Galileo's Paradox
in [Math]
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From: Virgil on 6 Dec 2006 14:17 In article <4tnmr2F142pnkU2(a)mid.individual.net>, Bob Kolker <nowhere(a)nowhere.com> wrote: > Virgil wrote:> > > EB's arguments give us a plethora of examples of both unfoundedness and > > the uselessness. > > That is why EB should be given the Zick Prize. > > Bob Kolker Isn't that pronounced "sick"?
From: David Marcus on 6 Dec 2006 14:18 Eckard Blumschein wrote: > On 12/5/2006 11:10 PM, Virgil wrote: > > In article <45753CB8.3080500(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> You all know that DA2 shows by contradiction that real numbers are > >> uncountable. > > > > WRONG! It is not, at least in Cantor's version, a proof by contradiction. > > He assumes that his list of all reals is complete and shows that this is > not the case. From this contradiction he was forced to conclude that the > reals are uncountable The proof by contradiction can be recast into a proof that is not by contradiction. It is a matter of taste. Yes, proofs usually "force" us to conclude things. > but he intentionally misinterpreted the outcome by > claiming there are more real than rational numbers. Please define the word "more" in this context. What exactly are you saying is not correct (or misinterpreted)? -- David Marcus
From: Virgil on 6 Dec 2006 14:29 In article <4576E233.7030802(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 11:10 PM, Virgil wrote: > > In article <45753CB8.3080500(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> Let's rank Tonico outside but David Marcus still inside mathematics. > >> > >> 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. > > > > WRONG! It is not, at least in Cantor's version, a proof by contradiction. > > He assumes that his list of all reals is complete and shows that this is > not the case. Not so. He assumes one has an arbitrary list, with no other qualifications than that it is a list of reals, and then shows it is not complete. There is no necessity to assume it complete and convert a straightforward direct proof into a proof by contradiction. In is only those who misread the proof that themselves supply that additional assumption that Cantor himself does not require. > From this contradiction he was forced to conclude that the > reals are uncountable but he intentionally misinterpreted the outcome by > claiming there are more real than rational numbers. In the sense in which Cantor defined "more", there are more. If EB wishes to use his own meanings for words in place of the clearly defined mathematical ones, he will, as here, repeatedly come a cropper. > > > > Since Cantor's first proof > > You perhaps refer to Cantor's first diagonal argument No, I refer to Cantor's first proof that there cannot be a surjection from the naturals to the reals. > > > was not in any way the same and IS valid for > > Dedekind cuts, any flaws in his second are irrelevant. > > I do not understand what you mean in this contex regarding Dedekind > cuts. I do not refer to any flaw in DA2 itself, just the interpretation > is wrong. It is the interpreter making that interpretation who is in error because he insists on using his own non-mathematical meanings for in place of mathematically standard definitions.
From: Virgil on 6 Dec 2006 14:33 In article <4576E3FF.9090803(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 11:14 PM, Virgil wrote: > > In article <457584A4.3000108(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> 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: > > > >> >> 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 did. And showed why with two separate proofs. > > I pointed to the 4th and only correct possibility of interpretation to DA2. That is your opinion, but not one accepted by the mathematically literate. > Do not hope for maintaining the proof concerning the power set. Do not hope to destroy what you do not understand. > The reason why the power set is also uncountable is quite simpel And Cantor showed it.
From: Virgil on 6 Dec 2006 14:39
In article <4576E61B.1070700(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 11:21 PM, Virgil wrote: > > In article <457586BB.9020406(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> 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. > > > > But they are measures of fictional qualities in EB's rubric, so must be > > uncountable. > > Get serious! Why need I be any less nonsensical than EB? > >> Who > >> introduced subsettedness, and is it also quantified? > > > > The subset relation, as a partial ordering of sets, has been around as > > long as sets. > > Well, I read the word subset (Teilmenge) but you are the first one who > offers subsettedness. In English, one is allowed to make up words to express new ideas. I understood that German allowed likewise. > > > > >> >> 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. > > > > What has that to do with whether one can well order a set of sets by > > inclusion? > > > > EB seems compelled to go off on tangents. > > I will perhaps never understand what you mean with well-ordered > subsettedness. When I wrote well-ordering of the irreals, I expressed > smiling. As I have no idea of what it would mean to well order the irreals, I supposed you were grimacing. |