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An uncountable countable set

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W. Mueckenheim
science forum Guru

Joined: 23 Apr 2005
Posts: 934

Posted: Mon Jul 17, 2006 8:56 pm Post subject: Note

Re: An uncountable countable set

Franziska Neugebauer schrieb:

Quote:

It reads: Either there is a column with only zeros, or there is at
least one 1 in each column spanned by the digit positions of 0.111...
, isn't it?
But you will "argue": Nice try but there is neither nor.

What you call a proof is an unproven proposition.

IF aleph_0 does exist, THEN 0.111... covers aleph_0 columns.
IF 0.111... covers aleph_0 columns, THEN aleph_0 columns do exist.
IF aleph_0 columns do exist THEN we can consider their contents.
IF we can consider the contents of each column, THEN we can ask how
many 1's are therein.
IF we can ask how many 1's are in each one, THEN the answer can be
"zero 1's" or "not zero 1's".
IF the answer is in each case is "not zero 1's", THEN in each column
at least one 1 must be present.

However, there is no natural numbers with this property, because
0,111... has more 1's than each natural number. Hence 0.111... itself
must be present among its disciples. Unheard and unseen but in the
midst among them.

----- Matheology.

Quote:

Is it in particular point 4 which excites your fury?

I will not read it.

The ultimate defense.

Self protection.

Such public confessions are unusual in matheology.

Regards, WM

Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Mon Jul 17, 2006 9:14 pm Post subject: Note

Re: An uncountable countable set

In article <1153144813.275459.154630@m73g2000cwd.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:

Quote:

David Hartley schrieb:

I cannot comment on the German text, but Dik's reading of the English
is
clearly correct, WM's wrong. To put it even more clearly

Question: Which transformations preserve ordinal number?

Question: What is an order-preserving transformation?

A function f: A --> B such that f(x) > f(y) if and only if x > y.

But that is not the issue here. It is the well-ordering of the set , not
its ordinary ordering that is to be preserved, and any transpostion will
destroy the original ordering, but not the well ordering.

Quote:

As I and others keep reminding you, it doesn't make any difference what
Cantor actually meant. Quoting Cantor does not constitute proof.

But outspoken wrong interpretations like that of Dik and yours must be
corrected.

Since you seem to object violently to having your own mistakes
corrected, one does not see that your attitude is at all justifiable.

Quote:

If you
want to make your supposed proof of inconsistency rigorous, you must:

a) define what you mean by the result of applying an infinite sequence
of transpositions,

Has been done.

Not to the satisfaction of anyone else, it hasn't.

Quote:

b) prove that your particular sequence re-orders a well-ordering of the
positive rationals to the usual ordering,

c) prove that such a sequence cannot alter the order-type.

It is impossible to have a dense order and a well order simultaneously.

Then you cannot do what you claim to have done.

Quote:

(Naturally, you will do this within a standard formulation of modern set
theory.)

Dik and I think you can do b but not c. Virgil thinks you can't do b,
but until you provide a your claims are "not even wrong", just
meaningless.

What would a proof mean? Dik would think this and Virgil would think
that and you would perhaps have another opinion, but nobody would
accept this as a proof, because all like set theory and then it would
be dead.

So far WM has provided no proof that is valid within any standard set
theory, i.e., that does not require assumptions not a part of ZF or NBG.

Quote:

Set theory is meaningless, but it has the advantage that it cannot be
wrong.

Which gives it a tremendous advantage over WM, who can rarely be right.

Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Mon Jul 17, 2006 9:20 pm Post subject: Note

Re: An uncountable countable set

In article <1153145345.281803.129520@35g2000cwc.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:

Quote:

Franziska Neugebauer schrieb:

mueckenh@rz.fh-augsburg.de wrote:

Franziska Neugebauer schrieb:

You may recall that every sequence member has trailing zeros.

Indeed. I recall. Therefore the linearity of the list numbers enforces
a column with zeros.

Nice try, but non sequitur. There is no such column. Otherwise show one
or prove its existence.

