Re: A consideration concerning the diagonal argument of G. Cantor
WM
> In article <b0cd8161-c7c0-4fa5-97d1-f69fdd71c...@34g2000hsh.googlegroups.com>,
> WM says...
>
>
>
>
>
>
>
> >On 2 Okt., 21:42, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
> >> WM says...
>
> >> >But as I already mentioned much earlier. Cantor's argument fails,
> >> >because it must cover any conceivable case. One simple counter-
> >> >argument spoiling it is the sequence
>
> >> >0.0
> >> >0.1
> >> >0.11
> >> >0.111
> >> >...
>
> >> >and the replacement rule 0 --> 1 which yields an anti-diagonal that is
> >> >in the sequence,
>
> >> What are you talking about? No, it is not.
>
> >By construction the list contains at least one entry that has at most
> >one digit less. Otherwise the infinite diagonal cannot exist. How much
> >is aleph_0 minus 1?
>
> Once again, in base 2, the numbers in your list are:
> 0, 1/2, 3/4, 7/8, etc. In general, the nth element in
> your list is f(n) = (1 - 1/2^n). The number 1 is
> not on the list: forall n, f(n) is not equal to 1.
> But the anti-diagonal is 1. So it is not on your list.
If the anti-diagonal is 1 = 0.111..., then every initial segment is in
the list.
If the anti-diagonal, has aleph_0 digits, then there is at least one
list entry with aleph_0 - 1 digits.
How much is that?
Look, this is trivial stuff. It's barely more than arithmetic. And it
shows that the *actual* existence of an infinity of digits is in
contradiction with mathematics. It shows that transfinite set theory
is nonsense - one of the biggest mistakes of human intellectual
history. It could even be considered the biggest one unless
transfinite set theory was so absolutely meaningless and
insignificant.
Regards, WM
Daryl McCullough
>> Once again, in base 2, the numbers in your list are:
>> 0, 1/2, 3/4, 7/8, etc. In general, the nth element in
>> your list is f(n) = (1 - 1/2^n). The number 1 is
>> not on the list: forall n, f(n) is not equal to 1.
>> But the anti-diagonal is 1. So it is not on your list.
>
>If the anti-diagonal is 1 = 0.111..., then every initial segment is in
>the list.
That's true, but irrelevant. You do agree that
the anti-diagonal is equal to 1? You do agree that
the numbers on the list are 1/2, 3/4, 7/8, etc.
You do agree that none of those is equal to 1?
--
Daryl McCullough
Ithaca, NY
Virgil
<05ad25a2-ec9a-4e64...@z72g2000hsb.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
Nonsense.
Consider the list:
0.0
0.10
0.110
...
Its anti-diagonal is 0.111..., by the list is far from complete even for
finite sequences.
> If the anti-diagonal, has aleph_0 digits, then there is at least one
> list entry with aleph_0 - 1 digits.
Claimed but not justified, unless entries can be extended with zeros, in
which case one just adds one more zero.
>
> Look, this is trivial stuff. It's barely more than arithmetic. And it
> shows that the *actual* existence of an infinity of digits is in
> contradiction with mathematics.
It doesn't show it to mathematicians, and physicists don't count.
Herbert Newman
>>
>> If the anti-diagonal is [bla bla] [WM]
>>
> Nonsense.
>
You might as well just write
Mückenheim.
Herb
WM
> WM says...
>
> >> Once again, in base 2, the numbers in your list are:
> >> 0, 1/2, 3/4, 7/8, etc. In general, the nth element in
> >> your list is f(n) = (1 - 1/2^n). The number 1 is
> >> not on the list: forall n, f(n) is not equal to 1.
> >> But the anti-diagonal is 1. So it is not on your list.
>
> >If the anti-diagonal is 1 = 0.111..., then every initial segment is in
> >the list.
>
> That's true,
Yes!
> but irrelevant.
No!
< You do agree that
> the anti-diagonal is equal to 1?
I do agree that no actual infinity exists. Therefore there is no
string 0.111... = 1!
Otherwise the same string had to exist as an entry of the list,
because an actually infinite set of elements retains the same
cardinality if you remove one element. But by the construction of the
list it is clear that the infinite diagonal cannot exist unless there
exist regular entries with the same number of 1's.
Probably you cannot understand that - nevertheless it is true.
> You do agree that
> the numbers on the list are 1/2, 3/4, 7/8, etc.
> You do agree that none of those is equal to 1?
