Re: Proof that counting is valid?

In article <ehfsgq$o2k$3@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
Dave Seaman <dseaman@xxxxxxxxxxxx> wrote:
On 22 Oct 2006 02:22:27 -0700, imaginatorium@xxxxxxxxxxxxx wrote:

bogus61847688@xxxxxxxxxxxxxx wrote:
I remember, many years ago, having seen a Sesame Street episode in
which Bert tries to show Ernie that four cookies are always four
cookies, no matter how they are arranged.

I have never actually seen a proof that "counting works": that is, that
no matter in what order you count a set of objects, the answer will
always be the same. I know that this is (to most of us) "self-evident",
yet mathematicians seem to feel that it is not a waste of time to prove
the "self-evident".

For the sake of completeness, can I see a mathematical proof that
"counting works"?

I really can't see that Christopher Heckman's post answers your
question. It's easy to go on about "here's a bijection",.. "here's
another one", but while it's easy to "see" that you can count the same
number when you rearrange the pile of beans, what I think you need is a
proof that you can't also count a different number.

This comes down to the pigeonhole principle: for any finite set there
does not exist a bijection from the set to a proper subset thereof. Try
proving that by induction - it should be very simple. (Might be better
to "start from 2", to make sure you're not falling down a "all humans
are the same age" pothole.)

Induction won't show it, but counting also works for infinite sets. No
matter how you count the reals, they always have cardinality c.

Counting cannot be done unless the set can be well-ordered, as
counting a set is a well-ordering. The integers can be well-ordered
as omega, omega+1, ... omega^2, ..., omega^omega, ... up to
all ordinals less than omega_1.

For all transfinite sets, counting, if possible, can always
be done so as to yield different values. A finite set is
one which can be counted, and such that the counting always
ends at the same ordinal number.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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