Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions. On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ. (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).) Before all fractions have been enumerated there, at least one fraction must have been enumerated in every unit interval. For this sake ℵo natural numbers are required. More are not available.

Regards, WM

On Friday, September 3, 2021 at 3:23:03 PM UTC+2, WM wrote:

> Cantor [...] is mistaken.

Of course. See:

Crank Dot Net | Cantor was wrong
| http://www.crank.net/cantor.html

On Friday, 3 September 2021 at 16:23:03 UTC+3, WM wrote:
> Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions.

I think they are systematically enumerable, but not "ALL".

If Cantor wrote "ALL", then he was simply boasting about something as completely unremarkable as a radix system.

A set is countable iff its members can be systematically named. This is why |N is countable, not because |N has any magical properties, but that the young, wily Jew Cantor realised this fact before he went insane.

> On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ. (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).)

They can't be "unit intervals" because there is nothing longer than a unit between any two consecutive natural numbers.

> Before all fractions have been enumerated there, at least one fraction must have been enumerated in every unit interval. For this sake ℵo natural numbers are required. More are not available.
>
> Regards, WM

On Friday, 3 September 2021 at 10:23:03 UTC-3, WM wrote:
> Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions. On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ. (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).) *Before* [my emphasis] all fractions have been enumerated there, at least one fraction must have been enumerated in every unit interval.. For this sake ℵo natural numbers are required. More are not available.

The very fact that Muckenheim is using the notion of "before" here shows that he has no clue about infinity and marks him as a crank.

Gus Gassmann schrieb am Freitag, 3. September 2021 um 15:51:08 UTC+2:
> On Friday, 3 September 2021 at 10:23:03 UTC-3, WM wrote:
> > Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions. On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ.. (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).) *Before* [my emphasis] all fractions have been enumerated there, at least one fraction must have been enumerated in every unit interval. For this sake ℵo natural numbers are required. More are not available.
>
> The very fact that Muckenheim is using the notion of "before" here shows that he has no clue about infinity

Can you understand that 2 comes before 6 in the sequence of natural numbers as Cantor used them?
Can you understand that in
1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...
1/3 is indexed before 5/1?

Regards, WM

Eram semper recta schrieb am Freitag, 3. September 2021 um 15:39:00 UTC+2:
> On Friday, 3 September 2021 at 16:23:03 UTC+3, WM wrote:
> > Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions.
> I think they are systematically enumerable, but not "ALL".
>
> If Cantor wrote "ALL", then he was simply boasting about something as completely unremarkable as a radix system.

Without enumerating all, ℵo is nonsense.

> > On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ. (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).)
> They can't be "unit intervals" because there is nothing longer than a unit between any two consecutive natural numbers.

(n-1, n] is a called unit interval because n - (n-1) = 1.

Regards, WM

Galileo's mistake.

Galileo claimed that all natural numbers can be enumerated. But consider the perfect squares. Clearly there are aleph_0 of them and so enumerating them uses aleph_0 natural numbers. More are not available to enumerate the natural numbers which are not perfect squares.

--
William Hughes

On Friday, 3 September 2021 at 13:36:20 UTC-3, WM wrote:
> Gus Gassmann schrieb am Freitag, 3. September 2021 um 15:51:08 UTC+2:
> > On Friday, 3 September 2021 at 10:23:03 UTC-3, WM wrote:
> > > Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions. On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ. (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).) *Before* [my emphasis] all fractions have been enumerated there, at least one fraction must have been enumerated in every unit interval. For this sake ℵo natural numbers are required. More are not available.
> >
> > The very fact that Muckenheim is using the notion of "before" here shows that he has no clue about infinity
> Can you understand that 2 comes before 6 in the sequence of natural numbers as Cantor used them?
> Can you understand that in
> 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...
> 1/3 is indexed before 5/1?
As I said, you have no clue about the issues. I have some doubts that you even comprehend your own drivel any more. Your description of the diagonal method certainly has *nothing* to do with the statement in your previous post.

On 9/3/2021 8:22 AM, WM wrote:
> Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions. On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ. (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).) Before all fractions have been enumerated there, at least one fraction must have been enumerated in every unit interval. For this sake ℵo natural numbers are required. More are not available.
>
> Regards, WM
>

wrong.

Proof;

Assume there is a rational a/b in (n-1, n] n ∈ ℕ
Cantors Enumeration includes a/b
therefore Cantors Enumeration includes all rationals in (n-1, n] n ∈ ℕ

(“There are so many different kinds of stupidity, and cleverness is one of the worst.”
― Thomas Mann, )

[ABBA REUNION !!!]

