Re: ....Infinity
- From: Dave Seaman <dseaman@xxxxxxxxxxxx>
- Date: Sat, 5 Nov 2005 13:58:25 +0000 (UTC)
On 5 Nov 2005 05:13:25 -0800, zuhair wrote:
> 12. Dave Seaman
>> In reality you didn't show the enumeration for socks explicitely you
>> only refered to the axiom of choice.
> That's because there is no way to write down an enumeration explicitly.
> That's exactly what I said when I was explaining the need for the axiom
> of choice in the first place.
>> Why not put one sock in each boot. and make an enumeration of sockes
>> marked by the boot they are put into.
> How do you decide which sock to put in the left boot, and which in the
> right one? You can decide that for one pair, or for any finite number
> of
> pairs, but not for all the pairs at once. That's why the axiom of
> choice
> is needed.
> ----------------------------------------------------------------------------------
> What do you mean how I decide, I will put ONE sock in ONE boot. the
> matter of left
> or right is not that important.
As I said, there is no problem in making that decision for one pair or
for any finite number of pairs.
Your assertion is basically what the axiom of choice says. According to
AC, we can be assured that a suitable mapping exists, even though we
can't actually describe it.
> Since their is ONE-ONE correspondance between
> boots and socks, then the number of all socks should be the same as all
> boots.
> This is obvious.
Lots of people think so. That's why AC is popular among mathematicians.
> I want to ask you ? then how Cantor worked out
> his diagonal proof? didn't he say if we change
> the n-th decimal in the n-th raw? .Then I can
> say the same thing against this argument
> we can only do this operation for finite number
> of terms ie for n. and not infinitely.
No, it's not the same. Cantor didn't need AC here, because he gave an
explicit formula for constructing the number on the diagonal, something
along the lines of:
Let the n-th digit of d be a 5, if the n-th digit of x_n is a 4,
and let the n-th digit of d be a 4, otherwise.
An explicit formula is exactly what is lacking in constructing a mapping
between the boots and the socks. Can you give one for determining which
sock to put in the n-th left boot, and which sock to put in the n-th
right boot, for all n?
Remember, this is the same question that you asked. You objected that I
had not given an explicit formula, merely citing the axiom of choice
instead. Have you now changed your position? You now seem to be
implicitly assuming that AC holds instead of trying to give a formula.
> In a similar manner we can put ONe sock in ONe
> boot.(like the n-th decimal in n-th raw).
If we were mapping boots to gloves, we wouldn't need AC, since each pair
has a left and a right member.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
.
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