Re: Cantor Confusion

In article <457d8cc0$1@xxxxxxxxxxxxxxxxxxx> Tony Orlow <tony@xxxxxxxxxxxxx> writes:
*** T. Winter wrote:
In article <1165761763.908889.34550@xxxxxxxxxxxxxxxxxxxxxxxxxxx> Han.deBruijn@xxxxxxxxxxxxxx writes:
...
> Let P(a) be the probability that an arbitrary natural is divisible by
> a fixed natural a. Then P(a) = 1/a . Forbidden by set theory.

No. Not specifically forbidden by set theory. Forbidden because there are
no appropriate definitions for the words you are using (they are not used
conforming to standard definitions, so you better supply definitions).
In probability theory (as is commonly use) you have to define how you
*select* your arbitrary natural. You have not done so, so probability
theory does not have an answer.

Why does that matter?

It does matter because if you do not properly define your problem,
mathematics is not able to give an answer.

This is the same thing as your stupid ball and
vase trick. Why do you need to label anything, or know what you're
choosing from the infinite set?

Because that is part of the problem setting. Giving that setten will
allow mathematics to model the question and give an answer.

And it is bad to think that because for a sequence of sets holds that
lim{n -> oo} |S_n| = k
with some particular value of k, that also
| lim{n -> oo} S_n | = k
because the latter statement contains something that has not been
defined in mathematics. But even when we define it, it is not certain
that it holds. Given the following (I think reasonable) definition:
lim{n -> oo} S_n = S
if:
(1) for every element a in S there is an n0 such that a is in each of
the sets S_n with n > n0
(2) for every element a not in S there is an n0 such that a is not in
each of the sets S_n with n > n0.
So from some particular point an element either remains in the sets in
the sequence or remains out of the sets.

With this definition (when we look at the rationals) we have that
lim{n -> oo} [0, 1/n] = [0]
and so:
lim{n -> oo} | [0, 1/n] | = aleph0 != 1 = | lim{n -> oo} [0, 1/n] |
(I am talking standard mathematics here).

So taking cardinality and limits can not be interchanged except in some
particular cases. But that is not unprecedented in mathematics.
limits and integrals can also not be interchanged except in particular
cases. And so can the interchange is not in general passoble if one
of the things you interchange is a limit. Even interchanging limits
is not in general possible. Consider:
lim{x -> oo} lim{y -> oo} (2x + 3y)/xy
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.