Re: ....Infinity

On 4 Nov 2005 07:38:54 -0800, zuhair wrote:
> Hellow every one.

> Amillionaire who bought a pair of socks whenever he bought a pair of
> boots , and never at any other time , and who had a passion for buying
> both that at last he had Aleph-0 pairs of boots and Aleph-0 pairs of
> socks . The problem is: How many boots had he,and how many socks?


> Bertrand Russell.

> Note: Some would think that the number of boots is Aleph-0 and of socks
> is Aleph-0
> But the author of this problem says it is not necessarily the case. The
> solution of the problem
> is related to the subject of selections and the multiplicative axiom.

> Can any body explain that?

It's easy to show that there are aleph_0 boots, given that there are
aleph_0 pairs. Let P be the set of pairs, and let {p_n} be an
enumeration of P. Let Then UP is the union of P, which is the set of
boots. f: N -> UP be given by

f(n) = the left boot of pair p_n, if n is even,
= the right boot of pair p_n, if n is odd.

Then f enumerates the boots in UP, which is to say that f is a choice
function on UP.

The problem is, you can't do this with the socks. Given S = set of pairs
of socks, and given an enumeration of the pairs, there is no way to
explicitly write down an enumeration of US, the set of socks. So, does
such an enumeration exist? It does, if you assume the axiom of choice.


--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
.

Relevant Pages

  • Re: ....Infinity
    ... >> Then f enumerates the boots in UP, which is to say that f is a choice ... you can't do this with the socks. ... >> of socks, and given an enumeration of the pairs, there is no way to ... >> Dave Seaman ...
    (sci.math)
  • Re: Boots fitting problem
    ... of walking socks instead of the usual one pair (it was snowing ... boots, so if you've sized for one, stick to that. ... ALt-Berg are the standard 'made to measure' walking boots folks, but you have to trot up to Yorkshire. ... Let me echo the comment on extra socks though - I have several pairs of boots, varying slightly in size, and the smallest ones will HURT with my normal thick smartwool socks. ...
    (uk.rec.walking)
  • Re: ....Infinity
    ... >> Dave Seaman wrote: ... >>> of socks, and given an enumeration of the pairs, there is no way to ... >>> explicitly write down an enumeration of US, the set of socks. ... > The same way I solved the problem for pairs of boots, ...
    (sci.math)
  • Re: Boots fitting problem
    ... of walking socks instead of the usual one pair (it was snowing ... boots, so if you've sized for one, stick to that. ... ALt-Berg are the standard 'made to measure' walking boots folks, ... The larger boots take the thick sock, ...
    (uk.rec.walking)
  • Re: ....Infinity
    ... Amillionaire who bought a pair of socks whenever he bought a pair of boots, and never at any other time, and who had a passion for buying both that at last he had Aleph-0 pairs of boots and Aleph-0 pairs of socks. ... The solution of the problem is related to the subject of selections and the multiplicative axiom. ... Let P be the set of pairs, and let be an enumeration of P. Let Then UP is the union of P, which is the set of boots. ...
    (sci.math)