Why?

[WM]

There are less prime numbers P than natural numbers N, less natural numbers N than fractions Q. That is easy to prove: P is a proper subset of N, N is a proper subset of Q. Or this way: In every interval [0, n] there are more natural numbers than prime numbers, more fractions than natural numbers.

Why have Galilei and Cantor claimed that there are as many square numbers as natural numbers? Why has Cantor claimed that a bijection exists between N and Q, although bijections have to include in fact every element and because of injectivity do contradict the proofs shown above?

The answer is simple: Both have not recognized that all usable and identifiable numbers in each set have the same "cardinality", namely there are potentially infinitely many, whereas the remaining sets of dark numbers have very different sizes. They have started to write down the first pairs of their mappings, but then they have used the "and so on". Formulas like f(x) = 2x however only feign bijections.

Why do nowadays so many stick to the clearly wrong idea that bijections between _all_ elements of the sets P, N, and Q were possible?

This question remains without an answer. But it has definitely nothing to do with mathematics.

Regards, WM

[zelos...@gmail.com]

Holy shit you are dumb as always.

>There are less prime numbers P than natural numbers N, less natural numbers N than fractions Q.

All false statements, provably false.

>That is easy to prove

Go ahead and try.

>P is a proper subset of N, N is a proper subset of Q

X being a subset of Y does not in any way shape or form demonstrate that the cardinalities are the same when it comes to infinite sets.

A complete non-sequitor, especially given that the definition of an infintie set is IT CAN BE PUT INTO A BIJECTION WITH A PROPER SUBSET!

>Or this way: In every interval [0, n] there are more natural numbers than prime numbers, more fractions than natural numbers.

Any finite set having less primes than natural numbers do not in any way imply that the set of all primes has smaller cardinality than natural numbers.

Thsi si again another fuckign non-sequitor you dishonest scumbag.

>Why have Galilei and Cantor claimed that there are as many square numbers as natural numbers?

Because there is a bijection between them, ergo they have the same cardinality.

>Why has Cantor claimed that a bijection exists between N and Q

Because he, and others after, can demonstrate such a bijection exists.

>although bijections have to include in fact every element and because of injectivity do contradict the proofs shown above?

Your "proofs" are not proofs, they are fallacious

>The answer is simple

Yes, they are provably so and you are as always wrong and don't understand mathematics.

>Both have not recognized that all usable and identifiable numbers in each set have the same "cardinality", namely there are potentially infinitely many

Horse shit, there is no such thing as "potential infinite" in mathematics.

>whereas the remaining sets of dark numbers have very different sizes

More horseshit because you cannot understand basic mathematics.

>They have started to write down the first pairs of their mappings, but then they have used the "and so on".

And thats fine because the pattern is obvious.

>Formulas like f(x) = 2x however only feign bijections.

No, they are bijections on certain sets.

>Why do nowadays so many stick to the clearly wrong idea that bijections between _all_ elements of the sets P, N, and Q were possible?

Because it is clearly possible, demonstrably done so and you are wrong.

>This question remains without an answer. But it has definitely nothing to do with mathematics.

It has an answer, the big question is why you're too stupid to understand this

[Sergio]

On 1/8/2021 5:27 AM, WM wrote:
> There are less prime numbers P than natural numbers N, less natural numbers N than fractions Q. That is easy to prove: P is a proper subset of N, N is a proper subset of Q.


Or this way: In every interval [0, n] there are more natural numbers
than prime numbers, more fractions than natural numbers.

that is a statement, not a proof.

>
> Why have Galilei and Cantor claimed that there are as many square numbers as natural numbers?

google Galileo's Paradox

Why has Cantor claimed that a bijection exists between N and Q, although
bijections have to include in fact every element and because of
injectivity do contradict the proofs shown above?

there are no proofs above.

how are you defining "injectivity" ?

>
> The answer is simple: Both have not recognized that all usable and identifiable numbers in each set have the same "cardinality",

how are you defining "usable" and "identifiable" ?

> namely there are potentially infinitely many, whereas the remaining sets of dark numbers have very different sizes. They have started to write down the first pairs of their mappings, but then they have used the "and so on". Formulas like f(x) = 2x however only feign bijections.

whoa! now back to counting rocks...

>
> Why do nowadays so many stick to the clearly wrong idea that bijections between _all_ elements of the sets P, N, and Q were possible?

how do you do a 1:1 with primes ?

>
> This question remains without an answer. But it has definitely nothing to do with mathematics.

if so, please post in alt.math.fiction

>
> Regards, WM
>

[WM]

zelos...@gmail.com schrieb am Freitag, 8. Januar 2021 um 13:35:38 UTC+1:


> >There are less prime numbers P than natural numbers N, less natural numbers N than fractions Q.
> All false statements, provably false.
> >That is easy to prove
> Go ahead and try.
> >P is a proper subset of N, N is a proper subset of Q
> X being a subset of Y does not in any way shape or form demonstrate that the cardinalities are the same when it comes to infinite sets.

It proves that when all elements of X are subtracted, then further elements of y remain.
>
> A complete non-sequitor,

A simple fact in classical mathematics with its powerful tool of limit. There are precisely twice as many natural numbers than even numbers.

Lim_{n--> oo} N ∩ [0, n] / E ∩ [0, n] = 2

That is a result of classical mathematics. Do you deny it?

> especially given that the definition of an infintie set is IT CAN BE PUT INTO A BIJECTION WITH A PROPER SUBSET!

Bijection shows _precisely_ same number of elements. If there exists a bijection between X and Y, then every injective mapping is surjective and vice versa.

> >Or this way: In every interval [0, n] there are more natural numbers than prime numbers, more fractions than natural numbers.
> Any finite set having less primes than natural numbers do not in any way imply that the set of all primes has smaller cardinality than natural numbers.

Why are you so insane? That is just the question.

>
> Horse shit, there is no such thing as "potential infinite" in mathematics.

If aleph_0 elements exist, then the aleph_0 elements of the set of reorderings exist too. Then the bijection between N and Q can be reordered such that Q appears as well-ordered by size. At least then such a well-ordering exist. Does it exist?

If not, why then do all aleph_0 natural numbers exist, but all bijections between N and Q do not exist?

Regards, WM

[WM]

Sergio schrieb am Freitag, 8. Januar 2021 um 13:57:28 UTC+1:
> On 1/8/2021 5:27 AM, WM wrote:
> > Or this way: In every interval [0, n] there are more natural numbers
> > than prime numbers, more fractions than natural numbers.

> that is a statement, not a proof.

Can you read mathematical formulas? Then try it:

Lim_{n--> oo} N ∩ [0, n] / E ∩ [0, n] = 2

> how are you defining "injectivity" ?

See any math book of your choice, e.g. my "Mathematik für die ersten Semester", 4th ed., De Gruyter, Berlin (2015).

Regards, WM

[FromTheRafters]

After serious thinking WM wrote :

> There are less prime numbers P than natural numbers N, less natural numbers N
> than fractions Q. That is easy to prove: P is a proper subset of N, N is a
> proper subset of Q.

You probably meant 'fewer' but it would still be wrong. These smallest
of inductive sets all have the same cardinality.

> Or this way: In every interval [0, n] there are more
> natural numbers than prime numbers, more fractions than natural numbers.

Intervals in Z don't have "fractional" numbers, only whole numbers. If
you insist on Q then the above regarding the smallest inductive set
holds.

Cardinality of finite sets which are from intervals with a finite
number of points, are equal to the number of members. If your superset
is the smallest inductive set, and your subset is inductive too, they
have equal cardinality.

> Why have Galilei and Cantor claimed that there are as many square numbers as
> natural numbers?

Because there are.

> Why has Cantor claimed that a bijection exists between N and Q,

Because it does, there is.

> although bijections have to include in fact every element and because of
> injectivity do contradict the proofs shown above?

You started with false assumptions, and the above is not proof only
your unconvincing mistaken intuitionism.

> The answer is simple: Both have not recognized that all usable and
> identifiable numbers in each set have the same "cardinality",

"Usable and identifiable numbers" is not defined. The numbers in an
inductive set are defined by the rules, so what are the rules for your
dark set which you intend to union with Q or N for your mixed dark set?

> namely there are potentially infinitely many, whereas the remaining sets of
> dark numbers have very different sizes. They have started to write down the
> first pairs of their mappings, but then they have used the "and so on".
> Formulas like f(x) = 2x however only feign bijections.

You apparently don't get the idea behind bijections, or counting.

> Why do nowadays so many stick to the clearly wrong idea that bijections
> between _all_ elements of the sets P, N, and Q were possible?

Because it is even more clearly correct.

> This question remains without an answer. But it has definitely nothing to do
> with mathematics.

Wrong, you just don't *like* the answer.

[WM]

FromTheRafters schrieb am Freitag, 8. Januar 2021 um 14:52:06 UTC+1:
> After serious thinking WM wrote :
> > There are less prime numbers P than natural numbers N, less natural numbers N
> > than fractions Q. That is easy to prove: P is a proper subset of N, N is a
> > proper subset of Q.
> You probably meant 'fewer'

No I meant less.

> > Or this way: In every interval [0, n] there are more
> > natural numbers than prime numbers, more fractions than natural numbers.
> Intervals in Z don't have "fractional" numbers,

But intervals on the real axis do.

> Cardinality of finite sets which are from intervals with a finite
> number of points, are equal to the number of members. If your superset
> is the smallest inductive set, and your subset is inductive too, they
> have equal cardinality.

But they have not equal number of elements. For a bijection however equal numbers of elements are required.

> You started with false assumptions, and the above is not proof only
> your unconvincing mistaken intuitionism.

In mathematics we can find for natural N and even E numbers:

Lim_{n--> oo} N ∩ [0, n] / E ∩ [0, n] = 2

This is a proof of not equal number of elements.

> You apparently don't get the idea behind bijections, or counting.

I got this idea: A bijection between sets requires and proves precisely the same number of elements.

This implies: If a bijection really exists, then _every_ injective mapping is surjective and vice versa.