The proof requires logic. Therefore I am afraid you will not accept it.

WM's version of logic is quite different from the sort used in ZF of NBG
or any other part of mathematics.

Quote:

Consider the columns spanned by the digit positions of 0.111... Either
there is a column with only zeros, or there is at least one 1 in each
column, or ?

If you mean to list 0.0, 0.1, 0.11,0,111,... so that the 0's line up
vertically, then every "column" has a least one 1. And each column to
the right of the '.' column has the same "number" of 1's as all the
others.

Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Mon Jul 17, 2006 9:26 pm Post subject: Note

Re: An uncountable countable set

In article <1153147907.669663.129980@75g2000cwc.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:

Quote:

Set theory lives by contradiction. Some exhaustions
of nfinite sets are accepted other are not. But except from tradition
there are no other reasons. Therefore, set theory is folklore.

Regards, WM

The difference is that our "traditions and folklore", which we chose to
call axioms and definitions, are logically consistent, as far as anyone
can tell, whereas WM's is full of self-contradictons.

Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Mon Jul 17, 2006 9:29 pm Post subject: Note

Re: An uncountable countable set

In article <1153148551.942037.97110@s13g2000cwa.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:

Quote:

This is the deep dilemma of
set theory: There is no actually infinite set of finite numbers.

But the existence of this "dilemma" can only be established by assuming
it.

So for those who do not chose to assume it, it does not exist.

Franziska Neugebauer
science forum addict

Joined: 23 Apr 2005
Posts: 52

Posted: Mon Jul 17, 2006 9:44 pm Post subject: Note

Re: An uncountable countable set

mueckenh@rz.fh-augsburg.de wrote:

Quote:

preliminaries:
--------------

aleph_0 def= | omega |
0.111... =def (a_i) having a_i = 1 A i e omega
a_ij means the well known matrix of figures

Quote:

IF aleph_0 does exist, THEN 0.111... covers aleph_0 columns.

This is as meaningful as
If i exists then sqrt(-1) is i.

Quote:

IF 0.111... covers aleph_0 columns, THEN aleph_0 columns do exist.

This is as meaningful as
If sqrt(-1) is i then i exists.

Quote:

IF aleph_0 columns do exist THEN we can consider their contents.

This is as meaningful as
If i exists then we can consider its value.

Quote:

IF we can consider the contents of each column, THEN we can ask how
many 1's are therein.

Lotta questions.

Quote:

IF we can ask how many 1's are in each one, THEN the answer can be
"zero 1's" or "not zero 1's".

We can.

Quote:

IF the answer is in each case is "not zero 1's", THEN in each column
at least one 1 must be present.

This is the case, since every a_jj = 1 j e N by definition.

Quote:

However, there is no natural numbers with this property,

Could you precicely _define_ which /property/ you are talking about?

For every column j e N a_mj has the 1 in position m(j) = j, since a_jj =
1 A j e N. Where exactly lies your problem?

Quote:

because 0,111... has more 1's than each natural number.

You are riding a dead horse.

F. N.
--
xyz

Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Mon Jul 17, 2006 9:57 pm Post subject: Note

Re: An uncountable countable set

In article <1153168957.805313.57460@p79g2000cwp.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:

Quote:

Dik T. Winter schrieb:

Now you use an entirely new term: "can be exhausted". I think you mean
that you can take out elements one by one and doing this at some stage
the infinite set becomes empty. Howver, I think, that if that can be
done, that there is a last element you can take out. And, according
to the axiom of infinity, that is not possible, so infinite sets can
not be exhausted in this sense.

But in another sense?

In the sense of having a set of all of them, as per the axiom of
infinity, an axiom does it.

Franziska Neugebauer
science forum addict

Joined: 23 Apr 2005
Posts: 52

Posted: Mon Jul 17, 2006 10:00 pm Post subject: Note

Re: An uncountable countable set

Franziska Neugebauer wrote:

Quote:

mueckenh@rz.fh-augsburg.de wrote:
Franziska Neugebauer schrieb:

[...]

