Re: Review of Mueckenheims book.

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.

Is 1/2 between 1/4 and 3/4 "in the order in which they are countable"?
Starting at 1/1 and traversing the standard table diagonally to make a sequence, no, 1/2 is reached before 1/4, which is reached before 3/4:

1/1 1/2 1/3 1/4 ...

2/1 2/2 2/3 2/4 ...

3/1 3/2 3/3 3/4 ...

4/1 4/2 4/3 4/4 ...

5/1 5/2 5/3 5/4 ...

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.

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.

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.

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

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?
.

Relevant Pages

  • Re: cantors theorem - the pieces of the puzzle
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  • Re: Review of Mueckenheims book.
    ... There is no such thing as /the/ order in which an infinite set is ... There are uncountably many other sequential orderings of those rationals ... That's the ordering used by Cantor to prove their countability. ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... For any other set, finite or infinite, there are at least two orders in which it many be counted. ... There are uncountably many other sequential orderings of those rationals which are in any interval of length greater than zero, ... That's the ordering used by Cantor to prove their countability. ...
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    ... > the rationals or reals here. ... and an infinite number of rationals for every unit. ... >>> substitute is mere handwaving and babbling. ...
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