> > Why do nowadays so many stick to the clearly wrong idea that bijections
> > between _all_ elements of the sets P, N, and Q were possible?
> Because it is even more clearly correct.

If aleph_0 elements exist, then the aleph_0 elements of the set of all reorderings of Q exist too. Then the bijection between N and Q can be reordered such that Q appears as well-ordered by size. At least then such a well-ordering exist. Does it exist? Is it accessible? Or is it dark?

Can you answer this question? Can you even think about it?

Regards, WM

[Gus Gassmann]

On Friday, 8 January 2021 at 09:51:01 UTC-4, WM wrote:
[...]

> Can you read mathematical formulas? Then try it:
> Lim_{n--> oo} N ∩ [0, n] / E ∩ [0, n] = 2

Is this a joke? Incompetence? Senility? Intentional deceit?

I will be charitable and assume that you are trying to angle for your tired cardinality argument, which, as usual, proves nothing. However, you have left out the cardinality markers, so everyone is going to have to guess at what you might have meant by your ill-formed symbol chain.

[Gus Gassmann]

On Friday, 8 January 2021 at 10:04:39 UTC-4, WM wrote:
> FromTheRafters schrieb am Freitag, 8. Januar 2021 um 14:52:06 UTC+1:
> > After serious thinking WM wrote :
> > > There are less prime numbers P than natural numbers N, less natural numbers N
> > > than fractions Q. That is easy to prove: P is a proper subset of N, N is a
> > > proper subset of Q.
> > You probably meant 'fewer'
> No I meant less.

Clearly, English is not your strong suit. It's not your native tongue, so that's excusable, but contradicting someone who is trying to correct your English makes you look, well, stupid. (But your stupidity has by now been well established, so you should not worry to much about that, either.)

[FromTheRafters]

on 1/8/2021, WM supposed :

> FromTheRafters schrieb am Freitag, 8. Januar 2021 um 14:52:06 UTC+1:
>> After serious thinking WM wrote :
>>> There are less prime numbers P than natural numbers N, less natural numbers
>>> N than fractions Q. That is easy to prove: P is a proper subset of N, N is
>>> a proper subset of Q.
>> You probably meant 'fewer'
>
> No I meant less.

That figures.

>>> Or this way: In every interval [0, n] there are more
>>> natural numbers than prime numbers, more fractions than natural numbers.
>> Intervals in Z don't have "fractional" numbers,
>
> But intervals on the real axis do.

The real axis has uncountably many members in every proper interval.

>> Cardinality of finite sets which are from intervals with a finite
>> number of points, are equal to the number of members. If your superset
>> is the smallest inductive set, and your subset is inductive too, they
>> have equal cardinality.
>
> But they have not equal number of elements.

They have just as many elements, because a bijection exists.

> For a bijection however equal numbers of elements are required.

And so it is, they do, in finite sets. For infinite sets we have the
countable and the uncountable (so far) and we distinguish between them
by the fact that they are either able to be placed in a bijection or
they are not.

>> You started with false assumptions, and the above is not proof only
>> your unconvincing mistaken intuitionism.
>
> In mathematics we can find for natural N and even E numbers:
>
> Lim_{n--> oo} N ∩ [0, n] / E ∩ [0, n] = 2
>
> This is a proof of not equal number of elements.

How so? Just because there is a two to one ratio at each step does not
imply that the infinite cardinalities share that same relation.

>> You apparently don't get the idea behind bijections, or counting.
>
> I got this idea: A bijection between sets requires and proves precisely the
> same number of elements.

Works for finite sets.

> This implies: If a bijection really exists, then _every_ injective mapping is
> surjective and vice versa.

Okay, that is bijection, so?

>>> Why do nowadays so many stick to the clearly wrong idea that bijections
>>> between _all_ elements of the sets P, N, and Q were possible?
>> Because it is even more clearly correct.
>
> If aleph_0 elements exist, then the aleph_0 elements of the set of all
> reorderings of Q exist too. Then the bijection between N and Q can be
> reordered such that Q appears as well-ordered by size. At least then such a
> well-ordering exist. Does it exist? Is it accessible? Or is it dark?
>
> Can you answer this question? Can you even think about it?

Ordering has nothing to do with size (cardinality) of sets other than
perhaps a way to show it is an inductive set - as Cantor did. Then
showing the bijection with the smallest such set, in the form of the
natural numbers.

Can you understand this answer?

[Wess Bay]

WM wrote:

> There are less prime numbers P than natural numbers N, less natural
> numbers N than fractions Q. That is easy to prove: P is a proper subset
> of N, N is a proper subset of Q. Or this way: In every interval [0, n]
> there are more natural numbers than prime numbers, more fractions than
> natural numbers.

Like the police state beating the corona out of kids and families own
home?? bill gates and the Dr. Fucksee of geormony Drosten, like Dr.
mengele, without the Dr's. They are not even doctors in anything, giving
fraud PhD diplomas to one another.

Coronavirus Scandal Breaking in Merkel’s Germany Over Drosten PCR Test
http://tapnewswire.com/2020/12/coronavirus-scandal-breaking-in-merkels-
germany-over-drosten-pcr-test/

They accuse Drosten and cohorts of “fatal” scientific incompetence and
flaws in promoting their test.

To begin with, as the critical scientists reveal, the paper that
established the Drosten PCR test for the Wuhan strain of coronavirus that
has subsequently been adopted with indecent haste by the Merkel government
along with WHO for worldwide use–resulting in severe lockdowns globally
and an economic and social catastrophe–was never peer-reviewed before its
publication by Eurosurveillance journal. The critics point out that,

on January 21 there were a world total of 6 deaths being attributed to
the Wuhan virus. They ask, “Why did the authors assume a challenge for
public health laboratories while there was no substantial evidence at that
time to indicate that the outbreak was more widespread than initially
thought?” Another co-author of the Drosten paper that gave a cover of
apparent scientific credibility to the Drosten PCR procedure was head of
the company who developed the test being marketed today, with the blessing
of WHO, in the hundreds of millions, Olfert Landt (see image),
of Tib-Molbiol in Berlin, but Landt did not disclose that pertinent fact
in the Drosten paper either.

Certainly nothing suspicious or improper here, or? It would be relevant to
know if Drosten, the Merkel chief scientific advisor for COVID-19,
Germany’s de facto “Tony Fauci,” gets a percentage for each test sold by
Tib-Molbiol in their global marketing agreement with Roche.

In simple English, the entire edifice of the Gates foundation, the Merkel
government, the WHO and WEF as well as the case for de facto forced
untested vaccines, rests on results of a PCR test for coronavirus that is
not worth a hill of beans. The test of Drosten and WHO is, more or less,
scientific crap.
Missing Doctor proof too?

This devastating critique from twenty three world leading scientists,
including scientists who have patents related to PCR, DNA Isolation and
Sequencing, and a former Pfizer Chief Scientist, is damning, but not the
only problem Professor Dr. Christian Drosten faces today. He and the
officials at Frankfurt’s Goethe University, where he claims to have
received his medical doctorate in 2003, are being accused of degree fraud.

According to Dr. Markus Kühbacher, a specialist investigating scientific
fraud such as dissertation plagiarism, Dr. Drosten’s doctor thesis, by law
must be deposited on a certain date with academic authorities at his
University, who then sign a legal form, Revisionsschein, verified with
signature, stamp of the University and date, with thesis title and author,
to be sent to the University archive. With it, three original copies of
the thesis are filed.

Kühbacher charges that the Goethe University is guilty of cover-up by
claiming, falsely, Drosten’s Revisionsschein, was on file. The University
spokesman later was forced to admit it was not filed, at least not
locatable by them. Moreover, of the three mandatory file copies of his
doctor thesis, highly relevant given the global importance of Drosten’s
coronavirus role, two copies have “disappeared,” and the remaining single
copy is water-damaged. Kühbacher says Drosten will now likely face court
charges for holding a fraudulent doctoral title.

[Jim Burns]

On 1/8/2021 6:27 AM, WM wrote:

> There are less prime numbers P than natural numbers N,

Well, it's a theory.
How does your theory explain each natural number being
associated with one and only one prime number?

You have referred to the pigeonhole principle in your own
arguments in the past, so I hope you won't object to
others using it, too. By the pigeonhole principle,
there cannot be more natural numbers than prime numbers.

( Suppose the first k natural numbers are associated
( with the first k prime numbers,
( 1 ~~ 2, 2 ~~ 3, 3 ~~ 5, ..., k ~~ pr(k)
(
( Either 1+pr(k)! is prime or there is a prime < 1+pr(k)!
( which divides 1+pr(k)!
( No prime =< p(k) divides 1+pr(k)! Therefore,
( there is at least one prime q', p(k) < q' < 1+pr(k)!
(
( The set of primes q, p(k) < q =< 1+pr(k)! is not empty,
( so it contains a first element, the first prime after p(k).
( k+1 is associated with the first prime after p(k).
( Call it p(k+1). k+1 ~~ p(k+1)
(
( There are no natural numbers which are NOT associated in this
( fashion with one and only one prime.
(
( Assume otherwise, that there is some natural number k without
( one and only one p(k).
( Then, somewhere in the finite sequence 1,2,...,k,
( there is some j+1 such that, for each i < j+1, there is one and
( only one p(i), but there is no p(j+1) for j+1.
( But that contradicts our definition of p(j+1) as
( the first prime q, p(j) < q =< 1+p(j)!
( Therefore, there is no natural number k without one and only
( one p(k).

> less natural numbers N than fractions Q.

p/q is at k = (p+q-1)*(p+q-2)/2 + p

Blah blah blah pigeonhole principle.
NOT less natural numbers N than fractions Q.

> That is easy to prove: P is a proper subset of N,
> N is a proper subset of Q.

You skipped over the "proof" part of your proof.
Surely just an oversight.

Take the set { 10,20,30,... }
Remove the '0' at the right of every numeral, getting
{ 1,2,3,... }

{ 10,20,30,... } is a proper subset of { 1,2,3,... }

You would have it be that removing '0' from the right
of every numeral gives us more numerals.

You know, there are actual reasons we don't do things your way.

