Galileo's Paradox
in [Math]
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From: Bob Kolker on 1 Dec 2006 13:27 Tony Orlow wrote: > > > Are you saying you can explicitly state the mapping mathematically > without any reference to the ordered coordinates which identify each > point? Not that you have to explicitly address each point individually, > but you do need to make reference to the coordinates in your mapping > formula, no, and those coordinates are each elements of an ordered set, no? The points in the disks are not ordered. We have a one one onto mapping from an unordered domain to an unordered codomain. The use of a two tuple to identify a vector is purely a naming convention. If you don't like vectors then use to slices of a right cone and do the mapping by putting a ray from the vertex of the cone thrugh the disk cross sections. No ordering anywhere. Neither the domain or the co-domain (range) has to be ordered at all. So in general what you say is not true, because I have presented a counter-example. Bob Kolker Bob Kolker
From: Bob Kolker on 1 Dec 2006 13:28 Tony Orlow wrote: > 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? There are no "points within a vector" and don't change the subject. Neither the domain nor the co-domain is ordered. Bob Kolker
From: Bob Kolker on 1 Dec 2006 13:29 Tony Orlow wrote: > Are x and y ordered? The Cartesian plane is ordered in two dimensions, > not a linear order, but a 2D ordered plane with origin. The plane is not a linearly ordered set of points. Bob Kolker
From: Lester Zick on 1 Dec 2006 13:29 On Fri, 01 Dec 2006 18:36:49 +0100, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >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. I have no critical opinion on these kinds of issues which is why I replied to Bob's comment instead of yours. I'm still trying get people to realize that a points is no more a unit of measure than zero is a metric. >>>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. > > > ~v~~
From: Bob Kolker on 1 Dec 2006 13:31
MoeBlee wrote: > > 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)? Without resorting to the axiom of choice or well ordering one can use the lexicographical ordering (alphabetic ordering of two letter words, so to speak) of two tuples. But this ordering has no arithmetic significance. Bob Kolker |