None of them including the diagonal can be 1.000...
Regards, WM
WM
>
> Consider the list:
> 0.0
> 0.10
> 0.110
> ...
> Its anti-diagonal is 0.111..., by the list is far from complete even for
> finite sequences.
>
> > If the anti-diagonal, has aleph_0 digits, then there is at least one
> > list entry with aleph_0 - 1 digits.
>
> Claimed but not justified,
Easy to see by the construction of the list, easy for anybody not
deteriorate by set theory.
< unless entries can be extended with zeros, in
> which case one just adds one more zero.
>
>
>
> > Look, this is trivial stuff. It's barely more than arithmetic. And it
> > shows that the *actual* existence of an infinity of digits is in
> > contradiction with mathematics.
>
> It doesn't show it to mathematicians, and physicists don't count.
Even a "mathematician" like you should know the basics of geometry
required to understand the list above.
Regards, WM
Virgil
<6f742f42-e3c7-4d57...@l76g2000hse.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 4 Okt., 14:09, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
> > WM says...
> >
> > >> Once again, in base 2, the numbers in your list are:
> > >> 0, 1/2, 3/4, 7/8, etc. In general, the nth element in
> > >> your list is f(n) = (1 - 1/2^n). The number 1 is
> > >> not on the list: forall n, f(n) is not equal to 1.
> > >> But the anti-diagonal is 1. So it is not on your list.
> >
> > >If the anti-diagonal is 1 = 0.111..., then every initial segment is in
> > >the list.
> >
> > That's true,
>
> Yes!
>
> > but irrelevant.
>
> No!
Yes!
>
> < You do agree that
> > the anti-diagonal is equal to 1?
>
> I do agree that no actual infinity exists. Therefore there is no
> string 0.111... = 1!
Then you deny that there is a ring of integers or a field of rationals
or a complete field of reals or any of the definitions and theorems of
analysis based upon the existence of such objects.
You can't have it both ways.
>
> Otherwise the same string had to exist as an entry of the list,
> because an actually infinite set of elements retains the same
> cardinality if you remove one element. But by the construction of the
> list it is clear that the infinite diagonal cannot exist unless there
> exist regular entries with the same number of 1's.
There can be a list in which the nth entry has a 0 at (or terminates
before) the nth position, in which case the anti-diagonal does exist
with all 1s.
>
> Probably you cannot understand that - nevertheless it is true.
Probably you cannot understand that - nevertheless it is true.
>
> > You do agree that
> > the numbers on the list are 1/2, 3/4, 7/8, etc.
> > You do agree that none of those is equal to 1?
>
> None of them including the diagonal can be 1.000...
What is the distance between such a diagonal and 1?
It appears to be smaller that every positive quantity.
If it is not zero, you are introducing infinitesimals into your
arithmetic.
Virgil
<b5242153-4044-42bb...@75g2000hso.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 4 Okt., 19:31, Virgil <Vir...@gmale.com> wrote:
>
> >
> > Consider the list:
> > 0.0
> > 0.10
> > 0.110
> > ...
> > Its anti-diagonal is 0.111..., by the list is far from complete even for
> > finite sequences.
> >
> > > If the anti-diagonal, has aleph_0 digits, then there is at least one
> > > list entry with aleph_0 - 1 digits.
> >
> > Claimed but not justified,
>
> Easy to see by the construction of the list, easy for anybody not
> deteriorate by set theory.
Until one has a reasonable definition of what "aleph_0 - 1" means, it is
nonsense. I know of no definition of "aleph_0 - 1" which is not
equivalent to "aleph_0" itself.
One definition of "the cardinality of set S is aleph_0" is
"Set S has a bijection with the set of all naturals".
Perhaps WM will provide as clear a definition of what he means by
"the cardinality of set S is aleph_0 - 1"
>
> < unless entries can be extended with zeros, in
> > which case one just adds one more zero.
> >
> >
> >
> > > Look, this is trivial stuff. It's barely more than arithmetic. And it
> > > shows that the *actual* existence of an infinity of digits is in
> > > contradiction with mathematics.
> >
> > It doesn't show it to mathematicians, and physicists don't count.
>
> Even a "mathematician" like you should know the basics of geometry
> required to understand the list above.
The existence of one list does not necessarily prevent the existence of
other lists. And a function from one set to another does not need to
iterate all its assignments individually if they can be covered by a
general rule, so that functions from N to an arbitrary non-empty set
are possible, even when that set is a set of functions from N to some
other set.
