Galileo's Paradox
in [Math]
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From: Bob Kolker on 4 Dec 2006 05:32 Eckard Blumschein wrote: > > Is it really justified to blame an allegedly insufficient definition of > the term set for obvious antinomies of set theory? Set theory has been modified so the antinomies cannot arise. No new contradictions have been found since the repairs to set theory have been done. > Notice, there is not even a valid definition of a set which includes > infinite sets. Cantor's definition has been declared untennable for > decades. That is simply not so. For example the set of integers. There is is. Tenable as ever. The set of real numbers. The set of functions from reals to reals. Bob Kolker
From: Bob Kolker on 4 Dec 2006 05:34 Tonico wrote: > Bob Kolker wrote: > >>>dr/dr = 1. >> >>in general d(k*r)/dr = k, where k is a constant. > > ****************************************************** > Since k^r = e^(r*lnk), d(k^r)/dr = (lnk)k^r.... > Tonio The asterixt * stand for multiplication, not exponentiation. Bob Kolker >
From: imaginatorium on 4 Dec 2006 07:31 Lester Zick wrote: > On Sat, 02 Dec 2006 13:26:13 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >Lester Zick wrote: > >> On Fri, 01 Dec 2006 11:39:57 -0500, Tony Orlow <tony(a)lightlink.com> > >> wrote: > >>> Lester Zick wrote: > >>>> On Thu, 30 Nov 2006 12:08:03 +0100, Eckard Blumschein > >>>> <blumschein(a)et.uni-magdeburg.de> wrote: > >>>>> On 11/29/2006 8:13 PM, Bob Kolker wrote: > >>>>>> Tony Orlow wrote: <snip all that> > Well, Tony, I don't really think any of this matters. These just seem > to be word strings having nothing to do with the matters at hand. Well, indeed. Yet curiously it manages to be entertaining from time to time. Brian Chandler http://imaginatorium.org
From: Eckard Blumschein on 4 Dec 2006 10:51 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." 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. 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. Continuum is a different quality. Tom some extent one may compare number and continuum with stone and mortar in between. >> > 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. A set in mathematical sense (including the infinite set) is something which has precisely no valid definition. Cantor's definition has been confessed "naive" (less merciful said untennable) and incorrigible. > Those who cannot deal with such precision should avoid mathematics. Or participate in standard 3rd class (i.e. arbitrarily axiomatised) 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. 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.
From: Eckard Blumschein on 4 Dec 2006 11:09
On 12/1/2006 10:41 PM, Virgil wrote: > 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. Surface declines with 1/x and does not converge with x-->0. Volume depends on 1/x^2. So the integral converges. You are knowledeable, know-it-all Virgil. |