Galileo's Paradox
in [Math]
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From: Bob Kolker on 30 Nov 2006 15:10 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. So in genaral one does not require an ordering. Bob Kolker
From: Bob Kolker on 30 Nov 2006 15:12 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
From: Virgil on 30 Nov 2006 15:56 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, then he also rejects the right to comment on the validity of Cantor's statement.
From: Virgil on 30 Nov 2006 16:00 In article <456EB544.3020005(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/29/2006 8:03 PM, Virgil wrote: > > In article <456D7544.8090000(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 11/28/2006 10:31 PM, Virgil wrote: > >> > >> > There is no such thing as "genuine" for numbers in mathematics. > >> > >> Maybe it will exist in genuine mathematics. > >> > >> > >> > So that EB has just refused to accept all of Analysis, including > >> > calculus, which is based on just the sort of sets that EB denies exist. > >> > >> This is perhaps a lie. I feel well served by pre-Cantorian analysis and > >> by modern mathematics which does not really rely on set theory. > > > > Since all of "pre-Cantorian analysis" is embeddable in "Cantorian > > analysis" without loss, and with some gains (e.g., point-set topology > > and measure theory), there is nothing to be gained by such retrograde > > devolution. > > 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. > I just see some imperfections. Get your eyes tested. > 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. > I wonder if they require aleph_2. Try learning enough of them to find out.
From: Virgil on 30 Nov 2006 16:02
In article <456EBC4D.5080608(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/29/2006 8:15 PM, Bob Kolker wrote: > > Eckard Blumschein wrote: > > > >> > >> No. Infinite quantities include e.g. an infinite amount of points. > >> Infinite means: The process of quantification has not been finished or > >> cannot be finished at all. > > > > > > A non-empty set is infinite if and only if it can be put in one to one > > correspondence with a proper subset of itself. That is the standard > > definition of infinite for sets. > > > 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. And utility is the measure of the value of a definition. |