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Topic: Matheology § 071
Replies: 55   Last Post: Jul 20, 2012 7:24 AM

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 Ki Song Posts: 221 Registered: 9/19/09
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 &lt;vir...@ligriv.com&gt; wrote:
> &gt; In article
> &gt; &lt;8b60b19d-ae20-4c27-9839-d21ceda4b...@m3g2000vbl.googlegroups.com&gt;,
> &gt;
> &gt;
> &gt;
> &gt;
> &gt;
> &gt;  WM &lt;mueck...@rz.fh-augsburg.de&gt; wrote:
> &gt; &gt; On 12 Jul., 16:40, David C. Ullrich &lt;ullr...@math.okstate.edu&gt; wrote:
> &gt; &gt; &gt; On Thu, 12 Jul 2012 00:01:56 -0700 (PDT), WM
> &gt;
> &gt; &gt; &gt; &lt;mueck...@rz.fh-augsburg.de&gt; wrote:
> &gt;
> &gt; &gt; &gt; &gt;Matheology § 071
> &gt;
> &gt; &gt; &gt; &gt;The Hausdorff Sphere Paradox [...] (here X, Y, Z are disjoint sets
> &gt; &gt; &gt; &gt;which nearly cover the sphere, and X is congruent to Y, in the sense
> &gt; &gt; &gt; &gt;that a rotation of the sphere makes X coincide with Y, and likewise Y
> &gt; &gt; &gt; &gt;is congruent to Z. But what is extraordinary is the claim that X is
> &gt; &gt; &gt; &gt;also congruent to the union of Y and Z, even though Y =/= Z). We are,
> &gt; &gt; &gt; &gt;like Poincaré and Weyl, puzzled by how mathematicians can accept and
> &gt; &gt; &gt; &gt;publish such results; why do they not see in this a blatant
> &gt; &gt; &gt; &gt;contradiction which invalidates the reasoning they are using?
> &gt;
> &gt; &gt; &gt; Maybe because it&#39;s not a contradiction? I bet that&#39;s it.
> &gt;
> &gt; &gt; &gt; See, if you want to say it&#39;s a contradiction it&#39;s not enough to say
> &gt; &gt; &gt; that it seems curious to you, or that you don&#39;t believe it.
> &gt;
> &gt; &gt; And it is not enough that some others believe it.
> &gt;
> &gt; &gt; &gt; You have
> &gt; &gt; &gt; to give a _proof_ that it&#39;s impossible, to go along with the proof
> &gt; &gt; &gt; that it&#39;s true.
> &gt;
> &gt; &gt; The proof is this: The Volume of each sphere is measurable and a fixed
> &gt; &gt; mathematical quantity:
> &gt; &gt; Therefore the axiom of choice leads to 1 V = 2 V, or, as V is not 0,
> &gt; &gt; 1 = 2.
> &gt;
> &gt; Only if one claims that the mapping must be measure preserving, which no
> &gt; one but WM seems to claim.
> &gt;
> &gt; Note that for a sphere centered at origin, the mapping
> &gt; (r, theta, phi) -&gt; (2*r, theta , phi) multiplies the volume of the
> &gt; sphere by 8.
>
> This is but another instant to observe that Cantor&#39;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?

Date Subject Author
7/12/12 mueckenh@rz.fh-augsburg.de
7/12/12 David C. Ullrich
7/13/12 mueckenh@rz.fh-augsburg.de
7/13/12 David C. Ullrich
7/13/12 mueckenh@rz.fh-augsburg.de
7/13/12 Fishcake
7/13/12 David C. Ullrich
7/13/12 mueckenh@rz.fh-augsburg.de
7/13/12 Virgil
7/14/12 David C. Ullrich
7/14/12 mueckenh@rz.fh-augsburg.de
7/14/12 Virgil
7/15/12 David C. Ullrich
7/15/12 mueckenh@rz.fh-augsburg.de
7/15/12 Ki Song
7/16/12 mueckenh@rz.fh-augsburg.de
7/16/12 Ki Song
7/16/12 Ki Song
7/16/12 mueckenh@rz.fh-augsburg.de
7/16/12 Ki Song
7/16/12 Virgil
7/17/12 David C. Ullrich
7/19/12 mueckenh@rz.fh-augsburg.de
7/19/12 David C. Ullrich
7/19/12 Frederick Williams
7/19/12 Gus Gassmann
7/20/12 Frederick Williams
7/19/12 Virgil
7/16/12 Virgil
7/17/12 Michael Stemper
7/17/12 Marshall
7/15/12 Virgil
7/17/12 Michael Stemper
7/13/12 Virgil
7/13/12 Virgil
7/14/12 mueckenh@rz.fh-augsburg.de
7/14/12 Ki Song
7/15/12 mueckenh@rz.fh-augsburg.de
7/15/12 YBM
7/15/12 mueckenh@rz.fh-augsburg.de
7/15/12 Ki Song
7/15/12 Virgil
7/15/12 YBM
7/15/12 Virgil
7/15/12 Ki Song
7/15/12 Virgil
7/15/12 Virgil
7/14/12 Virgil
7/14/12 David C. Ullrich
7/14/12 Virgil
7/15/12 David C. Ullrich
7/12/12 Virgil
7/13/12 mueckenh@rz.fh-augsburg.de
7/13/12 Virgil
7/13/12 Michael Stemper
7/12/12 FredJeffries@gmail.com

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