Re: An uncountable countable set

In article <45203282@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

Virgil wrote:
In article <1159711218.812268.276490@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

*** T. Winter schrieb:

In article <1159648393.632462.253170@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
mueckenh@xxxxxxxxxxxxxxxxx writes:

> (There are exactly twice so much
> natural numbers than even natural numbers.)

By what definitions? You never state definitions.
By the only meaningful and consistent definition: A n eps |N :
|{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|.
Do you challenge its truth?

I challenge the "truth" of its being the ONLY meaningful and consistent
definition.

"Mueckenh"'s claim is like that of a blind man claiming that colors are
imaginary.

They are, even for the seeing.

Then how is it that spectrographs can detect them?


I, and many others, find both meaning and consistency in the definition
of cardinality. That "Mueckenh" does not is more a measure of his
incapacity than of any lack of meaning and consistency.

WM, HdB, Finlayson, others and I see that the definition is lacking.
There's also Zuhair and Petry, and a slew more. This is the most
contentious of issues here. Perhaps you and I touched on the root of it,
The nature of logical implication.

The thing is that hundreds or even thousands agree with me and your
group do not even agree with each other.

Can you appreciate out striving for more.

Not when you are so ignorant of what is already there.


Are you so complacent in your position?

No, but I am reasonably complacent that any errors in my position will
be expressible with standard logic.




A specific proof of a general truth can be based on whatever it is based
on.

To the detriment of its generality.

Every proof is specific in some ways, but not necessarily in ways that
limit the generality of the theorem they prove.


There are other proofs , including Cantor's first proof, which do not
depend on any sort of representations of the reals.


Cantor's first is an interesting proof of the uncountability of the
continuum, and I consider it valid. It demonstrates that the notion
that, for numerical strings a, b, and c in set S containing
representation of all r in R, ((aeS ^ ceS ^ a<c) -> Eb ^ beS ^ a<b<c) ->
(Es ^ seS ^ A neN length(s)>n)

That is certainly nothing like the first proof that I was alluding to.
http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof
(quote)
The theorem
Suppose a set R is
1 linearly ordered, and
2 contains at least two members, and
3 is densely ordered, i.e., between any two members there is
another, and
4 is complete, i.e., if it is partitioned into two nonempty sets A
and B in such a way that every member of A is less than every member of
B, then there is a boundary point c (in R), so that every point less
than c is in A and every point greater than c is in B.
Then R is not countable.
The set of real numbers with its usual ordering is a typical example of
such an ordered set. The set of rational numbers (which is countable)
has properties 1-3 but does not have property 4.

The proof
The proof is by contradiction. It begins by assuming R is countable and
thus that some sequence x1, x2, x3, ... has all of R as its range.
Define two other sequences (an) and (bn) as follows:
Pick a1 < b1 in R (possible because of property 2).
Let an+1 be the first element in the sequence x which is between an and
bn (possible because of property 3).
Let bn+1 be the first element in the sequence x which is between an+1
and bn.
The two monotone sequences a and b move toward each other. By the
completeness of R, some point c must lie between them. (Define A to be
the set of all elements in R that are smaller than some member of the
sequence a, and let B be the complement of A; then every member of A is
smaller than every member of B, and so property 4 yields the point c.)
The claim is that c cannot be in the range of the sequence x, and that
is the contradiction. If c were in the range, then we would have c = xi
for some index i. But then, when that index was reached in the process
of defining a and b, then c would have been added as the next member of
one or the other of those two sequences, contrary to the assumption that
it lies between their ranges.
(\quote)


> You have not *shown* that, but defined it, erroneously. But if you had
> shown it, then 0.111... was in the list, which also would have been
> wrong.

Stated without proof at all. What is erroneous about my definition?
Do you assert that definitions can be erroneous? If so, why? Do you
think the definition
Let a be the number such that a = 4 and a = 5
is erroneous? I think not. It is a proper definition, but there is just
no 'a' that satisfies the definition.
It is erroneous, because you say let a *be* which is false, if a cannot
*be*.

There is no such thing as an "erroneous" definition, except possibly in
the sense of a grammatically incorrect one. A definition may lack any
instantiation, such as a 4 sided triangle, but as a definition is valid.

yada blabba flob. Sure Virgule.

TO at the peak of his rational reasoning again, I see.
.