Galileo's Paradox
in [Math]
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From: David Marcus on 5 Dec 2006 23:19 Eckard Blumschein wrote: > Let's rank Tonico outside but David Marcus still inside mathematics. What does that mean? > On 12/5/2006 8:55 AM, Tonico wrote: > > David Marcus wrote: > >> Eckard Blumschein wrote: > >> > >> > Reals, as indirectly defined with DA2, > >> > >> Why do you think that the diagonal argument defines the reals? > > You all know that DA2 shows by contradiction that real numbers are > uncountable. I carefully read how Cantor made sure that the numbers > under test are real numbers. He did not use Dedekind cuts, nested > intervals or anything else. Well, of course he did't use Dedeking cuts, etc. No one uses the construction of the real numbers in proofs (once you've finished the construction). You just use that the real numbers are a complete ordered field. > He assumed numbers with actually > indefinitely much rather than many e.g. decimals behind the decimal > point. Strictly speaking, he did not immediately show that the reals are > uncountable but that these representation like never ending decimals is > uncountable. That's because anyone who took an analysis course in college (or maybe even freshman calculus) can prove (starting from the properties of a complete ordered field) the existence of the decimal representation of real numbers. > Being uncountable is the common property of these numbers under test. > To my knowledge, sofar nobody was able to show that the numbers > allegedly defined by Dedekind's cut or nested intervals are uncountable. Saying that to your knowledge no one has proved that the set of Dedekind cuts is uncountable just demonstrates how little knowledge you have. This is done in many, many standard textbooks. I first learned it as a freshman from Spivak's Calculus. > If we need the notion real numbers at all, then in connetion with the > common property to be uncountable. > > You might wondwer that there is no chance to define the reals at will. > Cantor made a false promise when he said the essence of mathematics just > resides within its fredom. > > Do you still not yet understand why DA2 lets no room as to define the > reals accordingly? Of course I don't understand it! What does "no romm as to define" mean? > >> I mean,you say lots of nonsense, > > in the sense it did not yet make sense to you. Yes, I've always had that problem, i.e., nonsense doesn't make sense to me. > > but I don't see where you got this particular > >> nonsense from. Did you read it in a book? > > I read several original papers by Cantor. The rest is reasoning. Well, I guess that explains it. If you want to understand/learn mathematics, you pretty much have to take courses, read books, and do the exercises. Kind of arrogant to think you can rediscover centuries of mathematics on your own. Even Ramanujan read whatever books were available to him. -- David Marcus
From: David Marcus on 5 Dec 2006 23:27 Eckard Blumschein wrote: > On 12/5/2006 1:12 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> On 11/30/2006 4:41 AM, zuhair wrote: > >> > Six wrote: > >> > >> > and I think it was the > >> > idea before Cantor showed that there can be infinite sets of different > >> > sizes, > >> > >> He did not! Not! Not! He just misinterpreted uncountable as more than > >> countable. Is incorrect more than correct? > > > > Are you saying that the cardinality of the reals is not greater than the > > cardinality of the integers? > > While I do not consider the generalized size (= Maechtigkeit alias > cardinatity) justified, I nonetheless fully agree with the distinction > between countable and uncountable. > > The reals are as uncountable as is sauce because they are merely > fictitious numbers, no distinct elements. > On the other hand, the naturals, integers, and rationals are genuine, > i.e. discrete numbers. > > Look up and recall Galilei's insight: The relations smaller, equally > large, and larger are invalid among infinite quantities. > > So it would not make any difference when the order aleph_0 and aleph_1 > were reversed or more reasonably the correct denotation was preferred: > countable instead of aleph_0 and uncountable instead of aleph_1. Yes, yes, I know you like to take standard words and give them new (vague) meanings. But, I was asking you whether the cardinality of the reals is greater than the cardinality of the integers if we use the standard meanings for the words "cardinality", "reals", and "integers". > Of course, there is no room for aleph_2 or even higher alefs in my > reasoning. Therefore I asked: Is there any use of aleph_2? So far I did > not get any example for such use. Obviously, all alefs are fancies. Only > in case of a_0 and a_1 there is a reasonable but misinterpreted background. Depends on how you define "any use". Certainly the hierarchy of alephs is very useful in mathematics. Cantor's motivation for his work was his study of Fourier transforms. As for aleph_2 itself, because of the independence of the continuum hypothesis, it doesn't come up too often in mathematics. However, there were some interesting articles recently in the Notices of the AMS that discussed axioms to add to ZFC. There seemed to be good reasons to add an axiom which would make the cardinality of the reals equal to aleph_2. -- David Marcus