[Dan Christensen]

On Friday, January 8, 2021 at 6:27:49 AM UTC-5, WM wrote:
> There are less prime numbers P than natural numbers N, less natural numbers N than fractions Q. That is easy to prove: P is a proper subset of N, N is a proper subset of Q. Or this way: In every interval [0, n] there are more natural numbers than prime numbers, more fractions than natural numbers.
>

The set of even numbers is a proper subset of the set of natural numbers. Do you claim that there are more natural numbers than there are even numbers? Why or why not?

Dan

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

[WM]

Jim Burns schrieb am Samstag, 9. Januar 2021 um 00:30:03 UTC+1:
> On 1/8/2021 6:27 AM, WM wrote:
>
> > There are less prime numbers P than natural numbers N,
> Well, it's a theory.
> How does your theory explain each natural number being
> associated with one and only one prime number?

That is a delusion. There are no bijections between infinite sets because most elements are dark, i.e., not identifiable, i.e., not appearing in a mapping. If there were a true bijection and not only a Cantor delusion between two infinite sets, then this would prove precisely the same number of elements in both sets. Then every injection would be a surjection and vice versa.

> ( Assume otherwise, that there is some natural number k without
> ( one and only one p(k).

I do not assume but prove that.

If you remove all unit fractions which can be identified and therefore by theorem have infinitely many successors from (0, 1], then, be just this theorem, infinitely many will remain.
If however you remove all unit fractions, then none will remains. This proves the existence of infinitely many unit fractions which cnanot be identified and therefore cannot be in a bijections.

Regards, WM

[WM]

FromTheRafters schrieb am Freitag, 8. Januar 2021 um 15:54:08 UTC+1:
> on 1/8/2021, WM supposed :

> > I got this idea: A bijection between sets requires and proves precisely the
> > same number of elements.
> Works for finite sets.

It works for _every_ bijection. If it does not work for infinite stes, then there is no bijection between infinite sets.

> > This implies: If a bijection really exists, then _every_ injective mapping is
> > surjective and vice versa.
> Okay, that is bijection, so?

If there were any real bijection between infinite sets, then every permutation would be a bijection too.
Matheologians believe that infinite sets N and Q exist and that bijections between them exist. They do not believe however that the infinite set of all bijections between N and Q exists and that infinitely many permutations can be applied in every bijection. That is highly inconsistent. But otherwise the set Q could be well-ordered by size.

> > If aleph_0 elements exist, then the aleph_0 elements of the set of all
> > reorderings of Q exist too. Then the bijection between N and Q can be
> > reordered such that Q appears as well-ordered by size. At least then such a
> > well-ordering exist. Does it exist? Is it accessible? Or is it dark?
> >
> > Can you answer this question? Can you even think about it?
> Ordering has nothing to do with size

Therefore every ordering of Q must be possible without changing the bijection. This includes the well-ordering by size of the rationals.

Regards, WM

[Mostowski Collapse]

Bad start into the New Year. Even the first sentence shows
ignorance. Because of bijection n |-> pn, we have P| = |N|.

Or do you want to talk about asymptotic density?
If yes, what are the open questions that the Augsburg

Crank Institut doesn't understand?

WM schrieb am Freitag, 8. Januar 2021 um 12:27:49 UTC+1:
> There are less prime numbers P than natural numbers N

[WM]

Mostowski Collapse schrieb am Samstag, 9. Januar 2021 um 14:27:03 UTC+1:
> Because of bijection n |-> pn, we have P| = |N|.

There is no bijection. Are you also "unable" to understand the difference between identified and unidentified numbers? I am longing to know whether Mr. Christensen will find out the difference when doing his homework and answering the questions:

Is 17 larger than 10?
Is n larger than 10?

Can you answer both, one, or none of the questions?

> Or do you want to talk about asymptotic density?

The asymptotic density is the true measure in mathematics and mathematics is the true science of measuring. Since the measure of set theory deviates significantly it has to be discarded.

> If yes, what are the open questions that the Augsburg

How can an intelligent person accept two different measures? But again that is not mathematics but only a problem of psychology of religious sects.

Regards, WM

[Gus Gassmann]

On Sunday, 10 January 2021 at 07:47:34 UTC-4, WM wrote:
> Mostowski Collapse schrieb am Samstag, 9. Januar 2021 um 14:27:03 UTC+1:
> > Because of bijection n |-> pn, we have P| = |N|.
> There is no bijection. Are you also "unable" to understand the difference between identified and unidentified numbers? I am longing to know whether Mr. Christensen will find out the difference when doing his homework and answering the questions:
>
> Is 17 larger than 10?
> Is n larger than 10?
>
> Can you answer both, one, or none of the questions?

Yes, of course.

[Mostowski Collapse]

The bijection in set theory is between identifiable and unindentifiable
numbers. So lets say this is the WM horizon m, and n < m is identified,
and m =< n is not identified.

Then the bijection maps both:

/ n-the prime number if n < m
P(n) = <
\ n-the prime number if m =< n

Because you can form the sentence "n-the prime number" even for
not yet identified numbers "n". Thats probably the main error in
your theory WM. You don't understand how

variables work in mathematics and set theory.

[Sergio]

On 1/10/2021 5:47 AM, WM wrote:
> Mostowski Collapse schrieb am Samstag, 9. Januar 2021 um 14:27:03 UTC+1:
>> Because of bijection n |-> pn, we have P| = |N|.
>
> There is no bijection. Are you also "unable" to understand the difference between identified and unidentified numbers? I am longing to know whether Mr. Christensen will find out the difference when doing his homework and answering the questions:
>
> Is 17 larger than 10?

which base ? base 10 sure, base -7 no

> Is n larger than 10?

-oo < n < +oo so the answer is "allowed"

>
> Can you answer both, one, or none of the questions?
>
>> Or do you want to talk about asymptotic density?
>
> The asymptotic density is the true measure in mathematics and mathematics is the true science of measuring. Since the measure of set theory deviates significantly it has to be discarded.

density? the number of numbers per some interval ? do you have a
definition ?

>
>> If yes, what are the open questions that the Augsburg
>
> How can an intelligent person accept two different measures?

one can deal with lots of different measures, and convert between them

your knowledge of Mathematics seems limited.

I see Math as a collection of many different fields of math, some
overlap a lot, some overlap a little, some none at all, and some fields
of math are on the very edge of what is possible, like
encryption/decryption.

Seems to me your stuck in "Set Theory" vs "infinite series" the two
overlap some, the infinite case is hard to understand for many

[WM]

Sergio schrieb am Sonntag, 10. Januar 2021 um 17:56:15 UTC+1:


> > How can an intelligent person accept two different measures?
> one can deal with lots of different measures, and convert between them

One cannot accept that there are less integers than fractions. One cannot accept that there are as many.
There is nothing convertible. The measure "cardinality" is imprecise and wrong and therefore useless.

>
> your knowledge of Mathematics seems limited.

This judgement from a person, who did not even know what a well-order is, is an honour.

Regards, WM

[Jim Burns]

On 1/10/2021 6:47 AM, WM wrote:
> Mostowski Collapse schrieb
> am Samstag, 9. Januar 2021 um 14:27:03 UTC+1:

>> Because of bijection n |-> pn, we have P| = |N|.
>
> There is no bijection.

With each natural, there is associated a unique prime,
with different naturals associated with different primes.
With each prime, there is associated a unique natural,
with different primes associated with different naturals.

If that's what you mean by "bijection", you're provably
wrong.
If you don't mean that, you're not even wrong.

> Are you also "unable" to understand the difference
> between identified and unidentified numbers?

One difference they _don't_ have
(unless you are -- again -- not even wrong)
is that a natural is followed by more naturals and
a natural is finitely connected to 0.

> I am longing to know whether Mr. Christensen will find out
> the difference when doing his homework and answering
> the questions:
> Is 17 larger than 10?
> Is n larger than 10?

n is an indefinite reference to one of the naturals.

There are three flavors that the claim "n > 10" could have
(i)
"n > 10" could be TRUE _no matter which_ natural n refers to.
(ii)
"n > 10" could be FALSE _no matter which_ natural n refers to.
(iii)
It could _matter which_ natural n refers to, true for some,
false for others.

With claims certain to be type (i), we can apply truth-
-preserving inferences to produce new claims that we can be
just as certain are also type (i) -- that is, claims just as
certain to be true _no matter which_ natural n refers to.

These are facts about claims of type (i). They are not facts
about claims of type (iii), such as "n > 10".

Our type (i) arguments are analogous to claiming that
we can build _and fly in_ airplanes.
Your type (iii) argument, continuing the analogy, is
a claim that you've built an airplane out of butter, and
IT DOESN'T FLY.

So, yes, your butter airplane does not fly.
That says nothing about our non-butter airplanes.

> Can you answer both, one, or none of the questions?

You appear to have "overlooked" airplanes not made of butter.

>> Or do you want to talk about asymptotic density?
>
> The asymptotic density is the true measure in mathematics
> and mathematics is the true science of measuring. Since
> the measure of set theory deviates significantly it has
> to be discarded.

The asymptotic density of the primes in the naturals

pi(n)/n ~~ 1/log(n) --> 0

as n --> infinity.

Are you (WM) claiming that there are no primes?

>> If yes, what are the open questions that the Augsburg
>
> How can an intelligent person accept two different measures?

The next time you feel like you have a fever, check your
temperature by stepping on a bathroom scale. In order to
show us what an intelligent person you are.

> But again that is not mathematics but only a problem of
> psychology of religious sects.

I believe
all natural numbers are natural numbers.

There. I said it. Will I be struck by lightning?

[WM]

Mostowski Collapse schrieb am Sonntag, 10. Januar 2021 um 16:44:35 UTC+1:
> The bijection in set theory is between identifiable and unindentifiable
> numbers.

No. What cannot be identified cannot be indexed. Note that numbers are not material objects. Numbers exist only if someone can think them as individuals.

> Because you can form the sentence "n-the prime number" even for
> not yet identified numbers "n".

It denotes the typical element of the set, not an individualö that could be indexed.

> variables work in mathematics and set theory.

