Dear all,
I think Marc has hit the nail on the head: somehow the dictatorship of
journal editors and of rules (fetishes?) for filling tables and specifying
units has made everyone so insecure as to doubt even the fundmental notions
of set theory and of the cardinality of sets.
There is the axiomatic definition of integers by Peano's axioms, and
then there is Cantor's definition of the cardinality of sets where the
cardinal number of a set A is the class of all the sets B that can be put in
one-to-one correspondence with A. One can then show that integers are a
particular case of cardinal numbers: the cardinal number associated to the
integer 0 is the class of all sets having no members (e.g. the void set);
the cardinal number associated to the integer 1 is the class of all sets in
one-to-one correspondence with the set {0}; and given a cardinal number
associated to the integer m, one can get that associated to the successor of
m by considering the class of sets obtained by taking the disjoint union of
each of the sets in the class defining that cardinal and of {0}. Cardinals
are more powerful than integers because they can be infinite, and even
transfinite.
With this in mind, you can say that you have the same number of apples
as of oranges if you can associate one apple to each orange and vice-versa.
The set of apples and that of oranges have the same cardinal, and that
cardinal is uniquely associated to an integer, the "number" of both apples
and oranges. You cannot add apples and oranges, but you can add the integers
to which the cardinals of the two sets are associated, to get the cardinal
of a set to which both apples and oranges belong, e.g. of that of (pieces
of) fruit.
Marc was correct in pointing out the "anonymity" of numbers used to
"count things", i.e. of cardinal numbers: this anonymisation is produced by
the process of forgetting what things are, as long as you can put them in
one-to-one correspondence with each other. So indeed, the "unit" of a count
is the integer 1, i.e. the cardinal of the set {0}. Of course, if we say
that f"=7.8 this is not an integer; but the next chapter of any book on set
theory would explain how one progresses from integers to rational and real
numbers.
I apologise for this non-CCP4 answer to the initial question!
With best wishes,
Gerard.
--
On Sat, Feb 27, 2010 at 11:49:25AM +0100, [log in to unmask] wrote:
> Quoting Dale Tronrud <[log in to unmask]>:
>
>>
>> P.S. to respond out-of-band to Dr. Schiltz: On the US flag I see 7 "red
>> stripes",
>> 6 "white stripes", and 50 "stars". If I state "I see 7" I have conveyed
>> no
>> useful information.
>
>
> Yes, but cast in a mathematical equations one would write :
>
> Number of red stripes = 7
> Number of white stripes = 6
> Number of stars = 50
>
> i.e. without units
>
> one would not write :
>
> Number = 7 red stripes
> Number = 6 white stripes
> Number = 50 stars
>
>
> Marc
--
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