Quote:

However, there is no natural numbers with this property,

Could you precicely _define_ which /property/ you are talking about?

BTW: The /property/ _of_ _which_ object?

F. N.
--
xyz

Dik T. Winter
science forum Guru

Joined: 25 Mar 2005
Posts: 1359

Posted: Mon Jul 17, 2006 10:44 pm Post subject: Note

Re: An uncountable countable set

In article <1153145345.281803.129520@35g2000cwc.googlegroups.com> mueckenh@rz.fh-augsburg.de writes:

Quote:

Franziska Neugebauer schrieb:
mueckenh@rz.fh-augsburg.de wrote:
....
Indeed. I recall. Therefore the linearity of the list numbers enforces
a column with zeros.

Nice try, but non sequitur. There is no such column. Otherwise show one
or prove its existence.

The proof requires logic. Therefore I am afraid you will not accept it.
It reads: Either there is a column with only zeros, or there is at
least one 1 in each column spanned by the digit positions of 0.111... ,
isn't it?

Yes, right. And now the remainder of the proof, please?
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Dik T. Winter
science forum Guru

Joined: 25 Mar 2005
Posts: 1359

Posted: Mon Jul 17, 2006 11:35 pm Post subject: Note

Re: An uncountable countable set

In article <1153147907.669663.129980@75g2000cwc.googlegroups.com> mueckenh@rz.fh-augsburg.de writes:

Quote:

Dik T. Winter schrieb:

You do not believe in limits? Otherwise, why does the definition not make
sense? Quote the definition for precisely that notation:
0.999... = lim{n -> oo} sum{k = 1..n} 9.10^(-k)
what part of that definition makes no sense?

"n --> oo" because there is no natural number oo. n cannot become oo
whether there are infinitely many natural numbers or not. Each is
finite. Thus, for n e |N, the result is always different from 1.000...
.

n -> oo does *not* mean that n will become some oo, because n will not
become oo; it only means that n grows without bound. And as you should
know how the limit given above is defined, you ought to know that.
For each (real) epsilon > 0 there is an n0 such that for n > n0
|1 - sum{k = 1..n} 9.10^(-k)| < epsilon
at what place does n become oo? In the above, take n0 = log_10(-epsilon)
when epsilon < 1.0, else take n0 = 0.

n does not become oo. And therefore we have only the epsilon
definition. And therefore we have *always* undefined digits in any
irrational number and in the diagonal of Cantor's list.

Sorry, we are now talking about 0.999..., please remain with the argument.
You said that the definition is silly. Why is the definition above silly?
And that there are undefined digits in irrational numbers does not bother
me in the least. I have no idea why you have a problem with that.

But to go on:

Quote:

And therefore we have *always* undefined digits in any
irrational number and in the diagonal of Cantor's list. There are
*always* most of its digits unknown, i.e., always there are more digits
unknown than are known. No matter how small an epsilon you select.

For Cantor's diagonal no epsilon is needed, neither is their for irrational
numbers. The only thing we need to know is that the digits are all in the
range [0, 9], and that is sufficient (by minoration and majoration) that
the sequence converges and has (if we want a complete field) a real number
as limit. And for all the four different definitions of the real numbers
I know it can be shown that the definitions are equivalent (i.e. the
resulting fields are isomorphic) and that the resulting field is complete.

Quote:

Hence you cannot prove that the digits of the diagonal are all
different from those of the line numbers.

Why not? For the proof no exhaustion is needed. Just like (in another
thread) the proposition
For all p there is an n such that An[p] = K[p]
does not need exhaustion. In the case of the list and the diagonal the
similar statement is
For all p there is an n such that An[p] != D[p]
proving that D is different from all An. BTW, in the first case also the
proposition
For all p there is an n such that An[p] != K[p]
proving that K is also different from all An.
In the first and second case take n = p, in the third case take n = p + 1.
This is *not* a proof by exhaustion. It is simply stating a fact, with
an easy proof.

Quote:

You can prove it for each one
but you cannot prove it for all.

