Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 5 Dec 2006 09:07 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. 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.
From: Eckard Blumschein on 5 Dec 2006 09:29 On 12/5/2006 12:18 AM, Virgil wrote: (deleted) > In article <45747E16.6020904(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/1/2006 12:57 PM, Bob Kolker wrote: >> > Eckard Blumschein wrote:> >> >> Serious mathematicans have to know the pertaining confession. Dedekind >> >> wrote: "bin ausserstande irgendeinen Beweis f�r seine Richtigkeit >> >> beizubringen". In other words, he admitted being unable to furnish any >> >> mathematical proof which could substantiate his basic assumption. >> >> Consequently, any further conclusion does not have a sound basis. >> >> Dedekind's cuts are based on guesswork. >> > >> > Dedikind cuts are well defined objects which have exactly the algebraic >> > properties one wishes real numbers to have. So it is quite sensible to >> > identify the cuts with real numbers. >> > >> > We can define addition, mutliplication, subtraction and division for >> > cuts and the cuts satisfy the postulates for a an ordered field. >> > Furthermore every set of cuts (identified with real numbers) with an >> > upperbound has a least upper bound. Bingo! Just what we want. >> >> No wonder. Exactly this selfdelusion was the intention of Dedekind. Those who declared Cantor naive overlooked Dedekind. Learn German and enjoy his childish innocent style.
From: Eckard Blumschein on 5 Dec 2006 09:39 On 12/5/2006 12:16 AM, Virgil wrote: > In article <4574755B.4070507(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/1/2006 4:56 PM, Tony Orlow wrote: >> > Eckard Blumschein wrote: >> >> >> In case of two finite heaps of size a and b of numbers, a=b/2 implies a<b. >> > Generalize where possible. Why is this not true in the infinite case? >> >> 2*oo is not larger than oo. Infinity is not a quantum but a quality. > > Which "infinity" is that? We do not need different infinities. Cantor was correct when he considered infinity like something prefect. He called it actual because he believed that it is actually accessible. He called the so called potential infinity not really (uneigentlich) infinite. According to Spinoza, I refer to infinity like the quality to be neither enlargeable nor exhaustable. As a teacher of electrical engineering, I wrote or read this notion oo an estimated 10 000 times.
From: Eckard Blumschein on 5 Dec 2006 09:48 On 12/5/2006 12:09 AM, Virgil wrote: > In article <45745B16.40202(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/1/2006 9:59 PM, Virgil wrote: >> >> > Depends on one's standard of "size". >> > >> > Two solids of the same surface area can have differing volumes because >> > different qualities of the sets of points that form them are being >> > measured. >> >> Both surface and volume are considered like continua in physics as long >> as the physical atoms do not matter. >> Sets of points (i.e. mathematical atoms) are arbitrarily attributed. >> There is no universal rule for how fine-grained the mesh has to be. >> Therefore one cannot ascribe more or less points to these quantities. >> >> Look at the subject: Galileo's paradox: The relations smaller, equally >> large, and larger are pointless in case of infinite quantities. > > Then length and area and volume comparisons of size must be fictional > measurements. I do not think so. You deliberately mistook the sentence. No measured length, area, or volume is infinite. >> >> >> > Sets can have the same cardinality but different 'subsettedness' because >> > different qualities are being measured. >> >> I do not know a German equivalent to subsettedness. > > It is the quality of being a subset. It is one, but not the only, > measure of set "size". Cantor claimed to have discovered different sizes of infinite sets. Who introduced substtedness, and is it also quantified? >> >> TO>> You do? Do you mean that the addition of elements not already in a set >> >> doesn't add to the size of the set in any sense? >> > >> > In the subsettedness sense yes, in the cardinality sense, not >> > necessarily. In the sense of well-ordered subsettedness, not necessarily. >> > >> > TO seems to want all measures to give the same results, regardless of >> > what is being measured. >> >> Standard mathematics may lack solid fundamentals. At least it is >> understandable to me. However, I admit being not in position to likewise >> easily understand what you mean with well-ordered subsettedness. > > The subset relation does not provide a well ordering of arbtrary sets. Nobody does provide a well-ordering of the irrationals.
From: Bob Kolker on 5 Dec 2006 10:01
Eckard Blumschein wrote: > > > Those who declared Cantor naive overlooked Dedekind. Learn German and > enjoy his childish innocent style. He also gave a correct definition for the real numbers, too. There are other definitions which are equivalent. For example using limit points of Cauchy sequences of rational numbers which is the topological closure of the rational number space with interval topology and the usual metric. Bob Kolker |