On Friday, 3 September 2021 at 14:07:22 UTC-3, Serg io wrote:
> On 9/3/2021 8:22 AM, WM wrote:
> > Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions. On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ.. (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).) Before all fractions have been enumerated there, at least one fraction must have been enumerated in every unit interval. For this sake ℵo natural numbers are required. More are not available.
> >
> > Regards, WM
> >
> wrong.
>
> Proof;
>
> Assume there is a rational a/b in (n-1, n] n ∈ ℕ
> Cantors Enumeration includes a/b
> therefore Cantors Enumeration includes all rationals in (n-1, n] n ∈ ℕ
Even though this is of course correct, it does not address Muckenheim's blind spot. The truth is that every rational a/b in (n-1, n] has a position /k(a,b)/ in the sequence of enumerations and that there is an interval (M-1, M] whose *first* enumerated rational has a position that *exceeds* k(a,b). (In fact, there are infinitely many of these intervals.) That is the whole point of Cantor's enumeration. Muckenheim truly has no clue about infinity.

On Friday, September 3, 2021 at 7:06:16 PM UTC+2, Gus Gassmann wrote:
> On Friday, 3 September 2021 at 13:36:20 UTC-3, WM wrote:
>
> [bla bla]
>
> As I said, you have no clue about the issues. I have some doubts that you even comprehend your own drivel any more.

Agree.

On 9/3/2021 9:22 AM, WM wrote:

> Cantor claims that all positive fractions can be enumerated
> although he can prove this only for the first (less than ℵo)
> fractions.

Cantor (and others) begin with claims that are we already know
are true of each positive rational. Claims like these
| | x is a positive rational iff
| two FISONable naturals p,q exist such that q*x = p, q>0, p>0.
| | k is a FISONable natural iff
| a steppable set {0,...,k} of FISONable naturals exists
| in which, for all adjacent i,j, j = i+1.

Cantor (and others) extend what we already know by truth-
-preserving inference to further claims. Because we only use
truth-preserving inferences, for anything for which it is _true_
that p and q exist, q*x = p, it will also be _true_ (for example)
that a unique index k exists for x, k being a FISONable natural,
and that no other x' has the same index k.

In this sense, all positive rationals can be enumerated,
and this can be proven for all positive rationals.

> On the other hand it is easy to prove that he is mistaken:
> There is an actual infinity of ℵo positive unit intervals
> (n-1, n], n ∈ ℕ.
> (Without actual infinity transfinity is a long twaddle
> about nothing (Alexander Zenkin).)

Do you (WM) agree with Zenkin that potential infinity
is a long twaddle about nothing?

> Before all fractions have been enumerated there, at least one
> fraction must have been enumerated in every unit interval.

No.
Before all rationals have been enumerated, no more than
finitely-many rationals have been enumerated.
There are infinitely-many unit intervals, so your claim
cannot be true.

Your claim is the same as you made a little while ago.
We went around and around and, in order to continue asserting
your claim, you ended up, in effect, declaring arithmetic
and/or geometry inconsistent.

Consider the much simpler hypothesis that your claim here
is wrong. Before all rationals have been enumerated there,
at least one unit interval has zero rationals enumerated in it.

> For this sake ℵo natural numbers are required.
> More are not available.

I claim that all the integers Z = {...,-3,-2,-1,0,1,2,3,...}
can be enumerated by the naturals N = {0,1,2,3,...}.

Proof: We can re-order the integers this way
j 0, 1, -1, 2, -2, 3, -3, ...
k 0, 1, 2, 3, 4, 5, 6, ...

I can't list all the integers or all the naturals,
but, for each integer j, I can tell you index(j).
index(j) =
{ 2*j-1 for j >= 1
{ -2*j for j =< 0

For each integer j, a unique natural k exists, k = index(j).
For each natural k, a unique integer j exists, index(j) = k.
index(j) = index(j') iff j = j'.

Perhaps you (WM) will object to my claim that _all_ the integers
can be enumerated. But you might not. It's fairly easy to _imagine_
the number line being _folded_ at 1/3, giving us our new order.

I also claim that _all_ the positive rationals can be enumerated.
Any objection to this claim has a corresponding objection to
enumerating the integers.

There are many ways to enumerate the positive rationals.
My personal favorite folds _positive and negative_ powers of
each prime to correspond to _only positive_ powers of that prime.

2^j 1, 2, 1/2, 4, 1/4, 8, 1/8, ...
2^k 1, 2, 4, 8, 16, 32, 64, ...