Daryl McCullough
>
>On 4 Okt., 19:31, Virgil <Vir...@gmale.com> wrote:
>
>>
>> Consider the list:
>> 0.0
>> 0.10
>> 0.110
>> ...
>> Its anti-diagonal is 0.111..., by the list is far from complete even for
>> finite sequences.
>>
>> > If the anti-diagonal, has aleph_0 digits, then there is at least one
>> > list entry with aleph_0 - 1 digits.
>>
>> Claimed but not justified,
>
>Easy to see by the construction of the list, easy for anybody not
>deteriorate by set theory.
You reject set theory. You reject the notion of a "completed
infinity". But you think you can prove that a list has an
entry twith aleph_0 - 1 digits. Do you really think that
makes any sense, or do you know that you are speaking
nonsense?
If you want to prove something about *your* beliefs, then
you can't bring up "aleph_0", since you don't believe in
aleph_0. If you want to prove something about standard
set theory, then you have to actually know something
about set theory.
What are you doing? Do you even know?
Daryl McCullough
>I do agree that no actual infinity exists. Therefore there is no
>string 0.111... = 1!
>
>Otherwise the same string had to exist as an entry of the list,
>because an actually infinite set of elements retains the same
>cardinality if you remove one element. But by the construction of the
>list it is clear that the infinite diagonal cannot exist unless there
>exist regular entries with the same number of 1's.
Look, if you don't believe in "actual infinity", then why are
you talking about it? That's bizarre behavior.
If you want to talk about what you believe, then don't bring
up "actual infinity". If you want to talk about standard set
theory, then use the axioms of standard set theory. Your
arguments are an incoherent mishmash of statements that
*nobody* believes. You don't believe them. I don't believe
them. No standard mathematician believes them. What in the
world do you think you are doing?
WM
> WM says...
>
> >I do agree that no actual infinity exists. Therefore there is no
> >string 0.111... = 1!
>
> >Otherwise the same string had to exist as an entry of the list,
> >because an actually infinite set of elements retains the same
> >cardinality if you remove one element. But by the construction of the
> >list it is clear that the infinite diagonal cannot exist unless there
> >exist regular entries with the same number of 1's.
>
> Look, if you don't believe in "actual infinity", then why are
> you talking about it?
It's not believing. I have seen and I can show that actual infinity is
nonsense.
> That's bizarre behavior.
It is bizarre behaviour to keep on believing what has been proved
wrong.
Regards, WM
georgie
> WM says...
>
> >> Once again, in base 2, the numbers in your list are:
> >> 0, 1/2, 3/4, 7/8, etc.
Gee, how big does that sequence get? Does it get as big as 1/2 + 1/4 +
1/8 +... = 1? If not, where exactly does it fall short? Why does
that sequence fall short only when we talk about this topic?
WM
> WM says...
>
>
>
>
>
>
>
> >On 4 Okt., 19:31, Virgil <Vir...@gmale.com> wrote:
>
> >> Consider the list:
> >> 0.0
> >> 0.10
> >> 0.110
> >> ...
> >> Its anti-diagonal is 0.111..., by the list is far from complete even for
> >> finite sequences.
>
> >> > If the anti-diagonal, has aleph_0 digits, then there is at least one
> >> > list entry with aleph_0 - 1 digits.
>
> >> Claimed but not justified,
>
> >Easy to see by the construction of the list, easy for anybody not
> >deteriorate by set theory.
>
> You reject set theory. You reject the notion of a "completed
> infinity". But you think you can prove that a list has an
> entry twith aleph_0 - 1 digits. Do you really think that
> makes any sense,
If we assume that aleph_0 exists, then we get a contradiction. Yes, it
makes sense to assume, but not to believe in aleph_0.
>
> If you want to prove something about *your* beliefs,
After all you write here, I can guess what you believe, but there is
not one word about sober science.
> then
> you can't bring up "aleph_0", since you don't believe in
> aleph_0.
Have you ever seen a mathematical proof? One of the most elemntary
poofs is that of sqrt(2) not being a real number. Is it prohibited to
assume sqrt(2) = p/q if one does not believe it.
> If you want to prove something about standard
> set theory, then you have to actually know something
> about set theory.
I know that aleph_0 * aleph_0 = aleph_0. That's enough. If you want to
learn a bit more on set theory, you can read my book (if you
understand somethinmg else than English).
>
> What are you doing? Do you even know?