They hold the places where individual numbers can be inserted.

Note that a useful measure could never yield the same number of elements for |N and |Q. The same cardinality only shows what an imprecise measure the cardinality is. Every sober mind knows that there are more fractions than integers (although the sect of matheologians has corrupted thousands of students so much that they possible may be unable to think this fact). Cantors cardinality is the same for all infinite sets (also for the finitely definable reals) BECAUSE only the first few elements are definable as individuals and can be mapped as individuals on the first definable natnumbers. The remainders remain dark and not mapped.

It is one of the most perverse developments in human history that instead of recognizing and rejecting cardinality as an imprecise and useless measure a sect of believers have accepted it as the touchstone for their insane religion.

Regards, WM

[WM]

Jim Burns schrieb am Sonntag, 10. Januar 2021 um 20:08:16 UTC+1:
> On 1/10/2021 6:47 AM, WM wrote:
> > Mostowski Collapse schrieb
> > am Samstag, 9. Januar 2021 um 14:27:03 UTC+1:
>
> >> Because of bijection n |-> pn, we have P| = |N|.
> >
> > There is no bijection.
> With each natural, there is associated a unique prime,
> with different naturals associated with different primes.
> With each prime, there is associated a unique natural,
> with different primes associated with different naturals.
>
> If that's what you mean by "bijection", you're provably
> wrong.

No you are. Every such natural number is the last element of a FISON (1, 2, 3, ..., n), but all FISONs can be subtracted from |N without reducing its cardinality aleph_0.

> If you don't mean that, you're not even wrong.
> > Are you also "unable" to understand the difference
> > between identified and unidentified numbers?
> One difference they _don't_ have
> (unless you are -- again -- not even wrong)
> is that a natural is followed by more naturals and
> a natural is finitely connected to 0.

Wrong again. Why do you claim such obvious falsities? All FISONs can be subtracted from |N without reducing its cardinality aleph_0.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

Try to find a FISON that cannot be subtracted. Fail.

Regards, WM

[Jim Burns]

On 1/10/2021 2:00 PM, WM wrote:
> Sergio schrieb
> am Sonntag, 10. Januar 2021 um 17:56:15 UTC+1:

>> your knowledge of Mathematics seems limited.
>
> This judgement from a person, who did not even know
> what a well-order is, is an honour.

Do you (WM) still claim that
| for all j, exists k, k > j
implies that
| exists k, for all j, k >= j
?

[FredJeffries]

On Sunday, January 10, 2021 at 3:47:34 AM UTC-8, WM wrote:

> How can an intelligent person accept two different measures?

https://www.bipm.org/

[FredJeffries]

On Sunday, January 10, 2021 at 11:00:50 AM UTC-8, WM wrote:

> The measure "cardinality" is imprecise and wrong and therefore useless.

Indeed, here we see the real power of our Professor's MathRealism (or whatever brand name he is hawking this week).

Only an hour previously in the sci.logic thread 'Why?' our Professor proclaimed:

<quote>
With increasing n the function

|N ∩ [0, n]| / |E ∩ [0, n]|

comes closer and closer to 2, in that its deviations become smaller and smaller: it has as its limit precisely 2
</quote>

We therefore see that by dividing one 'imprecise and wrong and therefore useless' object by another 'imprecise and wrong and therefore useless' object and 'increase n', we can arrive at a 'precise' value. Indeed, a value which 'is not only mathematics but the result that every sensible thinker will obtain'.

Powerful magic!

[FredJeffries]

On Sunday, January 10, 2021 at 3:47:34 AM UTC-8, WM wrote:

> The asymptotic density is the true measure in mathematics and mathematics is the true science of measuring. Since the measure of set theory deviates significantly it has to be discarded.

> How can an intelligent person accept two different measures? But again that is not mathematics but only a problem of psychology of religious sects.

If the only thing a person sees is nails, the only tool he thinks he will ever need is a hammer.

[mitchr...@gmail.com]

On Friday, January 8, 2021 at 3:27:49 AM UTC-8, WM wrote:
> There are less prime numbers P than natural numbers N, less natural numbers N than fractions Q. That is easy to prove: P is a proper subset of N, N is a proper subset of Q. Or this way: In every interval [0, n] there are more natural numbers than prime numbers, more fractions than natural numbers.
>

> Why have Galilei and Cantor claimed that there are as many square numbers as natural numbers? Why has Cantor claimed that a bijection exists between N and Q, although bijections have to include in fact every element and because of injectivity do contradict the proofs shown above?
>
> The answer is simple: Both have not recognized that all usable and identifiable numbers in each set have the same "cardinality", namely there are potentially infinitely many, whereas the remaining sets of dark numbers have very different sizes. They have started to write down the first pairs of their mappings, but then they have used the "and so on". Formulas like f(x) = 2x however only feign bijections.

>
> Why do nowadays so many stick to the clearly wrong idea that bijections between _all_ elements of the sets P, N, and Q were possible?
>

> This question remains without an answer. But it has definitely nothing to do with mathematics.
>

> Regards, WM

The infinitesimal was first ratio...
as fundamental 1 divided by the unlimited.
Magnitude is built by fundamental...

Mitchell Raemsch

[Sergio]

On 1/10/2021 1:00 PM, WM wrote:
> Sergio schrieb am Sonntag, 10. Januar 2021 um 17:56:15 UTC+1:
>
>
>>> How can an intelligent person accept two different measures?
>> one can deal with lots of different measures, and convert between them
>
> One cannot accept that there are less integers than fractions.

the fractions are enumerated with the positive integers

> One cannot accept that there are as many.

both are infinite, it is usally folly to try to compare two infinite sets

> There is nothing convertible.

with infinite sets, one has to watch their math carefully.

> The measure "cardinality" is imprecise and wrong and therefore useless.

with infinities...

>>
>> your knowledge of Mathematics seems limited.
>
> This judgement from a person, who did not even know what a well-order is, is an honour.

step into my math realm of calculating the probabilities of undetected
error in in digital communications for different codes, new codes etc.

It is a type of Math that requires set theory,
combinations+permutations, abstract math, probability theory, and a few
others. only taught in only a few Universities, and not fully.


>
> Regards, WM
>

[Python]

Don't worry Sergio. And you definitely do not have to justify yourself
in front of a criminal student abuser like Wolfgang Mueckenheim.

You may have not known the meaning of "well order" and take it as
usual order relation. Big deal!

You've NEVER confused on purpose sequences and order relation in order
to support a fallacy, crank Wolfgang Mueckenheim did.

You've NEVER confused on purpose inclusion and membership in order to
build a sophistry, crank Wolfgang Mueckenheim did.

You've NEVER cut a quote in order to make it mean the opposite of
what the author meant. crank Wolfgang Mueckenheim did.

You've NEVER used a position of authority on students in order to
make them repeat as parrots lies, sophistries and fallacies, crank
Wolfgang Mueckenheim did.

[FromTheRafters]

on 1/10/2021, Python supposed :

I'm reasonably sure WM doesn't know anything about EFM and CIRC, there
is a list somewhere of stupid, wrong, and 'not even wrong' things he
has said here.

[Ben Bacarisse]

Jim Burns <james....@att.net> writes:

> On 1/10/2021 6:47 AM, WM wrote:
>> Mostowski Collapse schrieb
>> am Samstag, 9. Januar 2021 um 14:27:03 UTC+1:
>
>>> Because of bijection n |-> pn, we have P| = |N|.
>>
>> There is no bijection.
>
> With each natural, there is associated a unique prime,
> with different naturals associated with different primes.
> With each prime, there is associated a unique natural,
> with different primes associated with different naturals.
>
> If that's what you mean by "bijection", you're provably
> wrong.
> If you don't mean that, you're not even wrong.

That is indeed what he means by "bijection". From his textbook:

Eine Abbildung f von X nach Y heißt

surjektiv (oder Abbildung auf Y), wenn jedes y ∈ Y ein Bild ist,
injektiv (oder eindeutig), wenn aus f(x1) = f(x2) folgt x1 = x2,
bijektiv (oder eineindeutig), wenn f injektiv und surjektiv ist.

(My translation:

A mapping f from X to Y is called

surjective (or onto Y), if every y ∈ Y is an image under f,
injective (or one-to-one), if f(x1) = f(x2) implies that x1 = x2,
bijective (a one-to-one correspondence) if f is injective and surjective.)

He gives examples like f: R -> R with f(x) = x^3.

Basically (as I am sure you know), Usenet is a parour game for WM. He
will happily deny facts today that he stated only yesterday.

--
Ben.

[zelos...@gmail.com]

>It proves that when all elements of X are subtracted, then further elements of y remain.

It shows that there are elements not present in the otehr set, it doesn't show they have different cardinalities.

>A simple fact in classical mathematics with its powerful tool of limit. There are precisely twice as many natural numbers than even numbers

This demonstrates again your inability to understand that operations are not commutative. That does not prove tehre are twice as many, it only shows that in any FINITE set there are twice as many.

>Bijection shows _precisely_ same number of elements. If there exists a bijection between X and Y, then every injective mapping is surjective and vice versa.

Incorrect, the existence of A bijection does not mean every injective function is surjective, it only means THAT ONE is.

The property of every injective function being surjective is a property for FINITE sets. INFINITE sets don't have this.

>Why are you so insane? That is just the question

The insane one here is you whom do not understand basic anything.

>If aleph_0 elements exist, then the aleph_0 elements of the set of reorderings exist too. Then the bijection between N and Q can be reordered such that Q appears as well-ordered by size. At least then such a well-ordering exist. Does it exist?

By axiom of choice every set can be well-ordered, it doesn't however say how the well-ordering looks like.

The bijection between |N and Qs well ordering are independent of each other.

[Dan Christensen]

Still no reply, Mucke?