3^j 1, 3, 1/3, 9, 1/9, 27, 1/27, ...
3^k 1, 3, 9, 27, 81, 243, 729, ...

....

The powers of primes be folded the same way as
the integers were folded onto the naturals.
j 0, 1, -1, 2, -2, 3, -3, ...
k 0, 1, 2, 3, 4, 5, 6, ...

Because each index and each numerator and denominator
have unique prime factorizations, each prime power can be
folded separately and then multiplied together.

index(2/7) = index(2)*index(1/7) = 2*49 = 98
index(11/3) = index(1/3)*index(11) = 9*11 = 99
index(1/10) = index(1/2)*index(1/5) = 4*25 = 100

We can also reverse the process and calculate, for a given
index, what rational has that index.

William schrieb am Freitag, 3. September 2021 um 18:53:53 UTC+2:
> Galileo's mistake.
>
> Galileo claimed that all natural numbers can be enumerated. But consider the perfect squares. Clearly there are aleph_0 of them and so enumerating them uses aleph_0 natural numbers. More are not available to enumerate the natural numbers which are not perfect squares.

In this case the enumeration happened rather clumsy. Cantor's enumeration could be improved too. But I discuss what Cantor did.

Regards, WM

Jim Burns schrieb am Freitag, 3. September 2021 um 21:27:39 UTC+2:
> On 9/3/2021 9:22 AM, WM wrote:

> > Before all fractions have been enumerated there, at least one
> > fraction must have been enumerated in every unit interval.
> No.

So all rational numbers in all intervals can be enumerated without having enumerated at least one rationale number in every interval?

> Before all rationals have been enumerated, no more than
> finitely-many rationals have been enumerated.
> There are infinitely-many unit intervals, so your claim
> cannot be true.

It is Cantor's claim that all rationals can be enumerated.
My claim is minimalistic: If all ℵo rationals of each of all the ℵo intervals are enumerated, at least one rational in each interval has to be enumerated.

Proof: If not every interval has at least one enumerated rational, then there exists at least one interval without any enumerated rational. This interval cannot have all its rationals enumerated.

> Your claim is the same as you made a little while ago.

No, it much better and sharper.

> Consider the much simpler hypothesis that your claim here
> is wrong. Before all rationals have been enumerated there,
> at least one unit interval has zero rationals enumerated in it.

No, not "before" but after all rationals have been enumerated, one interval has no enumerated rationals in it. That is what you claim.

> > For this sake ℵo natural numbers are required.
> > More are not available.
> I claim that all the integers Z = {...,-3,-2,-1,0,1,2,3,...}
> can be enumerated by the naturals N = {0,1,2,3,...}.

This error is harder to detect, but it is not an excuse for the blatant mistake that I have discovered.

Regards, WM

On Friday, September 3, 2021 at 9:23:03 AM UTC-4, WM (aka Wolfgang Muckenheim (WM))wrote:
> Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions.

Wrong again, Mucke. Here again, is visual proof that ALL positive fractions can be enumerated: https://en.wikipedia.org/wiki/Rational_number#/media/File:Diagonal_argument.svg

> On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ.

True. But this does not prove your claim.

> (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).)

Wrong again, Mucke. There is actually no need to invoke "transfinity" here.

> Before all fractions have been enumerated there, at least one fraction must have been enumerated in every unit interval.

Pure gibberish!

> For this sake ℵo natural numbers are required. More are not available.
>

More gibberish! When will you learn, Mucke?

More absurd quotes from Wolfgang Muckenheim (WM):

“In my system, two different numbers can have the same value.”
-- sci.math, 2014/10/16

“1+2 and 2+1 are different numbers.”
-- sci.math, 2014/10/20

“1/9 has no decimal representation.”
-- sci.math, 2015/09/22

"0.999... is not 1."
-- sci.logic 2015/11/25

“Axioms are rubbish!”
-- sci.math, 2014/11/19

“Formal definitions have lead to worthless crap like undefinable numbers.”
-- sci.math 2017/02/05

“No set is countable, not even |N.”
-- sci.logic, 2015/08/05

“Countable is an inconsistent notion.”
-- sci.math, 2015/12/05

Slipping ever more deeply into madness...

