Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 5 Dec 2006 10:44 On 12/4/2006 11:47 PM, Virgil wrote: > In article <45744411.2060408(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/1/2006 10:50 PM, Virgil wrote: >> > In article <45706F34.1070809(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > >> >> The word completeness is misleading. >> > >> > "Complete" for an ordered set has a precise mathematical definition. >> > That mathematical meaning is the only relevant meaning in any >> > mathematical discussion of ordered sets. Most words used in technical >> > senses in mathematics mean something quite different from their common >> > meanings, >> >> >> Wiki: "In mathematics and related technical fields, >> a mathematical object is complete if nothing needs to be added to it." > > That is not the same as a complete ordering, either partial or total. > See: http://en.wikipedia.org/wiki/Completeness_%28order_theory%29 Yes. Infimum and supremum remind me of Weierstrass. Dedekinds "Vervollstaendigung" of the rationals takes almost the opposite point of view. > > >> What about the possibility to add numbers to "the" rational numbers, I >> argue: >> According to Archimedes, there is no limit for adding more precision. >> However, infinite precision is as fictitious as is infinity. > > But no more so that 1 or 2. A priori absolute precision is indeed a tricky question. > >> Consequently, the system of genuine (rational) numbers exhibits open >> ended acuity. It cannot at all be improved by adding genuine numbers >> which are not yet part of it. > > maybe EB's irrational rationals cannot, but standard rationals can. Rationals are p/q. This system cannot be improved by adding genuine (i.e. rational) numbers. One can merely move to the fictitious continuous alternative. > >> Continuum is a different quality. Tom some extent one may compare number >> and continuum with stone and mortar in between. > > Physical analogies are all ultimately false. I agree and remind of Dedekind and Cantor who made their fallacies when they referred to physical objects and a microscope. Analogies may nonetheless help to get the essence. >> > The meaning has been made precise by giving it a precise definition. >> >> A set in mathematical sense (including the infinite set) is something >> which has precisely no valid definition. > > In set theories "set" is a primitive, just like "point' and "line" in > Hilbert's completion of Euclidean geometry. A point has a clear definition at least since Euclid. >> >> Our disagreement is based on different interpretation. >> > >> > In mathematics, the operant mathematical definitions determine the >> > interpretations. To reject that is to reject mathematics entirely. >> >> Who does not obey rejects not just science entirely but even the holy >> mathematics. Lucky Gauss, lucky Leibniz. They cannot be forced to >> swallow stupid Virgilian definitions and pray to Georg the Lord. > > In order to discuss anything, there must be agreement on the meaning of > the terms to be used in that discussion. Thousands of mathematicians > have agreed on the meanings of certain mathematical terms, as did hundreds of millions on terminology of religious or other dogmas.
From: Tonico on 5 Dec 2006 10:46 Eckard Blumschein ha escrito: > On 12/4/2006 11:52 PM, Virgil wrote: > > >> I am an electrical engineer > > > > Shocking! > > Why? We love mathematics. *************************************************** Oh, I bet elec. eng. love maths; the problem seems to be that maths does not correspond AT ALL that love, at least in the case of several engineers...**sigh**...tough. Tonio
From: Eckard Blumschein on 5 Dec 2006 10:52 On 12/4/2006 11:23 PM, Lester Zick wrote: > On Mon, 04 Dec 2006 15:56:58 -0500, Bob Kolker <nowhere(a)nowhere.com> > wrote: > >>Eckard Blumschein wrote: >> >>> >>> >>> 2*oo is not larger than oo. Infinity is not a quantum but a quality. >> >>But aleph-0 is a quantity. > > It is? So which quantity is it? I mean could you show us an aleph-0 > quantity or at least prove that there is such a quantity? I am not sure if aleph_0 is considered a quantity in set theory. It is irrelevant. Since the alefs are obviously ordered by countable indices, they look like quantities. Cantor claimed to count the uncountable. Cantor was naive enough as to believe that infinity is a firm quantum.
From: Eckard Blumschein on 5 Dec 2006 10:56 On 12/4/2006 10:47 PM, David Marcus wrote: > Eckard Blumschein wrote: >> On 12/1/2006 9:59 PM, Virgil wrote: >> >> > Sets can have the same cardinality but different 'subsettedness' because >> > different qualities are being measured. >> >> I do not know a German equivalent to subsettedness. Wiki did not know >> subsettedness or at least subsetted either. So I have to gues what you >> possibly meant. Proper subset means included but not all-including. >> While I consider cardinality a rather illusory notion, I distinguish >> between counted (alias finite), countable (alias potentially infinite) >> and uncountable (alias fictitious). Well, the counteds are a subset of >> the countables. So far, the mislead bulk of mathematicians regard the >> countables (i.e. genuine numbers) a subset of the uncountables, too. >> >> 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. > > 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. Countably infinite refers to bijection. Uncountable is a correcting translation of the German nonsense word ueberabzaehlbar = more than countable. Meant is: there is no bijection to the naturals.
From: Eckard Blumschein on 5 Dec 2006 11:14
On 12/4/2006 9:58 PM, Bob Kolker wrote: > Eckard Blumschein wrote: > >> No wonder. Exactly this selfdelusion was the intention of dedekind. > > 'Twas no delusion. > > The Dedikind cut defining the square root of 2 is just as well defined > as the successor to the integer 2. > > Bob Kolker Dedekind's cuts did not create any new number. The square root of two is still what it was before. In order to understand what is illusory with Dedekind's cuts, read what I wrote last week. |