8 Dec 2006 08:08 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > Bob Kolker wrote: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >>> You cannot imagine the integer [pi*10^10^100]. >> >> That is not an integer, dummkopf. It is an irrational real number. > Dummkopf? Who? Doesn't "[ .. ]" stand for the "floor" function? > Han de Bruijn Not anywhere with which I am familiar. Stephen
From: Eckard Blumschein on 8 Dec 2006 08:48
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. > And why > does "old" make it any more correct than "new". As a rule new things get ripe and also reliable as older ones. 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. > 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". Cardinal numbers are not incorrect but the notion cardinality lacks a reasonable basis. >> > 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. > 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. >> 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. |