If something is true for each one, it is true for all. Let's reason
with the excluded middle (because that would complicate matters). Assume
it is not true for all, then there must be some for which it is not
true (say z). On the other hand, it is true for each one, so also
true for z. A contradiction, hence the assumption is false. And in
the cases above you can proof it for each one in one sweep, because
you prove it for an arbitrary element.

Quote:

That is the same problem as with the
*+ sum of my list. You can prove the sum is 1 for each column but you
cannot prove it for all columns.

But you can.

Quote:

Because stepwise exhaustion of
infinite sets is impossible. Otherwise my (and Cantor's) reordering
could be completed.

If stepwise exhaustion is impossible, your reordering could be completed.
But you can with your *+ operation give the proof in one sweep, but you
failed to answer to my question: "what happens at infinity?".

But, that was precisely what I intended when I asked you what you meant
with "*+ ..." in 0.1 *+ 0.11 *+ 0.111, because you gave no definition.
As you gave no meaning to that infinite "sum", only to finitely many
of them (and in that case going to infinity is impossible), there was
no way with your definitions.

Quote:

But the assertion
that the digits of these limits could be used to construct a diagonal
number is simply nonsense.

Only assertion and meaning. No content.

Proven by epsilon. Set theory lives by contradiction. Some exhaustions
of nfinite sets are accepted other are not.

You can exhaust an infinite set if you take out all elements at once.
That is why the function f(n) = n + 1 is a bijective mapping from
{1, 2, ...} to {2, 3, ...}, every definition works at once, you need
not to count to 100 to know what f(100) is. And the same is the case
with Cantor's diagonal number. You need not look at the first 100
elements to find the 101-st element to know that the 101-st digit
of the diagonal is. That is what the definition of a list is. A
mapping from N to the members of the list.

Your set of transpositions are different. They do not work at once, you
need sequencing (transpositions are in general non-commutative). In the
case of a sequence you need limiting procedures to show what would be
the "final" result (like lim{n -> oo} 1/n = 0). But the problem with
limiting procedures is that the "final" result has not necessarily the
same properties as each and every finite result.

Let me give an example with transpositions. Let's say that (a, b)
means that we interchange the a-th and b-th element in an ordering.
Let us start with the ordered set of natural numbers {1, 2, 3, ...}
(yes, not Bourbaki this time). Let's define a sequence of transpositions:
(1, 2)(2, 3)(3, 4)(4, 5)...
with a proper measure and a proper definition of limit, I think that we
can show that that leads to the ordered set {2, 3, ..., 1}. Now interchange
the first two elements of the transpositions, in this case we get
{1, 3, ..., 2}. So sequencing plays a crucial role.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Dik T. Winter
science forum Guru

Joined: 25 Mar 2005
Posts: 1359

Posted: Mon Jul 17, 2006 11:41 pm Post subject: Note

Re: An uncountable countable set

In article <1153148551.942037.97110@s13g2000cwa.googlegroups.com> mueckenh@rz.fh-augsburg.de writes:

Quote:

Dik T. Winter schrieb:

In article <1153052131.852951.273540@b28g2000cwb.googlegroups.com> muecke=
nh@rz.fh-augsburg.de writes:
Dik T. Winter schrieb:
....
(1): 0.111... is not *the* successor of anything, we may sloppily say that
it is *a* successor of all naturals, just like 10 is *a* successor of
2

It was Cantor who coined this expression saying " daß omega die erste
ganze Zahl sein soll, welche auf alle Zahlen nu folgt." (Works, p.
195) So it must be larger anyhow.

Yes. Not *the* successor, but *a* successor.

Quote:

The successor of all naturals is not a natural and, therefore, must be
larger (because it is not less).

Yes, it is larger than all naturals, but I would not call it *the*
successor, but *a* successor, or, if you wish, *the smallest* successor.

However, not all of its 1's in unary representation can be indexed by
natural numbers because they are smaller.

Are you again arguing that the statement
For all p, there is an n such that An[p] = K[p]
is false, or are you arguing that the statement
For all p, there is an n such that An[p] != K[p]
is false?

