Re: Review of Mueckenheims book.

In article <45fd601a@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

Virgil wrote:
In article <45fc8060@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

cbrown@xxxxxxxxxxxxxxxxx wrote:
On Mar 17, 8:59 am, Tony Orlow <tony@xxxxxxxxxxxxx> wrote:
cbrown@xxxxxxxxxxxxxxxxx wrote:
On Mar 13, 9:22 am, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:

In a countable set, there are only a finite number
of elements between any two specific elements.
Counterexample: The set of rationals in [0,1] is a countable set, and
there are an infinite number of elements between any two distinct
elements of that set.
Not in the order in which they are countable.


There is no such thing as /the/ order in which an infinite set is
countable.

The only finite sets for which there is /the/ order, are {} and {{}}.
For any other set, finite or infinite, there are at least two orders
in which it many be counted.

If an infinite set is countable at all, it has uncountably many "orders"
in which it is countable.

Please specify a sequential ordering of the rationals in which exist two
elements infinitely distant from each other.

Attempting that sort of impossibility is more in your line than mine.

You cannot state two rational numbers included in Cantor's diagonal
proof of their countability which are infinitely distant from each other
in that sequence.

There are uncountably many other sequential orderings of those rationals
which are in any interval of length greater than zero, including all of
them.

If you say so, but in none of them exists any element infinitely before
or after any other.

NO one has said otherwise.

Why don't you give me two
rational numbers, which are not finitely distant in the
pseudo-sequential non-quantitative (read, bogus) ordering of the
rationals through sparse diagonalization? Which comes infinitely beyond
any other, in that "countable" sequence? Hmmm....

So you yourself claim that your statement only makes sense if we use a
"bogus" ordering of the rationals? Hmmm indeed!

That's the ordering used by Cantor to prove their countability.

Lots of sequential orderings of rationals have been used, and any one of
them is valid.




In none of those ordering are two elements infinitely distant.

So?

What I
said was that no two elements in any countable set are infinitely
distant from each other in any linear ordering that makes them
countable.

Does TO mean to suggest that there is a linear ordering which makes them
uncountable?



The question of the size of the set of rationals is a little more
complicated than that.

The "size" of Q is quite simply Card(N).

What I have said is that there are sequences which are uncountable.

Then they are not sequences, at least in any standard meaning of
"seqeunce".

If this sequential order is valid for the rationals, then
this fact applies. If you have a problem with the fact that there are an
infinite number of rationals between any two in their natural
quantitative order, then you should consider the possibility that
non-quantitative orderings of subsets of the reals are not suitable for
relative measure of those subsets.

Since TO's argument here seems to be based on a false premise, perhaps
he should just forget the whole thing.

What premise is that?

That TO is capable of making sense.
.

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