Re: Review of Mueckenheims book.

Virgil wrote:
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.


If you agree it's impossible, what are you arguing about?

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.

Then why waste your time arguing, when that's all I've said?

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?

So, that is true of all countable sets, and can serve as a definition of a finite set, so that "uncountably infinite sets" can be more properly considered as "finite but unbounded structures".

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).

Card(Q)=card(N). So?

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".

They are nonstandard sequences, then.

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.

If one is ever to attempt to convey anything, one must always operate on that premise.
.

Relevant Pages

  • Re: Review of Mueckenheims book.
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  • Re: Review of Mueckenheims book.
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  • Re: Review of Mueckenheims book.
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