From: David Marcus on 5 Dec 2006 23:35 Eckard Blumschein wrote: > On 12/4/2006 11:57 PM, Virgil wrote: > > In article <45744E30.8090207(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/1/2006 10:33 PM, Virgil wrote: > >> > In article <45706AD1.808(a)et.uni-magdeburg.de>, > >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> > >> > You claimed many "imperfectins" but did not justify those claims with > >> > anything mathematically valid. > >> > >> When I performed Fourier transform back and forth for a function > >> stepping at t=a, cf. > >> http://iesk.et.uni-magdeburg.de/~blumsche/M283.html > >> I correctly returned to the original function iff I ignored the > >> intermediate value at t=a and decided to extend integration from t<a > >> instead. > > > > That Fourier transforms do not do precisely as you wish they might, does > > not constitute an imperfection in the transforms so much as an > > imperfection of your understanding of what they can do. > > > > If you choose to hammer with a wrench, you may not always get the result > > a hammer would produce. > > You did not understand that I am using Fourier transform as an example. Example of what? > I do not criticize FT but the integral tables, and I did not have a > problem myself but I recall several reported cases of unexplained error > by just the trifle of two. The integral tables suggest using the > intermediate value. You are criticizing integral tables? > Experienced mathematicians should indeed know that > they must avoid this use. Some tables give the intermedite value for the > sake of putative mathematical correctness. Please give an example. > Others omit it. > As long as one knows the result in advance, there is almost not risk. FT > and subsequent IFT may perform an ideal check of set theory. What do you mean "ideal check of set theory"? > Feel free to suggest a better one. A better what? -- David Marcus
From: David Marcus on 5 Dec 2006 23:40 Eckard Blumschein wrote: > On 12/4/2006 10:47 PM, David Marcus wrote: > > Eckard Blumschein wrote: > >> Standard mathematics may lack solid fundamentals. At least it is > >> understandable to me. > > > > If it is understandable to you, then convince us you understand it: > > Please tell us the standard definitions of "countable" and > > "uncountable". > > I do not like such unnecessary examination. Then prove it is unnecessary. All of us who went to school took tests. > Countably infinite refers to bijection. That is not a definition. You have to be more precise than "refers to". > Uncountable is a correcting translation of the German nonsense word > ueberabzaehlbar = more than countable. Meant is: there is no bijection > to the naturals. As Virgil pointed out, "no bijection to the naturals" is not a correct definition of "uncountable". Care to try again? -- David Marcus
From: David Marcus on 5 Dec 2006 23:48
Eckard Blumschein wrote: > On 12/4/2006 9:49 PM, David Marcus wrote: > > >> According to my reasoning, the power set is based on all elements of a > >> set. > > > > "Based"? > > Yes. The power set algorithm does not change what mathematicians still > used to call cardinality. 2^oo=oo. "still used to"? What does that mean? You still haven't explained what you meant by "based". > > > >> In case of an infinite set, there are nor all elements available. > > > > "Available"? > > Yes. You cannot apply the algorithm until you have all numbers. So, "available" means "have". What does "have" mean? > >> Nonetheless I can do so as if they would exist, and I am calling them a > >> fiction. > > > > "Exist"? "Fiction"? > > Exist means, they have their numerical address within a rational order. Define "numerical address" and "rational order", please. > Fiction means, they don't have it but it is reasonable to do so as if. So, "fiction" means "not exist, but reasonable to do so as if". What does "reasonable to do so as if" mean? > >> Fictions are uncountable. > > > > "Uncountable"? > > The continuum cannot really be resolved into countable elements. It can > just be thought to consist of an actually infinite amount of fictitious > elements. Doesn't seem to answer my question. Let's try again: What does "uncountable" mean? > >> So this power set has no chance but > >> to be also uncountable. > >> Try to get the title cardinal Kolker, or at least Bob the Builder and I > >> will possibly convert. > > > > When you tire of religion, you could always learn some math. > > My topic is not relegion but outdated quasi-religious mathematics. You are the one reading old papers by Cantor instead of reading modern mathematics texts. Why do you think that what you read has any relevance to modern, current, up-to-date mathematics? -- David Marcus |