Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 1 Dec 2006 12:48 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. >> 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. > >> I wonder if they require aleph_2. > > Try learning enough of them to find out. No. Aleph_2 is just fancy.
From: Eckard Blumschein on 1 Dec 2006 12:53 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. And utility is the measure of the value of a definition. Utility for what?
From: Eckard Blumschein on 1 Dec 2006 12:54 On 11/30/2006 10:09 PM, Virgil wrote: > In article <456EC9FA.5000603(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/29/2006 8:21 PM, Virgil wrote: >> >> > Does volume completely determine mass? Different measures measure >> > different things and need have any correlation. >> >> Since measure theory is mathematics, it should be in position to >> abstract from physics and have only one measure for let's say seven eleven. > > Nothing in measure theory requires that every quality of an object have > the same standard of measure. > > For a 3D object, one can simultaneously and independently have > measurements of surface area and volume. > > 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.
From: MoeBlee on 1 Dec 2006 13:04 Tony Orlow wrote: > The Cartesian plane is ordered in two dimensions, > not a linear order, but a 2D ordered plane with origin. An ordering is a set of ordered pairs such that the set of ordered pairs has certain properties among reflexivity, irreflexivity, asymmetry, antisymmetry, transitivity, connectedness, trichotomy, and least member of nonempty subsets of the field of the ordering, depending on what kind of ordering it is. The Cartesian plane is the set of ordered pairs of real numbers. An ordering of the Cartesian plane then would be a set of ordered pairs of ordered pairs of real numbers. What ordering do you claim to exist on the Cartesian plane? How do you prove its existence? What kind of ordering is it (what are its properties from those I mentioned)? MoeBlee
From: Eckard Blumschein on 1 Dec 2006 13:06
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. It suggest that there are genuine numbers that complete the rational ones. One can say there are holes alias impossible numbers outside the rationals, a sauce of inaccessiblity. > 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. Our disagreement is based on different interpretation. I prefer an interpretation which does not require fancy and illogism. My puzzle fits perfectly and explains very old paradoxa. |