Quote:

There is no actually infinite set of finite numbers.

Axiom of infinity.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Dik T. Winter
science forum Guru

Joined: 25 Mar 2005
Posts: 1359

Posted: Tue Jul 18, 2006 12:44 am Post subject: Note

Re: An uncountable countable set

In article <1153168957.805313.57460@p79g2000cwp.googlegroups.com> mueckenh@rz.fh-augsburg.de writes:

Quote:

Dik T. Winter schrieb:

In German it is "Umformungen", that means "change of the order". Cantor
says: those changes, which and only which can be traced back to
finitely many or infinitely many transpositions of elements, each
transposition including two elements.

So you claim that Cantor meant unrestricted re-ordering. If he meant that,
his statement is false. That is entirely possible, of course.

Of course.

But after reading the context today, I come to the conclusion that he does
*not* mean unrestricted re-ordering. The context is what ordinal numbers
can a set of particular cardinality have when it is well-ordered. His
transformations are (in my opinion) only transformations that change order
(and not arbitrary transformations), and also that the question ("which
transformations") refers to transformations that change a well-ordered
set to a well-ordered set.

Quote:

His other statements on infinity are wrong too. In
particular that which says that 0.111... is different from every
natural number but no digit is different from every digit of every
natural number.

Strange enough, when I read what he wrote, he does not state that, I think
quite contrary. Pray read page 213. Especially the sentence:
"während die Mengen erster Mächtigkeit nur durch (mit Hilfe von) Zahlen
der zweiten Zahlenklasse abgezählt werden können..."
translated:
"while sets of the first cardinality only can be counted through (with
the help of) numbers from the second class of numbers..."
I think (but have not looked it up for a thorough study) that the first
cardinality refers to aleph-0, and that (of this I am sure) second class
numbers are numbers of the form w, w+1, etc.. So we can infer that he
claims here that to count the numbers of the set of natural numbers
(cardinality aleph-0) you need at least w. I think not very dissimilar
from what you are arguing. And I think that you are agreeing with Cantor.
It may be noted that since that time quite a bit of research has been
done to give better foundation. This has resulted in redefinitions of
terms by Cantor and sometimes different (contradicting) formulations.
It was a field of research in progress, and even published results could
be shown to be (partially) wrong using later insight.

Another thing that did strike me: on that page he assumed the axiom of
choice (that is, every set can be well-ordered).

Quote:

Now you use an entirely new term: "can be exhausted". I think you mean
that you can take out elements one by one and doing this at some stage
the infinite set becomes empty. Howver, I think, that if that can be
done, that there is a last element you can take out. And, according
to the axiom of infinity, that is not possible, so infinite sets can
not be exhausted in this sense.

But in another sense?

All elements at once?

Quote:

You use a contradiction of the axiom of infinity (can be exhausted), so
there is no disproof. Now try the same with the assumptions that
infinite sets do exist and can not be exhausted.

Then there is no evidence that they exist.

Again philosophical and hence quite a bit of matheology. What means "exist"?
I have no idea. As a mathematician I am given a set of axioms and show the
results from the set of axioms. Some have their uses in practical sense,
some have no use in a practical sense at all. Give me a different set of
axioms, and I will do the same. That is what happens all the time. Start
with the Eclidean postulates (axioms) and show that the sum of the angles
of a triangle is 180. That is a result that may or may not have its uses.
Now change the parallel postulate and find that the sum of the angles of
a triangle is in general not 180. But also in this case the results may
or may not have practical implications.

Quote:

But his position was also, that infinite sets can be
exhausted if for each element there is a precise definition like that
leading to the well-order of the rationals. The well order of the
ordinals is another example. And the transformations consisting of
transpositions is a third.

You should better read his positions thoroughly, compare it to current versions
of set theory, note the differences, and, after that, hit at the wrong-doer.
But this is of course not mathematics.

Quote:

Well, I think *any* re-ordering can be accomplished by a sequence of
transpositions of two elements.

Cantor shared your opinion, but only in case of finite sets.