[Sergio]

For the readers;

basic problem in discussing infinities...

assume some function/operation/choosing# that results in infinity and
call it FOP()

so infinity = oo = k/0 = FOP() where k is some konstant

therefore oo * 0 = k = FOP()

so it does not matter what FOP() is at all.

therefore one can use many different FOP()s which arrive at the same
answer, the risk is drawing wrong conclusions about various FOP()s.


like WM did above, it seems like common sense in areas, but not when
taken to infinity.


and it is no big deal, there are methods in Math to avoid the infinity
traps, google line integrals of complex functions

[Stone B Fusco]

Sergio wrote:

> For the readers; basic problem in discussing infinities...
> assume some function/operation/choosing# that results in infinity and
> call it FOP() so infinity = oo = k/0 = FOP() where k is some konstant
> therefore oo * 0 = k = FOP()

If true communism fails in your country, is solely because you guys are
stupid. And nothen else. In true communism you own everything without
*DEBT* to the bankers and the government your entire life. Debt is
slavery, learn this meine gutte manner.

[Sergio]

Nein, mein Herr,

in true communism
you own nothing it all belongs to the state
you are told how many potatoes to grow per acre
you cannot disagree with the central party ever
Watch 1984, it is all there, pupa.

[Stone B Fusco]

Sergio wrote:

>> If true communism fails in your country, is solely because you guys are
>> stupid. And nothen else. In true communism you own everything without
>> *DEBT* to the bankers and the government your entire life. Debt is
>> slavery, learn this meine gutte manner.
>>
>>
> Nein, mein Herr, in true communism
> you own nothing it all belongs to the state you are told how many
> potatoes to grow per acre you cannot disagree with the central party
> ever Watch 1984, it is all there, pupa.

Is there an echo inhere, I just said if fails since you are stupid.
That's not communism, meine guten herren. Everything belongs to people,
not government. Humans cannot even survive *off-grid*, no matter how big
a prepper you might think you are. Even Jesus went off-grid in 40 days,
then he came back socialising among the people. You are getting
everything wrong.

[Stone B Fusco]

Dan Christensen wrote:

> Still no reply, Mucke?

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

Translation: the initial vaccinations will mostly be saline water until
word circulates there were few reactions. Then when massive amounts of
capitalist idiots line up, we give them the toxic soup, they shouldn't
reproduce anyway if they're that stupid.

[WM]

Sergio schrieb am Sonntag, 10. Januar 2021 um 21:33:31 UTC+1:
> On 1/10/2021 1:00 PM, WM wrote:
> > Sergio schrieb am Sonntag, 10. Januar 2021 um 17:56:15 UTC+1:
> >
> >
> >>> How can an intelligent person accept two different measures?
> >> one can deal with lots of different measures, and convert between them
> >
> > One cannot accept that there are less integers than fractions.
> the fractions are enumerated with the positive integers

All those used indexes can be subtracted from |N without reducing the cardinality of |N. This can be proven by induction and disproves your claim. Can you understand the proof?

> with infinite sets, one has to watch their math carefully.

It is simple to obtain a firm result concerning two infinite sets in mathematics. There are precisesly half ac many even numbers than integers. If a theory cannot reproduce this result, then it is in contradiction with mathematics and therefore useless for all mathematical tasks.

> step into my math realm of calculating the probabilities of undetected
> error in in digital communications for different codes, new codes etc.
>
> It is a type of Math that requires set theory,
> combinations+permutations, abstract math, probability theory, and a few
> others. only taught in only a few Universities, and not fully.

You should have learnt at least that it requires only finite set theory. Never a computer can handle actually infinite sets. "And G. Cantor's 'diagonal' argument for proving higher order infinity can not be simulated by finite memory computers as R. Penrose claims. Real computers only generate a finite number of rational numbers or repeating integers" [Bob Massey in "Cantor's transfinite numbers", sci.math (3 Nov 1996)]

Regards, WM

[Python]

Crank Wolfgang Mueckenheim, aka WM wrote:
...

> You should have learnt at least that it requires only finite set theory. Never a computer can handle actually infinite sets. "And G. Cantor's 'diagonal' argument for proving higher order infinity can not be simulated by finite memory computers as R. Penrose claims. Real computers only generate a finite number of rational numbers or repeating integers" [Bob Massey in "Cantor's transfinite numbers", sci.math (3 Nov 1996)]

They can. They also can formally proof Cantor's diagonal argument.

http://muaddibspace.blogspot.com/2009/10/cantors-diagonalization-proof-in-coq.html

You can quote a fallacy, it's not proving anything. Just like quoting
yourself, crank Wolfgang Mueckenheim, from Hochschule Augsburg.

[WM]

Python schrieb am Montag, 11. Januar 2021 um 19:21:51 UTC+1:
> WM wrote:
> ...
> > You should have learnt at least that it requires only finite set theory. Never a computer can handle actually infinite sets. "And G. Cantor's 'diagonal' argument for proving higher order infinity can not be simulated by finite memory computers as R. Penrose claims. Real computers only generate a finite number of rational numbers or repeating integers" [Bob Massey in "Cantor's transfinite numbers", sci.math (3 Nov 1996)]
> They can. They also can formally proof Cantor's diagonal argument.

They can also formally prove that undistinguishable elements can be well-ordered and that there are 17 even prime nunbers, if having been instructed by matheologians or other fools. No computer can treat all elements of infinite sets. And no computer or program needs transfinite set theory.

When a serious mathematician programs a computer to prove that

Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2

then a honest computer would claim that Cantor has been contradicted.

Regards, WM

[WM]

I claim that infinitely many natnumbers exist which are larger than all definable natnumbers. It is a simple fact and does not bother whether some fools of matheology talk about quantifiers.

If you can get rid of your pavlovian reflexes then try to empty the interval (0, 1/n) of unit fractions by identifying all of them. Fail.

Regards, WM

[WM]

FredJeffries schrieb am Sonntag, 10. Januar 2021 um 20:54:58 UTC+1:

> <quote>
> With increasing n the function
> |N ∩ [0, n]| / |E ∩ [0, n]|
> comes closer and closer to 2, in that its deviations become smaller and smaller: it has as its limit precisely 2
> </quote>
>
> We therefore see that by dividing one 'imprecise and wrong and therefore useless' object by another 'imprecise and wrong and therefore useless' object and 'increase n', we can arrive at a 'precise' value.

No. Every division concerns only finitely many elements.

Regards, WM

[WM]

zelos...@gmail.com schrieb am Montag, 11. Januar 2021 um 07:19:07 UTC+1:
> >It proves that when all elements of X are subtracted, then further elements of y remain.
>
> It shows that there are elements not present in the otehr set, it doesn't show they have different cardinalities.

That means that cardinality is a useless measure.

>
> >A simple fact in classical mathematics with its powerful tool of limit. There are precisely twice as many natural numbers than even numbers
>

> That does not prove tehre are twice as many, it only shows that in any FINITE set there are twice as many.

Wrong. Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2 proves the result for the infinite sets. That shows that cardinality is contradicted by mathematics.

Regards, WM

[Gus Gassmann]

On Monday, 11 January 2021 at 15:20:44 UTC-4, WM wrote:
[...]

> Wrong. Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2 proves the result for the infinite sets. That shows that cardinality is contradicted by mathematics.

Bullshit, as usual. You have no clue about infinity, and you display your ignorance daily.

Take Galileo's problem, and let S = set of square numbers.

Compute Lim_{n--> oo} |N ∩ [0, n]| / |S ∩ [0, n]| = 0. What do you think this means? That S is empty?

[zelos...@gmail.com]

>That means that cardinality is a useless measure.

Nope, it is very useful, there are other measures that are used to.

>Wrong. Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2 proves the result for the infinite sets.

Nope, because none of those sets are infinite.'

>That shows that cardinality is contradicted by mathematics.

Nope, this shows however you are an idiot that doesn't understand mathematics

For your statement to be true we would need ||N|/|2|N|=2, but division here is not defined for non-finite cardinalities so you cannot say it one way or another and we can easily show that the cardinality is equal between them with a simple bijection.

[WM]

zelos...@gmail.com schrieb am Dienstag, 12. Januar 2021 um 07:50:59 UTC+1:
> >That means that cardinality is a useless measure.
> Nope, it is very useful, there are other measures that are used to.

The cardinality shows that there are as many points in a quark as in the universe. What yould that be good for?

> >Wrong. Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2 proves the result for the infinite sets.
> Nope, because none of those sets are infinite.'

That's why we take the limit. In mathematics this concerns the infinite.

> For your statement to be true we would need ||N|/|2|N|=2, but division here is not defined for non-finite cardinalities so you cannot say it one way or another and we can easily show that the cardinality is equal between them with a simple bijection.

That means in "the infinite set" which is remaining when all finite sets have been subtracted, the even numbers are more dense than the natural numbers. Are the numbers populating this remaining set finite?

Regards, WM

[FromTheRafters]

WM presented the following explanation :

> zelos...@gmail.com schrieb am Dienstag, 12. Januar 2021 um 07:50:59 UTC+1:
>>> That means that cardinality is a useless measure.
>> Nope, it is very useful, there are other measures that are used to.
>
> The cardinality shows that there are as many points in a quark as in the
> universe. What yould that be good for?

That was proably pints in a quart and you got confused.

[zelos...@gmail.com]

>The cardinality shows that there are as many points in a quark as in the universe. What yould that be good for?

Nope, quarks are physical, mathematics is abstract.

There are as many points in (0,1)as there as there is in (0,oo), however.

It is one of its quirks, cardinality has its uses, it is not useful in ALL instances.

>That's why we take the limit. In mathematics this concerns the infinite.

Taking the limit does not imply what something is "AT" infinity so no, you cannot take a limit and say anything about infinity.
The fact that the limit for arbitrary large FINITE sets is somethign doesn't that mean it is RELEVANT or much less TRUE for the INFINITE case.

>That means in "the infinite set" which is remaining when all finite sets have been subtracted, the even numbers are more dense than the natural numbers. Are the numbers populating this remaining set finite?

Fuck sake are you this stupid?

[WM]

Can you understand that Cantor enumerates as many fractions in (0, 1) as in (1, oo)?
Have you ever seen the bijection by inversion for all real numbers: 1/r <--> r.
Can you choose 1 as quark radius?

