Galileo's Paradox
in [Math]
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From: Tony Orlow on 1 Dec 2006 11:45 Bob Kolker wrote: > Bob Kolker wrote: > >> Tony Orlow wrote: >> >>> >>> >>> To establish an explicit bijection between infinite sets does require >>> an ordering on the sets, at least in general. >> >> >> Not true. One can map the disk of radius one one onto the disk of >> radius two without ordering points in either disk. Hint: Use a cone. >> Or if you like vectors map the vector V of unit length into 2*V which >> has length 2. No ordering in sight. > > > Oops. Map the vector V of length <= 1 to vector 2*V which has length <= 2. > > Sorry about that. > > Bob Kolker > How do you distinguish the points within each vector, if they are not ordered. If you claim you're not mapping points within the vector, then the vector isn't really an infinite set, is it? TOny
From: Tony Orlow on 1 Dec 2006 11:48 Virgil wrote: > In article <456f334d$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <456e4621(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: > > >>>> Where standard measure is the same, there still may be an infinitesimal >>>> difference, such as between (0,1) and [0,1], if that's what you mean. >>> The outer measure of those two sets is exactly the same. >> Right, and yet, the second is missing two elements, and is therefore >> infinitesimally smaller in measure. > > Except that in outer measure there are no infinitesimals, and the outer > measure of the difference set, {0,1} is precisely and exactly zero. So, you're saying infinitesimals cannot be considered? You're saying one is not ALLOWED to consider the removal of a finite set from an ifninite set to make any difference in measure? I say you're wrong. Tony
From: Tony Orlow on 1 Dec 2006 11:52 Virgil wrote: > In article <456f3385(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <456e46b7(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Any uncountable >>>> set with element indexes expressed as digital numbers will require >>>> infinitely long indexes for most elements, such as is the case for the >>>> reals in any nonzero interval. >>> >>> There is nothing in being a set, including being a set of reals, that >>> requires its members to be indexed at all. >> To establish an explicit bijection between infinite sets does require an >> ordering on the sets, at least in general. > > It requires a function between the sets, but neither set need be ordered. > > For example, let P = R^2 be the Cartesian plane, then for any reals a > and b with a^2 + b^2 > 0, (x,y) |--> (a*x + b*y, -b*x + a*y) bijects > P to itself, but P is not an ordered set. Are x and y ordered? The Cartesian plane is ordered in two dimensions, not a linear order, but a 2D ordered plane with origin.
From: Eckard Blumschein on 1 Dec 2006 12:36 On 11/30/2006 11:40 PM, Lester Zick wrote: > On 30 Nov 2006 11:46:08 -0800, "Tonico" <Tonicopm(a)yahoo.com> wrote: >>Lester Zick wrote: >>> >Eckard Blumschein wrote: >>> >> I consider Dedekind wrong, and he admitted to have no evidence in order >>> >> to justify his basic idea. > I don't know what sense Eckard is trying to convey. Both Dedekind and Cantor concluded from erroneous reasonong according to their naive intuition that there must be more rationals than reals. Dedekind fabricated his belonging idea concerning the definition of real numbers by means of the so called Dedekind cut already in 1858 but hesitate until 1872 to publish them. His pretended reasons for that were: "erstens die Darstellung nicht ganz leicht, zweitens die Sache so wenig fruchtbar". While the latter aspect has proven correct, the hidden reason was perhaps: Dedekind understood that his cut was based on nothing than pure speculation. He did not have any provable justification. >>Cantor "not having evidence" for his idea (what stupid this sounds!) >>means that he (cantor) never foiund an aleph_null under his bed, I clearly explained that Cantor's only seemingly convincing DA2 was a correct demonstration of the uncountability of the reals which was misinterpreted as evidence for his claim that there are more reals than rationals. This interpretation was cyclic and ignored the so called 4th possibility. When Cantor allegedly proved Aristotele, Spinoze, Gauss, etc. wrong, then he did never provide compelling arguments, always pure assertions. It's a pity that those who warned of the charlatan remained unheared as were those who warned of certain political heroes.
From: Eckard Blumschein on 1 Dec 2006 12:42
On 11/30/2006 9:56 PM, Virgil wrote: > In article <456EB22F.70703(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/29/2006 7:59 PM, Virgil wrote: >> > In article <456D7417.30000(a)et.uni-magdeburg.de>, > >> Cantor himself was the victim of his own stupid notion of infinity. >> He wrote: There are not more points in a cube than in a line. >> I see it but I cannot believe it. > > Cantor defined his sense of "more" quite precisely, and according to > *that* definition what he wrote is precisely true. > > If EB wishes to reject that definition, Cantor did know that his fancy was rejected from all important figures even those hundreds or even thousands of years ago. He was not more than correct. He was wrong. > then he also rejects the right > to comment on the validity of Cantor's statement. |