Ki Song
Posts:
221
Registered:
9/19/09
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Re: Matheology § 071
Posted:
Jul 14, 2012 11:05 AM
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On Saturday, July 14, 2012 4:46:13 AM UTC-4, WM wrote:
> On 13 Jul., 22:40, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <8b60b19d-ae20-4c27-9839-d21ceda4b...@m3g2000vbl.googlegroups.com>,
> >
> >
> >
> >
> >
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 12 Jul., 16:40, David C. Ullrich <ullr...@math.okstate.edu> wrote:
> > > > On Thu, 12 Jul 2012 00:01:56 -0700 (PDT), WM
> >
> > > > <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > > >Matheology § 071
> >
> > > > >The Hausdorff Sphere Paradox [...] (here X, Y, Z are disjoint sets
> > > > >which nearly cover the sphere, and X is congruent to Y, in the sense
> > > > >that a rotation of the sphere makes X coincide with Y, and likewise Y
> > > > >is congruent to Z. But what is extraordinary is the claim that X is
> > > > >also congruent to the union of Y and Z, even though Y =/= Z). We are,
> > > > >like Poincaré and Weyl, puzzled by how mathematicians can accept and
> > > > >publish such results; why do they not see in this a blatant
> > > > >contradiction which invalidates the reasoning they are using?
> >
> > > > Maybe because it's not a contradiction? I bet that's it.
> >
> > > > See, if you want to say it's a contradiction it's not enough to say
> > > > that it seems curious to you, or that you don't believe it.
> >
> > > And it is not enough that some others believe it.
> >
> > > > You have
> > > > to give a _proof_ that it's impossible, to go along with the proof
> > > > that it's true.
> >
> > > The proof is this: The Volume of each sphere is measurable and a fixed
> > > mathematical quantity:
> > > Therefore the axiom of choice leads to 1 V = 2 V, or, as V is not 0,
> > > 1 = 2.
> >
> > Only if one claims that the mapping must be measure preserving, which no
> > one but WM seems to claim.
> >
> > Note that for a sphere centered at origin, the mapping
> > (r, theta, phi) -> (2*r, theta , phi) multiplies the volume of the
> > sphere by 8.
>
> This is but another instant to observe that Cantor's bijection
> technique is not suitable to derive mathematical results.
>
> Regards, WM
If you are looking for bijections that does not preserve measure, you don't even have to look at the Banach-Tarski Paradox. You don't even have to look at spheres.
Simply take the bijection from
[0,1] to [0,2]
f(x) = 2x
Holy crap, a bijection that doesn't preserve the (lebesgue) measure!
I have another one.
Let S = {1,2,3}, T = {2,3,4} be subsets of N, with the measure defined to be m(X) = 1 whenever X contains 1 and 0 when X does not.
The measure of set S is 1.
The measure of set T is 0.
There is a bijection between S and T.
By your logic, this would imply 0 = 1. (Note: no one else but you are saying that.)
Do you see how ridiculous your logic is Wolfgang Mückenheim?
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