Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)

From: Jesse F. Hughes (jesse_at_phiwumbda.org)
Date: 11/25/04 

Date: Thu, 25 Nov 2004 10:16:14 +0100

Han de Bruijn <Han.deBruijn@DTO.TUDelft.NL> writes:

> Jesse F. Hughes wrote:
>
>> Cardinality is well-motivated to capture the size of a set. When a
>> child counts a collection of pencils, he is creating a bijection
>> between an initial segment of N and the set of pencils. For finite
>> sets, assigning a size clearly involves a bijection.
>>
>> I've seen no reason why this isn't the essence of counting.
>
> Except for a couple of nasty details. When "creating" the bijection,
> the pencils must be labeled. Otherwise you wouldn't know if a pencil
> has already be counted or not.

This is nonsense, of course.

It is nonsense on the literal reading because counting does not
involve any literal act of labeling. We can count the ships at sea
that we see from the beach without labeling them, moving them about or
anything else.

It is nonsense on every other reading that I can see. Creating a
bijection always requires checking that each element of the codomain
is represented exactly once. There is nothing special about whether
we call the bijection "counting" (i.e., if its domain is an ordinal)
or not.

There is no more need to talk about this creation of sets already
counted and to be counted in the "counting" case than there is in the
case of an arbitrary bijection. The essence of counting is
establishing that a bijection exists. I don't see anything more going
on at all.

> Labeling the pencils actually means that you are creating a
> different set: that of the pencils already counted. Moreover, you
> are destroying the original set: that of the pencils still to be
> counted. The process stops when the latter set has become empty.
> Replacing the pencils by apples:
>
> http://hdebruijn.soo.dto.tudelft.nl/fototjes/appels.htm 

-- 
But in our enthusiasm, we could not resist a radical overhaul of the
system, in which all of its major weaknesses have been exposed,
analyzed, and replaced with new weaknesses.
   -- Bruce Leverett (presumably with apologies to Ambrose Bierce)

Relevant Pages

  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... >> between an initial segment of N and the set of pencils. ... assigning a size clearly involves a bijection. ... This is nonsense, of course. ... It is nonsense on the literal reading because counting does not ...
    (sci.math)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... child counts a collection of pencils, ... between an initial segment of N and the set of pencils. ... assigning a size clearly involves a bijection. ... > Since there is the observed antinomy of the infinitely big, ...
    (comp.theory)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... child counts a collection of pencils, ... between an initial segment of N and the set of pencils. ... assigning a size clearly involves a bijection. ... > Since there is the observed antinomy of the infinitely big, ...
    (sci.math)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... > child counts a collection of pencils, ... > between an initial segment of N and the set of pencils. ... assigning a size clearly involves a bijection. ...
    (comp.theory)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... > child counts a collection of pencils, ... > between an initial segment of N and the set of pencils. ... assigning a size clearly involves a bijection. ...
    (sci.math)