I have seen that you needed three post to get straight what 0.111...
is in binary. You should first learn elementary arithmetic.
Regards, WM
>
WM
> > < You do agree that
> > > the anti-diagonal is equal to 1?
>
> > I do agree that no actual infinity exists. Therefore there is no
> > string 0.111... = 1!
>
> Then you deny that there is a ring of integers or a field of rationals
> or a complete field of reals or any of the definitions and theorems of
> analysis based upon the existence of such objects.
>
> You can't have it both ways.
I can't change that. It is as it is. Nevertheless we can act "as if"
in arithmetic.
>
>
>
> > Otherwise the same string had to exist as an entry of the list,
> > because an actually infinite set of elements retains the same
> > cardinality if you remove one element. But by the construction of the
> > list it is clear that the infinite diagonal cannot exist unless there
> > exist regular entries with the same number of 1's.
>
> There can be a list in which the nth entry has a 0 at (or terminates
> before) the nth position, in which case the anti-diagonal does exist
> with all 1s.
For every 1 of the anti-diagonal there must exist an entry with a
string of 1's reaching to that position.
> > > You do agree that
> > > the numbers on the list are 1/2, 3/4, 7/8, etc.
> > > You do agree that none of those is equal to 1?
>
> > None of them including the diagonal can be 1.000...
>
> What is the distance between such a diagonal and 1?
> It appears to be smaller that every positive quantity.
It depends on the precision of your calculator.
> If it is not zero, you are introducing infinitesimals into your
> arithmetic.-
No. I recognized the practical limitations of calculating.
Regards, WM
WM
> In article
> <b5242153-4044-42bb-80a9-c52dd2c71...@75g2000hso.googlegroups.com>,
>
>
>
>
>
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 4 Okt., 19:31, Virgil <Vir...@gmale.com> wrote:
>
> > > Consider the list:
> > > 0.0
> > > 0.10
> > > 0.110
> > > ...
> > > Its anti-diagonal is 0.111..., by the list is far from complete even for
> > > finite sequences.
>
> > > > If the anti-diagonal, has aleph_0 digits, then there is at least one
> > > > list entry with aleph_0 - 1 digits.
>
> > > Claimed but not justified,
>
> > Easy to see by the construction of the list, easy for anybody not
> > deteriorate by set theory.
>
> Until one has a reasonable definition of what "aleph_0 - 1"
Take a set of aleph_0 elements and remove one element. That is aleph_0
- 1.
> means, it is
> nonsense. I know of no definition of "aleph_0 - 1" which is not
> equivalent to "aleph_0" itself.
Therefore an anti- diagonal with aleph_0 digits 1 requires at least
one line with as many digits 1.
> One definition of "the cardinality of set S is aleph_0" is
> "Set S has a bijection with the set of all naturals".
Another one is: The set consists of aleph_0 countable sets.
> Perhaps WM will provide as clear a definition of what he means by
> "the cardinality of set S is aleph_0 - 1"
See above.
Regards, WM
Daryl McCullough
>> Look, if you don't believe in "actual infinity", then why are
>> you talking about it?
>
>It's not believing. I have seen and I can show that actual infinity is
>nonsense.
What you are doing is making up a theory of infinity, and then showing
that your made-up theory is nonsense. Yes, that's true. You invented a
nonsensical theory. But your invented theory is *not* the theory described
by ZF. It is not Cantor's theory of infinity.
If a 7 year old makes a mistake doing subtraction, does that show that
subtraction is nonsense? Of course not. Your remarks about "actual
infinity" have no more significance.
Daryl McCullough
>
>On Oct 4, 7:09=A0am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
>> WM says...
>>
>> >> Once again, in base 2, the numbers in your list are:
>> >> 0, 1/2, 3/4, 7/8, etc.
>
>Gee, how big does that sequence get?
It's easy enough to prove the following facts about the sequence:
1. No entry is greater than or equal to 1.
2. If x is any real strictly smaller than 1, then there is an
entry that is greater than x.
The second fact is interesting, but irrelevant to the claim that
no entry is equal to 1.
Daryl McCullough
>> You reject set theory. You reject the notion of a "completed
>> infinity". But you think you can prove that a list has an
>> entry twith aleph_0 - 1 digits. Do you really think that
>> makes any sense,
>
>If we assume that aleph_0 exists, then we get a contradiction.
YOU get a contradiction, because you are using inconsistent
reasoning. There is no contradiction that has been derived
from standard set theory (as codified in ZF, for example).