Regards, WM

[WM]

zelos...@gmail.com schrieb am Mittwoch, 13. Januar 2021 um 07:28:47 UTC+1:
> >The cardinality shows that there are as many points in a quark as in the universe. What yould that be good for?
> Nope, quarks are physical, mathematics is abstract.

But they cover part of the space.

>
> There are as many points in (0,1)as there as there is in (0,oo), however.
>
> It is one of its quirks, cardinality has its uses, it is not useful in ALL instances.

Better so: in all instances it is not useful.

> >That's why we take the limit. In mathematics this concerns the infinite.
> Taking the limit does not imply what something is "AT" infinity

Of course it is. Do you think the limit lies before infinity?

> so no, you cannot take a limit and say anything about infinity.

The limit of the series Sum 1/2^n is 1. It is not reached before infinity but precoisely at infinity.

> The fact that the limit for arbitrary large FINITE sets is somethign doesn't that mean it is RELEVANT or much less TRUE for the INFINITE case.

Chuckle. Of course there are precisely twice as many integers than even integers. Not the least deviation of this mathematical theorem is possible. Set theory is in contradiction with this simple result.

Regards, WM

[Ernest Huffstatler]

WM wrote:

> zelos...@gmail.com schrieb am Mittwoch, 13. Januar 2021 um 07:28:47
> UTC+1:
>> >The cardinality shows that there are as many points in a quark as in
>> >the universe. What yould that be good for?
>> Nope, quarks are physical, mathematics is abstract.
>
> But they cover part of the space.

yet never detected. You guys talk in parables.

[Gus Gassmann]

On Wednesday, 13 January 2021 at 13:52:12 UTC-4, WM wrote:
> zelos...@gmail.com schrieb am Mittwoch, 13. Januar 2021 um 07:28:47 UTC+1:
> > >The cardinality shows that there are as many points in a quark as in the universe. What yould that be good for?
> > Nope, quarks are physical, mathematics is abstract.
> But they cover part of the space.
> >
> > There are as many points in (0,1)as there as there is in (0,oo), however.
> >
> > It is one of its quirks, cardinality has its uses, it is not useful in ALL instances.
> Better so: in all instances it is not useful.
> > >That's why we take the limit. In mathematics this concerns the infinite.
> > Taking the limit does not imply what something is "AT" infinity
> Of course it is. Do you think the limit lies before infinity?

Ah, no, your exaltedness. Reading comprehension was never your strong suit. Behaviour "near" infinity does not imply behaviour "AT" infinity, just as the behaviour of the signum function sgn near zero tells you nothing about the value of sgn(0).

Go back to your drawing board^H^H^H^H^H^H^H^H^H^H^H^H^H sandbox and try again.

> > so no, you cannot take a limit and say anything about infinity.
> The limit of the series Sum 1/2^n is 1. It is not reached before infinity but precoisely at infinity.

The limit of the sequence {sgn(1/n) + (1/n): n in N} is 1. It is not reached for any n, either. This tells you nothing about sgn(0), however.

> > The fact that the limit for arbitrary large FINITE sets is somethign doesn't that mean it is RELEVANT or much less TRUE for the INFINITE case.
> Chuckle. Of course there are precisely twice as many integers than even integers.

Of course NOT, you imbecile. card is not a continuous function. The cardinalities of N and E = 2*N, the set of even natural numbers, are equal.

Note, perhaps also, that card(2*N) =/= 2*card(N)

[FromTheRafters]

WM wrote :

> FromTheRafters schrieb am Dienstag, 12. Januar 2021 um 23:01:44 UTC+1:
>> WM presented the following explanation :
>>> zelos...@gmail.com schrieb am Dienstag, 12. Januar 2021 um 07:50:59 UTC+1:
>>>>> That means that cardinality is a useless measure.
>>>> Nope, it is very useful, there are other measures that are used to.
>>>
>>> The cardinality shows that there are as many points in a quark as in the
>>> universe. What yould that be good for?
>> That was proably pints in a quart and you got confused.
>
> Can you understand that Cantor enumerates as many fractions in (0, 1) as in
> (1, oo)?

Of course. Q is countable.

> Have you ever seen the bijection by inversion for all real numbers:
> 1/r <--> r. Can you choose 1 as quark radius?

R is uncountable.

I'll let your quarkiness slide.

[zelos...@gmail.com]

>But they cover part of the space.

Again, they are not mathematics so who cares?

>Better so: in all instances it is not useful.

Nope, it has uses where it is useful.

>Of course it is. Do you think the limit lies before infinity?

No it isn't and no I don't. You fucking imbecile.

>The limit of the series Sum 1/2^n is 1. It is not reached before infinity but precoisely at infinity.

Some cases it works, but others it do not.

You cannot take it as a universal rule.

>Chuckle. Of course there are precisely twice as many integers than even integers. Not the least deviation of this mathematical theorem is possible. Set theory is in contradiction with this simple result.

Except there isn't. That is an INTUITIVE feeling and there are ways to formalize it. But it ISN'T true in mathematics.

[WM]

Gus Gassmann schrieb am Mittwoch, 13. Januar 2021 um 20:14:12 UTC+1:
> Behaviour "near" infinity does not imply behaviour "AT" infinity, just as the behaviour of the signum function sgn near zero tells you nothing about the value of sgn(0).

That is nonsense, but it is the only way to avoid surrender. Of course the claim that McDuck gets infinitely rich and at infinity is bankrupt or that the set of not-indexed fractions grows by aleph_0 in every unit interval from n to n+1 but in the limit is empty is the summit of foolishness. Only a severely damaged brain can believe this and try to defend it. Note that McDuck can only lose one dollar per day. If all days see him rich, then he will stay rich in the limit.

>
> > Chuckle. Of course there are precisely twice as many integers than even integers.

> Of course NOT. card is not a continuous function.

That's the reason why it is nonsense.

> The cardinalities of N and E = 2*N, the set of even natural numbers, are equal.

The mathematical result contradict this. If for every n there are about twice as many integers, then there is no chance to egalize.

Regards, WM

[WM]

FromTheRafters schrieb am Mittwoch, 13. Januar 2021 um 21:04:31 UTC+1:
> WM wrote :
> > FromTheRafters schrieb am Dienstag, 12. Januar 2021 um 23:01:44 UTC+1:
> >> WM presented the following explanation :
> >>> zelos...@gmail.com schrieb am Dienstag, 12. Januar 2021 um 07:50:59 UTC+1:
> >>>>> That means that cardinality is a useless measure.
> >>>> Nope, it is very useful, there are other measures that are used to.
> >>>
> >>> The cardinality shows that there are as many points in a quark as in the
> >>> universe. What yould that be good for?
> >> That was proably pints in a quart and you got confused.
> >
> > Can you understand that Cantor enumerates as many fractions in (0, 1) as in
> > (1, oo)?
> Of course.

There are twice as many fractions in (2, 4) as in (2, 3).

> Q is countable.

You have been taught that, but it is wrong. Understand why Fraenkel has compared it with Tristram Shanys diary.
Since I am just supervising the exam finishing my lectures, I have the time to explain it to you in detail:

Buy a box that can store fractions.
Take all fractions of (0, 1] into your box, enumerate one of them by 1.
Add all fractions of (1, 2], into your box, enumerate another one of them by 2.
Add all fractions of (2, 3], into your box, enumerate another one of them by 3.
...
Add all fractions of (n, n+1], into your box, enumerate another one of them by n+1.
...

Continue in infinity until you have added all fractions of all unit intervals. At no finite number n the contents of your box will decrease. Az none! How should your box get empty "in the limit". This a such a foolish claim that it sound really incredible for every person confronted with it who is not introduced into set theory without considering this.

Of course set theorists will claim that everything happens simultaneously. But that was true, then it could be analyzed in the way that I showed above.

Regards, WM

[Gus Gassmann]

On Thursday, 14 January 2021 at 10:06:47 UTC-4, WM wrote:
> Gus Gassmann schrieb am Mittwoch, 13. Januar 2021 um 20:14:12 UTC+1:

[...] No need to belabor your delusions any further.

> > The cardinalities of N and E = 2*N, the set of even natural numbers, are equal.
> The mathematical result contradict this. If for every n there are about twice as many integers, then there is no chance to egalize.

I actually wrote 2*N on purpose. Don't you think that card(2*N) = 2*card(N)? If not, why not?

[WM]

zelos...@gmail.com schrieb am Donnerstag, 14. Januar 2021 um 07:44:06 UTC+1:

> >The limit of the series Sum 1/2^n is 1. It is not reached before infinity but precoisely at infinity.
> Some cases it works, but others it do not.
>
> You cannot take it as a universal rule.

Mathematics is universal.

> That is an INTUITIVE feeling and there are ways to formalize it. But it ISN'T true in mathematics.

lim_{n->oo} |N ∩ [0, n]| / |E∩ [0, n]| = 2 is no intuition.

But we can also analyse it like a detective:

[FromTheRafters]

WM expressed precisely :

> FromTheRafters schrieb am Mittwoch, 13. Januar 2021 um 21:04:31 UTC+1:
>> WM wrote :
>>> FromTheRafters schrieb am Dienstag, 12. Januar 2021 um 23:01:44 UTC+1:
>>>> WM presented the following explanation :
>>>>> zelos...@gmail.com schrieb am Dienstag, 12. Januar 2021 um 07:50:59
>>>>> UTC+1:
>>>>>>> That means that cardinality is a useless measure.
>>>>>> Nope, it is very useful, there are other measures that are used to.
>>>>>
>>>>> The cardinality shows that there are as many points in a quark as in the
>>>>> universe. What yould that be good for?
>>>> That was proably pints in a quart and you got confused.
>>>
>>> Can you understand that Cantor enumerates as many fractions in (0, 1) as in
>>> (1, oo)?
>> Of course.
>
> There are twice as many fractions in (2, 4) as in (2, 3).

No, there are just as many set elements in terms of cardinality because
the rational numbers are dense. The first interval is almost an octave
and the second is almost a perfect fifth. There are countably many
other intervals besides these.

>> Q is countable.
>
> You have been taught that, but it is wrong.

