Re: "Number" of elements; was: Distinct linear orderings on Z
From: Wolf Kirchmeir (wwolfkir_at_sympatico.ca)
Date: 03/24/05
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Date: Thu, 24 Mar 2005 14:52:18 -0500
Albert Wagner wrote:
> Daryl McCullough wrote:
>
>> In article <A1A0e.15159$Fy.12391@okepread04>, Albert Wagner says...
>>
>>> Stephen J. Herschkorn wrote:
>>>
>>>> There are two commonly accepted definitions of "infinite":
>>>>
>>>> a) A set is finite iff it can be put in bijection with a set of the
>>>> form {0, 1, 2,..., n-1}, where n is a natural number. A set is
>>>> infinite iff it is not finite.
>>>>
>>>> b) A set is infinite iff there exists an nonsujrective injection
>>>> from the set to itself. (A set is finite iff it is not infinite.)
>>>
>>>
>>> Neither of these is a definition, but rather each is simply a
>>> prescription for determining if a given set may be classified as
>>> infinite.
>>
>>
>>
>> That's what a definition *is*.
>
>
> No, it isn't. The fact that you believe it *is* is why you are given no
> respect as a mathematician.
>
> This is *not* a definition of 'red': A car is red, if there exists on
> it's surface a coat of red paint.
It seems, Albert, that you are using an accidental property as an
example to refute a definition of an essential property. NB that if a
set is not-finite, it's infinite; if it's not-infinite, then it's
finite. Finiteness is a property of sets in the same sense that charge
is a property of an electron (and note that an electron can have a
positive chrage, in which case for convenience it's given a different
name, but it can't have zero charge, because then it's not an
electron-type thing at all.) There have been some attempts in this
thread to get at this essential property of sets by using "size" and
"number of elements" as the property that is characterised as either
finite or infinite, but since "size" has many other meanings (most of
them very distinct from each other), it's a bad term to use. That's why
mathematicians have a distinct term for it: cardinality. It may not be
the happiest of choices, but it's the one they use.
OTOH, if a car is not-red, it's - what, exactly? Which of the colours a
car could be is the opposite of red? "Red" is a descriptive adjective,
that is, it doesn't have an opposite.
Our language mirrors this distinction between essential and accidental
properties by including a class of adjectives that whose "opposite" is
created by adding a negative prefix: im/mortal, un/dying,
dys/functional, and so on. NB also that pairs like large/small,
hot/cold, refer to measures of an essential property (size, temperature.)
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