>> If you want to prove something about standard
>> set theory, then you have to actually know something
>> about set theory.
>
>I know that aleph_0 * aleph_0 = aleph_0.
But you don't even know what that means. You don't know
what aleph_0 is. You are in the position of a seven year
old trying to prove that negative numbers are nonsense.
You make up your own theory of infinity, and you proceed
to show that *your* theory is nonsense. That's true. Your
theory of infinity is nonsense, because you are a very
incompetent mathematician. You are *playing* at being
a mathematician.
If you want to show that *standard* set theory is nonsense,
then derive a contradiction using *only* standard axioms of
set theory. If you derive a contradiction using *your*
mixed-up notion of set theory, you've just proved that
*you* are inconsistent.
Why can't you understand that? The way to prove that a theory
is inconsistent is to *use* that theory (and nothing more) and show
that it leads to a contradiction. You can't do that; you always
make claims that don't follow from standard mathematics. So your
claim that standard set theory is contradictory is just nonsense.
You don't know what you are talking about.
Daryl McCullough
>Take a set of aleph_0 elements and remove one element. That is aleph_0
>- 1.
You have no idea what you are talking about. Do you know what
aleph_0 is? No, you don't. You are a child talking about topics
that you know nothing about, pretending that you are doing something
important.
georgie
No part of your answer is relevent to the question I asked: How big
does the sequence get?
Herbert Newman
> WM says...
>
>> Take a set of aleph_0 elements and remove one element. That is aleph_0
>> - 1.
>>
> You have no idea what you are talking about.
>
How many posts did it need to get that straight?
Herb
Herbert Newman
>>
>> ... definition of "aleph_0 - 1" [???]
>>
> Take a set of aleph_0 elements and remove one element. That is aleph_0 - 1.
>
Huh?
I guess you meant (or not, who knows)
Take a set of aleph_0 elements and remove one element.
Then the "resulting" set has aleph_0 - 1 elements.
With other words: For any M and a:
|M| = aleph_0 & a e M -> |M \ {a}| = aleph_0 - 1.
Well, fine. In this case
aleph_0 - 1 = aleph_0.
(Proof left as an exercise.)
>>
>> I [can't imagine any] definition of "aleph_0 - 1" which is not
>> equivalent to "aleph_0" itself. [Virgil]
>>
And here Virgil is damn right.
>
> Therefore an anti-diagonal with aleph_0 digits 1 requires at least
> one line with as many digits 1.
>
No it doesn't. It requires just that the list is countable infinitely long;
i.e. has aleph_0 many entries. With other words, it requires that the
number of lines of the list is aleph_0. (Each and every line of the list is
"related" to exactly one digit of the diagonal and/or anti-diagonal.)
>>
>> One definition of "the cardinality of set S is aleph_0" is
>> "Set S has a bijection with the set of all naturals".
>>
Well, maybe we should not consider this to be a "definition" proper. But
clearly we have
|S| = aleph_0 iff there is a bijection between S and IN.
>
> Another one is: The set consists of aleph_0 countable sets.
>
Only in WM's strange anti-math world.
Herb
WM
His answers turn more and more into personal insults only whereas (or
because) he cannot even do the simplest mathematics. Therefore I quit
this discussion.
Regards, WM
Herbert Newman
>
> I quit this discussion.
>
Great!
Herb
Phil Carmody
Well, WM has made aleph_0 posts, and Daryl finally worked
out his stupidity on the final one, so he overlooked it
for aleph_0 - 1 posts. Duh!
Phil
--
The fact that a believer is happier than a sceptic is no more to the
point than the fact that a drunken man is happier than a sober one.
The happiness of credulity is a cheap and dangerous quality.
-- George Bernard Shaw (1856-1950), Preface to Androcles and the Lion
georgie
So we're supposed to believe that you know all about how big that
sequence gets except that you don't know how big it gets.
Virgil
<711f11d0-86af-4223...@26g2000hsk.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
That makes WM bizarre.
Virgil
<ed59c1c1-80c6-463f...@u46g2000hsc.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 6 Okt., 12:51, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
> > WM says...
> >
> >
> >
> >
> >
> >
> >
> > >On 4 Okt., 19:31, Virgil <Vir...@gmale.com> wrote:
> >
> > >> Consider the list:
> > >> 0.0
> > >> 0.10
> > >> 0.110
> > >> ...