You are teaching that it isn't true, but that is wrong.

> Understand why Fraenkel has
> compared it with Tristram Shanys diary. Since I am just supervising the exam
> finishing my lectures, I have the time to explain it to you in detail:
>
> Buy a box that can store fractions.
> Take all fractions of (0, 1] into your box, enumerate one of them by 1.
> Add all fractions of (1, 2], into your box, enumerate another one of them by
> 2. Add all fractions of (2, 3], into your box, enumerate another one of them
> by 3. ...
> Add all fractions of (n, n+1], into your box, enumerate another one of them
> by n+1. ...
>
> Continue in infinity until you have added all fractions of all unit
> intervals.

Define "all" as used here by you. Keep in mind that all you do above is
guarantee a one to be labeled 1 for the first and two for the second
etcetera. Always a next ordinal number.

> At no finite number n the contents of your box will decrease.

Sure, all you have been doing is adding positive rational numbers. Why
should they ever decrease?

(0,1]
(0,2] Concatenated (added) (0,1](1,2]
(0,3] Concatenated (added) (0,1](1,2](2,3]
(0,4] Concatenated (added) (0,1](1,2](2,3](3,4]
...

Growing an infinite list of finite elements, just like the natural
numbers. Who knew?

> Az none! How should your box get empty "in the limit".

Maybe if you didn't keep adding positive rational numbers to it, it
would diminish.

> This a such a foolish claim

Yet, you keep on making it.

> that it sound really incredible for every person confronted with it who
> is not introduced into set theory without considering this.
>
> Of course set theorists will claim that everything happens simultaneously.
> But that was true, then it could be analyzed in the way that I showed above.

I think you need to try another way, without addition. Consider
replacing the set's contents each time with the elements of the next
(shorter?) interval. As the shortness of the successive intervals
approaches zero, there are always infinitely many rational numbers in
there, that does not change. just like every octave has a perfect fifth
(and a fourth and a third etc.) for instance no matter the endpoints.

[Sergio]

On 1/14/2021 8:24 AM, WM wrote:
> FromTheRafters schrieb am Mittwoch, 13. Januar 2021 um 21:04:31 UTC+1:
>> WM wrote :
>>> FromTheRafters schrieb am Dienstag, 12. Januar 2021 um 23:01:44 UTC+1:
>>>> WM presented the following explanation :
>>>>> zelos...@gmail.com schrieb am Dienstag, 12. Januar 2021 um 07:50:59 UTC+1:
>>>>>>> That means that cardinality is a useless measure.
>>>>>> Nope, it is very useful, there are other measures that are used to.
>>>>>
>>>>> The cardinality shows that there are as many points in a quark as in the
>>>>> universe. What yould that be good for?
>>>> That was proably pints in a quart and you got confused.
>>>
>>> Can you understand that Cantor enumerates as many fractions in (0, 1) as in
>>> (1, oo)?
>> Of course.
>
> There are twice as many fractions in (2, 4) as in (2, 3).
>

obvious mistake on your part, trying to compare infinities.

2 * oo = oo

1 + oo = oo

>> Q is countable.
>
> You have been taught that, but it is wrong. Understand why Fraenkel has compared it with Tristram Shanys diary.
> Since I am just supervising the exam finishing my lectures, I have the time to explain it to you in detail:
>
> Buy a box that can store fractions.
> Take all fractions of (0, 1] into your box, enumerate one of them by 1.
> Add all fractions of (1, 2], into your box, enumerate another one of them by 2.
> Add all fractions of (2, 3], into your box, enumerate another one of them by 3.
> ...
> Add all fractions of (n, n+1], into your box, enumerate another one of them by n+1.
> ...
>
> Continue in infinity

that statement, "continue to infinity" nukes your math.

>until you have added all fractions of all unit intervals. At no finite number n the contents of your box will decrease. Az none! How should your box get >empty "in the limit". This a such a foolish claim that it sound really incredible for every person confronted with it who is not introduced into set theory >without considering this.
>
> Of course set theorists will claim that everything happens simultaneously. But that was true, then it could be analyzed in the way that I showed above.
>
> Regards, WM
>

Classic mistake in comparing infinities.

This is Math 101 stuff.

You should know better.

WM is Miss Leader, a troll.

[WM]

Of course it is. Why? Because only a pot. inf. subset is available for any bijection between infinite sets. All infinite sets have the same cardinality. Even the set of definable real numbers. And others are delusions.

"The possible combinations of finitely many letters form a countable set, and since every determined real number must be definable by a finite number of words, there can exist only countably many real numbers – in contradiction to Cantor's classical theorem and its proof." [H. Weyl: "Das Kontinuum", Veit, Leipzig (1918) p. 18]

"If we pursue the thought that each real number is defined by an arithmetical law, the idea of the totality of real numbers is no longer indispensable, and the axiom of choice is not at all evident." [P. Bernays: "On Platonism in mathematics" (1934) p. 7]

"If we define the real numbers in a strictly formal system, where only finite derivations and fixed symbols are permitted, then these real numbers can certainly be enumerated because the formulas and derivations on the basis of their constructive definition are countable." [K. Schütte: "Beweistheorie", Springer (1960)]

Regards, WM

[Gus Gassmann]

On Thursday, 14 January 2021 at 11:51:32 UTC-4, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 15:27:42 UTC+1:
> > On Thursday, 14 January 2021 at 10:06:47 UTC-4, WM wrote:
> > > Gus Gassmann schrieb am Mittwoch, 13. Januar 2021 um 20:14:12 UTC+1:
> > > > The cardinalities of N and E = 2*N, the set of even natural numbers, are equal.
> > > The mathematical result contradict this. If for every n there are about twice as many integers, then there is no chance to egalize.
> > I actually wrote 2*N on purpose. Don't you think that card(2*N) = 2*card(N)? If not, why not?
> Of course it is.

Bingo! Just go back to see what you agreed to: card(E) = card(2*N) = 2*card(N). So there are *twice* as many even naturals as there are naturals. And yet you also believe that there are half as many even naturals as there are naturals. Now, which is it? A half or twice? Or maybe 0.5 = 2???

Your stupidity is breathtaking.

[Sergio]

On 1/14/2021 8:06 AM, WM wrote:
> Gus Gassmann schrieb am Mittwoch, 13. Januar 2021 um 20:14:12 UTC+1:
>> Behaviour "near" infinity does not imply behaviour "AT" infinity, just as the behaviour of the signum function sgn near zero tells you nothing about the value of sgn(0).
>
> That is nonsense, but it is the only way to avoid surrender. Of course the claim that McDuck gets infinitely rich and at infinity is bankrupt or that the set of not-indexed fractions grows by aleph_0 in every unit interval from n to n+1 but in the limit is empty is the summit of foolishness. Only a severely damaged brain can believe this and try to defend it. Note that McDuck can only lose one dollar per day. If all days see him rich, then he will stay rich in the limit.

the emergance of the very rare McDuck Ant!

This underscores the seriousness + immediate coverup of/by WMs
misleading and false statements that "Behaviour "near" infinity DOES
imply behaviour "AT" infinity"

>>
>>> Chuckle. Of course there are precisely twice as many integers than even integers.
>> Of course NOT. card is not a continuous function.
>
> That's the reason why it is nonsense.
>
>> The cardinalities of N and E = 2*N, the set of even natural numbers, are equal.
>
> The mathematical result contradict this. If for every n there are about twice as many integers, then there is no chance to egalize.

nope. you are caught in infinite corner.

>
> Regards, WM
>

[Ernest Huffstatler]

WM wrote:

>> That is an INTUITIVE feeling and there are ways to formalize it. But it
>> ISN'T true in mathematics.
>
> lim_{n->oo} |N ∩ [0, n]| / |E∩ [0, n]| = 2 is no intuition.
> But we can also analyse it like a detective:
> Take all fractions of (0, 1] into your box, enumerate one of them by 1.
> Add all fractions of (1, 2], into your box, enumerate another one of
> them by 2.

Give the stupid a bit of authority, and he will tax the shit out of you.
The capitalism is conditioning morons, for taxing the air they inhale.
Wake the feck up. That's why permanent mouth suffocators, "new-normal"
and everything.

Runner Stopped By Covid Marshal For "Breathing Too Heavily"
https://www.brighteon.com/dc702151-db75-4782-b2da-4593e0f37a92

[WM]

Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 17:58:39 UTC+1:
> On Thursday, 14 January 2021 at 11:51:32 UTC-4, WM wrote:
> > Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 15:27:42 UTC+1:
> > > On Thursday, 14 January 2021 at 10:06:47 UTC-4, WM wrote:
> > > > Gus Gassmann schrieb am Mittwoch, 13. Januar 2021 um 20:14:12 UTC+1:
> > > > > The cardinalities of N and E = 2*N, the set of even natural numbers, are equal.
> > > > The mathematical result contradict this. If for every n there are about twice as many integers, then there is no chance to egalize.
> > > I actually wrote 2*N on purpose. Don't you think that card(2*N) = 2*card(N)? If not, why not?
> > Of course it is.
> Bingo! Just go back to see what you agreed to: card(E) = card(2*N) = 2*card(N).

I did because cardinality does it this way and therefore is nonsense.

> So there are *twice* as many even naturals as there are naturals.

Wrong, cardinality has nothing to do with "what is". In mathematics the contrary is true.

Regards, WM

[Sergio]

when miss applied, yes


>
> Regards, WM
>

[Gus Gassmann]

On Thursday, 14 January 2021 at 18:13:02 UTC-4, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 17:58:39 UTC+1:
> > On Thursday, 14 January 2021 at 11:51:32 UTC-4, WM wrote:
> > > Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 15:27:42 UTC+1:
> > > > On Thursday, 14 January 2021 at 10:06:47 UTC-4, WM wrote:
> > > > > Gus Gassmann schrieb am Mittwoch, 13. Januar 2021 um 20:14:12 UTC+1:
> > > > > > The cardinalities of N and E = 2*N, the set of even natural numbers, are equal.
> > > > > The mathematical result contradict this. If for every n there are about twice as many integers, then there is no chance to egalize.
> > > > I actually wrote 2*N on purpose. Don't you think that card(2*N) = 2*card(N)? If not, why not?
> > > Of course it is.
> > Bingo! Just go back to see what you agreed to: card(E) = card(2*N) = 2*card(N).
> I did because cardinality does it this way and therefore is nonsense.