Perhaps, I have seen no evidence of that.

Quote:

But when I write:
The question, though which froobles the ordinal number of a
well-ordered set is changed, and through with it is not changed, is
easily answered, those, and only those froobles leave the ordinal
number unchanged that can be rewritten as a finite or infinite
set of transpositions, that is, of interchanges of two elements.
you do not need the definition of froobles? I can tell you that my
statement is entirely correct. (I can state that your sequence of
transpositions does not form a frooble.)

If you exchange "can be rewritten as" by "can be traced back to" or by
"consist of", as is the meaning of Cantor's German statement, then
"those froobles" is obviously merely another name for " fin. or inf.
set of transpositions".

You are wrong. Let's suppose that a "frooble" is a transformation that
interchanges the first two elements of an ordered set (I do not say it is,
but let us just assume for the argument).

Quote:

Eh? Infinite sets do exist, but they can not be exhausted. So why do they
not have cardinality? It is entirely possible to define cardinality for
them. What is your problem with that?

That it is not possible to do so without well-ordering all elements.
That is nothing but an exhaustion.

Cardinality does not need well-ordering. It is ordinality that needs
well-ordering. And, well-ordering is not an exhaustion.

Quote:

One should have seen that earlier, then Bourbaki would not have
succedeed to define 0 as natural number, even in political decisions.

Pray explain the last part "even in political decisions".

It is laid down in the guide lines of the European Community that zero
was a natural number. I am indebted to your compatriots that they have
dismissed the constitution of this disastrous association.

Perhaps. I have no idea. There were quite a few other reasons that I
voted against it, but this is not the forum to discuss that.

Quote:

But I can rewrite
it in non-Bourbaki:
Reorder the naturals to (2, 3, 4, ..., 1) and see that the order type is
now w+1, so it did change from w.

That would be nice, if omega could be exhausted. But it cannot. Hence,
there is no omega + 1.

You mean that the above ordered set does not have an ordinal number?
Why is it not possible to define such?

Quote:

You are fighting the mathematical use of limits. On what basis do you
allow the computational use?

On practical basis. As precise as any computation can be the usual
definitions are correct.

Do you always use a full error-analysis on your computations?
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Dik T. Winter
science forum Guru

Joined: 25 Mar 2005
Posts: 1359

Posted: Tue Jul 18, 2006 1:16 am Post subject: Note

Re: An uncountable countable set

In article <1153169497.380974.32190@i42g2000cwa.googlegroups.com> mueckenh@rz.fh-augsburg.de writes:

Quote:

Dik T. Winter schrieb:

0 was invented less than 2000 years ago. Natural numbers need not be
invented. However, Bourbaki and Halmos tried this trick in order to
prevent set theory from too easily been demasked as inconsistent.

Eh? I was not talking about invention but about definition. What problems
do you have with definitions?

I don't like definitions which define nonsense like the corner of a
circle.

Check the manhattan-measure and you will find square circles. But whatever,
if you do not like definitions that define nonsense in your opinion, you
should have a hard time with mathematics.

Quote:

Yes, it was sloppy terminology. What happens when n grows without bound?

Nothing happens with the *+ sum.

What is lim{n -> oo} SUM{i = 1 .. n} A_i ? How do you define that?
Without such a definition I have no idea what the result is when I *+
all An.

If there is at least one 1 in a column then the *+sum is 1. You need
not investigate how many 1's are there to follow if you want to
calculate the *+ sum.

That is not an answer to my question. But I will take it in good faith,
so
lim{n -> oo} SUM{i = 1 .. n} A_i = 0.111...
tell me where I have gone wrong.

Quote:

Define: If any case includes at least one 1 the the *+ sum is 1.

Again, in the finite case. You have not defined what you mean with the
infinite sum.

Hell and devil! Can't you read? The definition for *+ sum = 1 is: at
least one 1 must be encountered. That is enough in any case, finite or
not.