> > >> Its anti-diagonal is 0.111..., by the list is far from complete even for
> > >> finite sequences.
> >
> > >> > If the anti-diagonal, has aleph_0 digits, then there is at least one
> > >> > list entry with aleph_0 - 1 digits.
> >
> > >> Claimed but not justified,
> >
> > >Easy to see by the construction of the list, easy for anybody not
> > >deteriorate by set theory.
> >
> > You reject set theory. You reject the notion of a "completed
> > infinity". But you think you can prove that a list has an
> > entry twith aleph_0 - 1 digits. Do you really think that
> > makes any sense,
>
> If we assume that aleph_0 exists, then we get a contradiction.
Is that a royal "WE"? Since it does not seem to apply to more than one
person here, "we" seems a bit inappropriate.
Yes, it
> makes sense to assume, but not to believe in aleph_0.
> >
> > If you want to prove something about *your* beliefs,
>
> After all you write here, I can guess what you believe, but there is
> not one word about sober science.
The sort of science you seem to favor does not seem all that sober.
>
> I know that aleph_0 * aleph_0 = aleph_0.
Do you know what it means?
It means that the cardinality of NxN, the Cartesian product of N with
itself, or , equivalently, the set of all ordered pairs,
{(m,n): m in N and n in N} can be sequentially ordered so as to be order
isomorphic to (N,<).
Do you deny that such bijection is possible?
> That's enough. If you want to
> learn a bit more on set theory, you can read my book (if you
> understand somethinmg else than English).
Never have I been happier not to know German.
Virgil
<b389bd81-a4b4-4421...@q9g2000hsb.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 5 Okt., 23:00, Virgil <Vir...@gmale.com> wrote:
>
> > > < You do agree that
> > > > the anti-diagonal is equal to 1?
> >
> > > I do agree that no actual infinity exists. Therefore there is no
> > > string 0.111... = 1!
> >
> > Then you deny that there is a ring of integers or a field of rationals
> > or a complete field of reals or any of the definitions and theorems of
> > analysis based upon the existence of such objects.
> >
> > You can't have it both ways.
>
> I can't change that. It is as it is. Nevertheless we can act "as if"
> in arithmetic.
You mean you do things knowing they are false? Bad mathematics!
One can do things without knowing whether they are false, but "knowing
they are false"?
Naughty, naughty!
> >
> >
> >
[snip]
> For every 1 of the anti-diagonal there must exist an entry with a
> string of 1's reaching to that position.
False, one can have a list of strings with exactly one 1 in each row and
still have lots of 1's in an anti-diagonal.
Virgil
<635f3bfe-17d2-48fd...@u28g2000hsc.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 5 Okt., 23:12, Virgil <Vir...@gmale.com> wrote:
> > In article
> > <b5242153-4044-42bb-80a9-c52dd2c71...@75g2000hso.googlegroups.com>,
> >
> >
> >
> >
> >
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 4 Okt., 19:31, Virgil <Vir...@gmale.com> wrote:
> >
> > > > Consider the list:
> > > > 0.0
> > > > 0.10
> > > > 0.110
> > > > ...
> > > > Its anti-diagonal is 0.111..., by the list is far from complete even for
> > > > finite sequences.
> >
> > > > > If the anti-diagonal, has aleph_0 digits, then there is at least one
> > > > > list entry with aleph_0 - 1 digits.
> >
> > > > Claimed but not justified,
> >
> > > Easy to see by the construction of the list, easy for anybody not
> > > deteriorate by set theory.
> >
> > Until one has a reasonable definition of what "aleph_0 - 1"
>
> Take a set of aleph_0 elements and remove one element. That is aleph_0
> - 1.
So is the new set bijectable with the original or not?
>
> > means, it is
> > nonsense. I know of no definition of "aleph_0 - 1" which is not
> > equivalent to "aleph_0" itself.
>
> Therefore an anti- diagonal with aleph_0 digits 1 requires at least
> one line with as many digits 1.
Non-sequitur.
Consider a list in which all but the (n+1)st character is 0.
Its anti-diagonal is all 1's.
>
>
> > One definition of "the cardinality of set S is aleph_0" is
> > "Set S has a bijection with the set of all naturals".
>
> Another one is: The set consists of aleph_0 countable sets.
That is not a definition, but is a theorem.
>
> > Perhaps WM will provide as clear a definition of what he means by
> > "the cardinality of set S is aleph_0 - 1"
>
> See above.
If the above is your definition, then what is the point of your claim?