“There is no actually infinite set |N.”
-- sci.math, 2015/10/26

“|N is not covered by the set of natural numbers.”
-- sci.math, 2015/10/26

“The set of all rationals can be shown not to exist.”
--sci.math, 2015/11/28

“Everything is in the list of everything and therefore everything belongs to a not uncountable set.”
-- sci.math, 2015/11/30

"'Not equal' and 'equal can mean the same.”
-- sci.math, 2016/06/09

“The set of numbers will get empty after all have numbers been used..”
-- sci.math, 2016/08/24

“I need no set theory.”
-- sci.math, 2016/09/01

A special word of caution to students: Do not attempt to use WM's “system” (MuckeMath) in any course work in any high school, college or university on the planet. You will fail miserably. MuckeMath is certainly no shortcut to success in mathematics.

Using WM's “axioms” for the natural numbers, he cannot even prove that 1=/=2. His goofy little system is truly a dead-end.

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

STUDENTS BEWARE: Don't be a victim of JG's fake math

On Friday, September 3, 2021 at 9:39:00 AM UTC-4, I am Super Rectum (aka John Gabriel (JG), Troll Boy) wrote:
> On Friday, 3 September 2021 at 16:23:03 UTC+3, WM wrote:
> > Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions.
> I think they are systematically enumerable, but not "ALL".
>

Yes, ALL of them, Mr. Rectum.

JG here claims to have a discovered a shortcut to mastering calculus without using limits. Unfortunately for him, this means he has no workable a definition of the derivative of a function. It blows up for functions as simple f(x)=|x|. Or even f(x)=0. As a result, he has had to ban 0, negative numbers and instantaneous rates of change rendering his goofy little system quite useless. What a moron!

Forget calculus. JG has also banned all axioms because he cannot even derive the most elementary results of basic arithmetic, e.g. 2+2=4. Such results require the use of axioms, so he must figure he's now off the hook. Again, what a moron!

Even at his advanced age (60+?), John Gabriel is STILL struggling with basic, elementary-school arithmetic. As he has repeatedly posted here:

"There are no points on a line."
--April 12, 2021

"Pi is NOT a number of ANY kind!"
--July 10, 2020

"1/2 not equal to 2/4"
--October 22, 2017

“1/3 does NOT mean 1 divided by 3 and never has meant that”
-- February 8, 2015

"3 =< 4 is nonsense.”
--October 28, 2017

"Zero is not a number."
-- Dec. 2, 2019

"0 is not required at all in mathematics, just like negative numbers."
-- Jan. 4, 2017

“There is no such thing as an empty set.”
--Oct. 4, 2019

“3 <=> 2 + 1 or 3 <=> 8 - 5, etc, are all propositions” (actually all are meaningless gibberish)
--Oct. 22, 2019

No math genius our JG, though he actually lists his job title as “mathematician” at Linkedin.com. Apparently, they do not verify your credentials.

Though really quite disturbing, interested readers should see: “About the spamming troll John Gabriel in his own words...” (lasted updated March 10, 2020) at https://groups.google.com/forum/#!msg/sci.math/PcpAzX5pDeY/1PDiSlK_BwAJ

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog a http://www.dcproof.wordpress.com

On Friday, September 3, 2021 at 12:53:53 PM UTC-4, William wrote:
> Galileo's mistake.
>
> Galileo claimed that all natural numbers can be enumerated. But consider the perfect squares. Clearly there are aleph_0 of them and so enumerating them uses aleph_0 natural numbers. More are not available to enumerate the natural numbers which are not perfect squares.

There are many infinite subsets of N. The set of perfect squares. The set of even numbers. The set of numbers greater than 2. All have the same cardinality as N. Deal with it.

>
> --
> William Hughes

An impostor???

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

On Friday, 3 September 2021 at 19:39:47 UTC+3, WM wrote:
> Eram semper recta schrieb am Freitag, 3. September 2021 um 15:39:00 UTC+2:
> > On Friday, 3 September 2021 at 16:23:03 UTC+3, WM wrote:
> > > Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions.
> > I think they are systematically enumerable, but not "ALL".
> >
> > If Cantor wrote "ALL", then he was simply boasting about something as completely unremarkable as a radix system.
> Without enumerating all, ℵo is nonsense.

I know. My point is I don't think he meant "ALL", but rather systematic enumeration.

> > > On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ. (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).)
> > They can't be "unit intervals" because there is nothing longer than a unit between any two consecutive natural numbers.
> (n-1, n] is a called unit interval because n - (n-1) = 1.

Sorry, I think I misunderstood what you meant.

>
> Regards, WM

On 9/3/2021 11:53 AM, William wrote:
> Galileo's mistake.
>
> Galileo claimed that all natural numbers can be enumerated. But consider the perfect squares.

having trouble ? here you go;

1 1
2 4
3 9
4 16
.. .
.. .
.. .