Damnation leaving aside, you defined what SUM{i = 1 .. n} An is. You did
*not* define what happens when n grows without bound. Now you state that
in each column, whenever there is a 1, the final result should be 1. But
what you are now meaning is that lim{n -> oo} SUM{i = 1 .. n} An = 0.111... .

Quote:

So the statement
For all n, An[n] = K[n]
is true? As is the statement
For all p there is an n such that An[p] = K[p]
also true?

No. Then 0.111... would not differ from every n.

No to which question? Is the first statement false? And if so, why?
Is the second question false, and if so why? And how do you come at
your conclusion? I think your logic is lacking.

Quote:

Why than do you write that it is false?

Because it is not correct. According to the axiom of infinity [...]
infinite sets can not be exhausted in this sense. (Dik T. Winter)
Therefore, a unary representation of a natural number can never reach
the line next to the unary representation of aleph_0.

You are arguing on two different lines at the same time, confusing one
with the other. When we consider the An as unary representations of
natural numbers the result is aleph_0, not a natural number. When we
consider them as being decimal numbers, the result is 1/9. In both
cases the resulting number is not in the list. What is the problem?

Quote:

Therefore, a unary representation of a natural number can never reach
the line next to the unary representation of aleph_0. The latter does
exist according to set theory.

Can you prove that? If the line next to the unary representation of
aleph_0 does exist (and I think you mean preceding line) you have to
show that aleph_0 does have a predecessor. But it has not. Unless
you show how it can be proven through set theory.

Quote:

The natural next to it does not. Hence,
the sum of

0.1
0.11
0.111
...
is *not* 0.111...

With your definition (finally extracted) above (if in any column there is
a 1, there is a 1 in the final result), it is. If you think that is false,
please show me a column where there are only 0's.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Tue Jul 18, 2006 3:09 am Post subject: Note

Re: An uncountable countable set

In article <1153169497.380974.32190@i42g2000cwa.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:

Quote:

Then stop making them.

Quote:

Yes, it was sloppy terminology. What happens when n grows without bound?

Nothing happens with the *+ sum.

As "*+" does not have the common values of a "sum", it is WM making
square circles again.

Quote:

the line next to the unary representation of aleph_0. The latter does
exist according to set theory. The natural next to it does not. Hence,
the sum of

0.1
0.11
0.111
...
is *not* 0.111...

There is a difference between appending one new value at a time to a
finite list and collecting all members of an endless list in one step.

The first is not possible to reach an end of, the second ends when it
starts.

Quote:

If exhaustion were possible, then I offered the natural oder by size
and simultaneoulsy the well-order of the rationals.

WM presents us with another of those corners of a circle.

Virgil
science forum Guru

Joined: 24 Mar 2005
Posts: 5536

Posted: Tue Jul 18, 2006 3:16 am Post subject: Note

Re: An uncountable countable set

In article <1153169803.310772.70530@s13g2000cwa.googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:

Quote:

Franziska Neugebauer schrieb:

It reads: Either there is a column with only zeros, or there is at
least one 1 in each column spanned by the digit positions of 0.111...
, isn't it?
But you will "argue": Nice try but there is neither nor.

What you call a proof is an unproven proposition.

IF aleph_0 does exist, THEN 0.111... covers aleph_0 columns.
IF 0.111... covers aleph_0 columns, THEN aleph_0 columns do exist.
IF aleph_0 columns do exist THEN we can consider their contents.
IF we can consider the contents of each column, THEN we can ask how
many 1's are therein.
IF we can ask how many 1's are in each one, THEN the answer can be
"zero 1's" or "not zero 1's".
IF the answer is in each case is "not zero 1's", THEN in each column
at least one 1 must be present.

In fact, in each "column" infinitely many 1's are present.

Quote:

However, there is no natural numbers with this property

Precisely.

Quote:

because
0,111... has more 1's than each natural number.

As does each column.

Quote:

Hence 0.111... itself
must be present among its disciples.

Non-sequitur.

Except for the leading "0" and "." there will be
"111..." in each column. But excluding the first column of all "0"s
there re no others and except for the 2nd column of all "."s there are
no others.

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