Galileo's Paradox
in [Math]
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From: Virgil on 1 Dec 2006 16:33 In article <45706AD1.808(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/30/2006 10:00 PM, Virgil wrote: > > >> > >> While colonies were changed when embedded in a Commonwealth, Cantorian > >> distemper did effectively almost not at all affect mathematics. > > > > EB's distempers will not have any effect on mathematics, though set > > theory has had effects. > > Worries enough, what else? > > > >> I just see some imperfections. > > > > Get your eyes tested. > > I listed many. You claimed many "imperfectins" but did not justify those claims with anything mathematically valid. > > > >> I guess, point-set topology and measure > >> theory do not require the claim of set theory to rule all mathematics. > > > > They cannot exist without a foundation of set theory. > > In this case they could not exist. Set theory does not have a solid > basis. So I doubt. There are a lot of textbooks on point-set, and other, topologies and on measure theory. I have yet to see one of them that is not based on set theory. If EB claims these books do not exist, he is even more foolish than usual. > > > > >> I wonder if they require aleph_2. > > > > Try learning enough of them to find out. > > No. Aleph_2 is just fancy. So EB chooses to remain ignorant in order to support his claims.
From: Virgil on 1 Dec 2006 16:38 In article <45706BFD.7090506(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/30/2006 10:02 PM, Virgil wrote: > > >> I consider Dedekind wrong, and he admitted to have no evidence in order > >> to justify his basic idea. > > > > The fact that Dedekind's definition of infiniteness of sets has been > > widely adopted indicates that many others have found it to be a useful > > definition. > > It was appealing even to Peirce. BTW, I referred to the lacking basis of > his cuts. What about the definition of an infinite set, I alredy > explained somewhere here why it tacitly implies an illusion. When you "explain" why 2 = 1, I am not persuaded. > > And utility is the measure of the value of a definition. > > Utility for what? Mathematical definitions are abbreviations, they shorten things. If they are useful enough to be used often, they can save a great deal of time and space in mathematical writings. They are useful for that.
From: Virgil on 1 Dec 2006 16:41 In article <45706C65.2040504(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/30/2006 10:09 PM, Virgil wrote: > > There are even such anomalies as "Gabriel's horn" of finite volume but > > infinite surface area: > > > > http://mathworld.wolfram.com/GabrielsHorn.html > > Do not try impressing me withb old hats. As EB seems not to have a head big enough to to put one on, I wouldn't think of it.
From: Virgil on 1 Dec 2006 16:50 In article <45706F34.1070809(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/30/2006 10:14 PM, Virgil wrote: > > In article <456EF429.6080201(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 11/30/2006 6:25 AM, Virgil wrote: > >> > In article <456e475e(a)news2.lightlink.com>, > >> > Tony Orlow <tony(a)lightlink.com> wrote: > >> > > >> > > >> >> Given a > >> >> set density, value range determines count. > >> > > >> > Compare the "set densities" of the set of naturals, the set of > >> > rationals, the set of algebraics, the set of transcendentals, the set of > >> > constructibles, and the set of reals. > >> > >> Either discrete or continuous. Nothing in between. > > > > The natural ordering of the set of rationals is neither continuous not > > discrete, at least in the mathematical meanings of those words. > > Any rational number has a numerical address. Just the entity of all > rational numbers is a fiction. > > > It is dense, but not complete, and both density and completeness are > > required for the set to have a continuous ordering. > > 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, and those who conflate the common with the technical meanings demonstrate their mathematical incompetence in so doing. > > > So of the sets mentioned above, the reals and only the reals are > > continuous in amy mathematically acceptable sense. > > I agree with the caveat that the meaning of the term set has been made > dubious. The meaning has been made precise by giving it a precise definition. Those who cannot deal with such precision should avoid mathematics. > Our disagreement is based on different interpretation. In mathematics, the operant mathematical definitions determine the interpretations. To reject that is to reject mathematics entirely.
From: Michael Press on 1 Dec 2006 19:49
In article <virgil-773272.13591501122006(a)comcast.dca.giganews.com> , Virgil <virgil(a)comcast.net> wrote: > In article <45704f2b(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > Eckard Blumschein wrote: > > > On 11/29/2006 6:37 PM, Bob Kolker wrote: > > >> Tony Orlow wrote: > > >>> It has the same cardinality perhaps, but where one set contains all the > > >>> elements of another, plus more, it can rightfully be considered a larger > > >>> set. > > >> Not necessarily so, if it is an infinite set. > > >> > > >> Bob Kolker > > > > What, is it not necessarily so that it CAN rightfully be considered a > > larger set, with some justification? It doesn't have a right to be > > considered that way? Why? Because it contradicts theoretical > > transfinitology? > > 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. > > Sets can have the same cardinality but different 'subsettedness' because > different qualities are being measured. > > > 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. Procrustes meet TO. TO, Procrustes. -- Michael Press |