EZ Cheesy

> Clearly there are aleph_0 of them and so enumerating them uses aleph_0 natural numbers.

what do you mean by "uses" ? Used Up? an infinite set ?

> More are not available to enumerate the natural numbers which are not perfect squares.

bullshit.

and you will say since all naturals are USED UP, then you cant number all the even numbers one to one,

double bullshit.

advice: try to be more clear about what you are stating by using MATH statements

Dan Christensen schrieb am Samstag, 4. September 2021 um 05:43:48 UTC+2:
> On Friday, September 3, 2021 at 9:23:03 AM UTC-4, WM (aka Wolfgang Muckenheim (WM))wrote:
> > Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions.
> Here again, is visual proof that ALL positive fractions can be enumerated: https://en.wikipedia.org/wiki/Rational_number#/media/File:Diagonal_argument.svg

Two statemenets contradicting each other. This is called a contradiction.

> > On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ.
> True. But this does not prove your claim.

This proves that if every interval is hit at least once, then all indices have been exhausted.

> > Before all fractions have been enumerated there, at least one fraction must have been enumerated in every unit interval.
> Pure gibberish!

Simplest logic. The contrary would be: All fractions have been enumerated, but in at least one interval no first one has been enumerated.

Regards, WM

Eram semper recta schrieb am Samstag, 4. September 2021 um 07:46:59 UTC+2:
> On Friday, 3 September 2021 at 19:39:47 UTC+3, WM wrote:
> > Eram semper recta schrieb am Freitag, 3. September 2021 um 15:39:00 UTC+2:
> > > On Friday, 3 September 2021 at 16:23:03 UTC+3, WM wrote:
> > > > Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions.
> > > I think they are systematically enumerable, but not "ALL".
> > >
> > > If Cantor wrote "ALL", then he was simply boasting about something as completely unremarkable as a radix system.
> > Without enumerating all, ℵo is nonsense.
> I know. My point is I don't think he meant "ALL", but rather systematic enumeration.

"Wenn zwei wohldefinierte Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für Element, einander zuordnen lassen" vollständig = complete means all.

Regards, WM

fredag 3 september 2021 kl. 15:23:03 UTC+2 skrev WM:
> Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions. On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ. (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).) Before all fractions have been enumerated there, at least one fraction must have been enumerated in every unit interval. For this sake ℵo natural numbers are required. More are not available.
>
> Regards, WM
The product of two countable set is countable. So your argument does not hold.

On Saturday, 4 September 2021 at 09:18:08 UTC-3, WM wrote:
> "Wenn zwei wohldefinierte Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für Element, einander zuordnen lassen" vollständig = complete means all.
What an arsehole you are! "eindeutig und vollständig, Element für Element" is simply Cantor's way of describing a bijection. (Did he have the term when he wrote that passage?) Every element in M has a single associated element in N, and vice versa. And, yes, "all" means that none are left behind, either in N or in M.
So, nothing at all to see here, folks.

markus...@gmail.com schrieb am Samstag, 4. September 2021 um 14:40:02 UTC+2:
> fredag 3 september 2021 kl. 15:23:03 UTC+2 skrev WM:
> > Cantor claims that all positive fractions can be enumerated although he can prove this only for the first (less than ℵo) fractions. On the other hand it is easy to prove that he is mistaken: There is an actual infinity of ℵo positive unit intervals (n-1, n], n ∈ ℕ.. (Without actual infinity transfinity is a long twaddle about nothing (Alexander Zenkin).) Before all fractions have been enumerated there, at least one fraction must have been enumerated in every unit interval. For this sake ℵo natural numbers are required. More are not available.

> The product of two countable set is countable. So your argument does not hold.

Here we have not two countable sets but only one set of indices mapped to the rationals. One could use many infinite subsets of ℕ but Cantor didn't. (And it would not help, because we could ask for the point when at least two fractions were indexed in every unit interval and so on.)

Regards, WM

Gus Gassmann schrieb am Samstag, 4. September 2021 um 14:45:01 UTC+2:
> On Saturday, 4 September 2021 at 09:18:08 UTC-3, WM wrote:
> > "Wenn zwei wohldefinierte Mannigfaltigkeiten M und N sich eindeutig und vollständig, Element für Element, einander zuordnen lassen" vollständig = complete means all.
> "eindeutig und vollständig, Element für Element" is simply Cantor's way of describing a bijection.

Of course.

> (Did he have the term when he wrote that passage?)

Of course not.

> Every element in M has a single associated element in N, and vice versa. And, yes, "all" means that none are left behind, either in N or in M.

That is what I said. It means "complete" or "all".

Regards, WM