Cardinality does not do it that way. You have absolutely no clue about infinite cardinalities. Everything you believe about infinity is wrong, and everything you think is right is as dead wrong as it can be. You are as deluded as Smullyan's red knight.

> > So there are *twice* as many even naturals as there are naturals.
> Wrong, cardinality has nothing to do with "what is". In mathematics the contrary is true.

Bullshit, again. You used a cardinality argument to "prove" that there are half as many even naturals as there are naturals. You then proceeded to trumpet that as a fact --- NOTE: not as an incorrect statement, as a fact. I am well aware that you are going to deny that, spin it, twist it six ways from here to Sunday, but the fact is that you are wrong, you have been wrong for fifteen years, and you will continue to be wrong until the end of your miserable life.

[zelos...@gmail.com]

>That's the reason why it is nonsense.

Counterintuitive doesn't mean it is non-sense.

>The mathematical result contradict this. If for every n there are about twice as many integers, then there is no chance to egalize.

The twice as many is only true for finite sets, not infinite ones.

Again, you cannot from the finite, extrapolate the infinite.

>There are twice as many fractions in (2, 4) as in (2, 3).

There isn't, they are exactly equal as many.

>You have been taught that, but it is wrong.

It is true, because there exists a surjection N->Q and anther surjection Q->N, that means there is a bijection between them, hence they have the same cardinality.

>Mathematics is universal.

But not everything in mathematics applies to everything in it.

Not all binary functions are commutative, etc.

>lim_{n->oo} |N ∩ [0, n]| / |E∩ [0, n]| = 2 is no intuition.

Nope but it does not however in any way shape or form imply anything about the relative cardinality between the two sets N and 2N.

>Of course set theorists will claim that everything happens simultaneously. But that was true, then it could be analyzed in the way that I showed above.

As always, you're a fucking imbecile. You cannot do these in a step by step fashion and ask for at whichstep something is when your step by step idea requires all steps to be done for it to equal the original thing where its all done at once.

[WM]

Cantor’s cardinality is simply a misapplications. The "equinumerosity of naturals, rationals, algebraics, etc. is caused by the fact that every “bijection” is not a surjection but covers only the potentially infinite part of individually identifiable elements of the paired sequences. That his result is in contradiction with mathematics is proved by the fact that there are precisely twice as many integers as even integers. This is the mathematical limit obtained from the sequence of all intervals (0, n].

Regards, WM

[WM]

Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 23:53:55 UTC+1:
> On Thursday, 14 January 2021 at 18:13:02 UTC-4, WM wrote:

> > Wrong, cardinality has nothing to do with "what is". In mathematics the contrary is true.

> You used a cardinality argument to "prove" that there are half as many even naturals as there are naturals.

Wrong. I use mathematics.

Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2

has nothing to do with cardinality.

> the fact is that you are wrong,

The fact is that mathematics proves the limit 2 for every sequence intervals (0, n], whatever stepwidth may be chosen, when n --> oo. There cannot be the least doubt.

And I can explain why Cantor went wrong: His result of same cardinality of naturals, rationals, algebraics, etc. is simply caused by the fact that every “bijection” is not a surjection but covers only the potentially infinite part of individually identifiable elements of the paired sequences.

I find it difficult to understand how such a situation should have been capable of persisting in mathematics (after Bridgman).

Regards, W M

[WM]

zelos...@gmail.com schrieb am Freitag, 15. Januar 2021 um 07:25:13 UTC+1:
> >That's the reason why it is nonsense.
> Counterintuitive doesn't mean it is non-sense.

But wrong means nonsense. And Cantor went wrong, and I can explain why he went wrong: Cantor’s result of same cardinality of naturals, rationals, algebraics, etc. is simply caused by the fact that every “bijection” is not a surjection but covers only the potentially infinite part of individually identifiable elements of the paired sequences.

The simplest proof is this: In the interval (0, 1/n] there remain, for every identified n, ℵ0 unit fractions which cannot be identified as individuls. Therefore they cannot be applied for enumerating purposes.

> >The mathematical result contradict this. If for every n there are about twice as many integers, then there is no chance to egalize.
> The twice as many is only true for finite sets, not infinite ones.
>
> Again, you cannot from the finite, extrapolate the infinite.

That is just what Cantor did. Otherwise he could not have found any result about the infinite.

> >There are twice as many fractions in (2, 4) as in (2, 3).
> There isn't, they are exactly equal as many.

Every schoolboy could tell you the truth. Mathematics derives precisely the limit 2 for every sequence intervals (0, n], whatever stepwidth may be chosen, when n --> oo. There cannot be the least doubt.


To deny this means to throw away the achievements of mathematics in order to keep the religious nonsense that has been explained as such above.

You must claim that all unit fractions in (0, 1] must be individually definable and that analytical limits are wrong.

> >You have been taught that, but it is wrong.
> It is true, because there exists a surjection N->Q and anther surjection Q->N, that means there is a bijection between them, hence they have the same cardinality.

No, that is not a bijection between actually infinite sets. It concerns only definable parts.

>
> >Mathematics is universal.
>
> But not everything in mathematics applies to everything in it.

There is no doubt that every interval (0, n] contains infinitely many fractions and only finitely many natnumbers. Since after every n there is no further natnumber, this ratio cannot be egalized.

>
> >lim_{n->oo} |N ∩ [0, n]| / |E∩ [0, n]| = 2 is no intuition.
>
> Nope but it does not however in any way shape or form imply anything about the relative cardinality between the two sets N and 2N.

It does not concern cardinality, it concerns mathematical fact.

> >Of course set theorists will claim that everything happens simultaneously. But that was true, then it could be analyzed in the way that I showed above.

> You cannot do these in a step by step fashion

In mathematics every step can be investigated. There we do not need to apply tricks.

Regards, WM

[Gus Gassmann]

On Friday, 15 January 2021 at 06:20:06 UTC-4, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 23:53:55 UTC+1:
> > On Thursday, 14 January 2021 at 18:13:02 UTC-4, WM wrote:
>
> > > Wrong, cardinality has nothing to do with "what is". In mathematics the contrary is true.
> > You used a cardinality argument to "prove" that there are half as many even naturals as there are naturals.
> Wrong. I use mathematics.
> Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2
> has nothing to do with cardinality.

??? What the fuck do you think these funny vertical bars represent that I asked you to put in and you agreed should be there??? Mueckenheim, you are growing more senile every day. It is actually quite frightening to watch.

> > the fact is that you are wrong,
> The fact is that mathematics proves the limit 2 for every sequence intervals (0, n], whatever stepwidth may be chosen, when n --> oo. There cannot be the least doubt.

The fact is that you can't write down any argument cleanly so that other people can actually follow it.

> And I can explain why Cantor went wrong: His result of same cardinality of naturals, rationals, algebraics, etc. is simply caused by the fact that every “bijection” is not a surjection but covers only the potentially infinite part of individually identifiable elements of the paired sequences.

Which is it? Potentially infinite or light and dark, with only finitely many light numbers? Believe it or not, it does make a difference

> I find it difficult to understand how such a situation should have been capable of persisting in mathematics (after Bridgman).

Blablabla. I find it difficult to understand how a doddering idiot like you can still be allowed to spout his nonsense in front of a class of students, and force them to regurgigate it for credit. That has nothing to do with academic freedom.

[zelos...@gmail.com]

>But wrong means nonsense. And Cantor went wrong, and I can explain why he went wrong: Cantor’s result of same cardinality of naturals, rationals, algebraics, etc. is simply caused by the fact that every “bijection” is not a surjection but covers only the potentially infinite part of individually identifiable elements of the paired sequences.

This is entirely wrong. The bijections are surjective adn injective, that is what a fucking bijection means.

>The simplest proof is this: In the interval (0, 1/n] there remain, for every identified n, ℵ0 unit fractions which cannot be identified as individuls. Therefore they cannot be applied for enumerating purposes.

We can identify all of them and we can make functions for all of them. You are simply wrong.

>That is just what Cantor did. Otherwise he could not have found any result about the infinite.

The theorems related to cardinality and such do NOT do this.

>Every schoolboy could tell you the truth. Mathematics derives precisely the limit 2 for every sequence intervals (0, n], whatever stepwidth may be chosen, when n --> oo. There cannot be the least doubt.

No one disagrees that any finite set of natural numbers like that, there are twice the cardinality to only the even ones up to that point.

However, that in no way say anything about the infinite set.

>No, that is not a bijection between actually infinite sets. It concerns only definable parts.

There is a bijection, just get over it.

>It does not concern cardinality, it concerns mathematical fact.

Except it is about cardinality

>In mathematics every step can be investigated. There we do not need to apply tricks

No one is doing tricks but you, you are being decietful as always.

[Python]

Gus Gassmann wrote:
> On Friday, 15 January 2021 at 06:20:06 UTC-4, WM wrote:
>> Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 23:53:55 UTC+1:

...

>>> the fact is that you are wrong,
>> The fact is that mathematics proves the limit 2 for every sequence intervals (0, n], whatever stepwidth may be chosen, when n --> oo. There cannot be the least doubt.
>
> The fact is that you can't write down any argument cleanly so that other people can actually follow it.

Well, he's not even able to follow his own arguments.

[Sergio]

On 1/15/2021 4:35 AM, WM wrote:
> zelos...@gmail.com schrieb am Freitag, 15. Januar 2021 um 07:25:13 UTC+1:
>>> That's the reason why it is nonsense.
>> Counterintuitive doesn't mean it is non-sense.
>
> But wrong means nonsense. And Cantor went wrong, and I can explain why he went wrong: Cantor’s result of same cardinality of naturals, rationals, algebraics, etc. is simply caused by the fact that every “bijection” is not a surjection but covers only the potentially infinite part of individually identifiable elements of the paired sequences.

this is where it is broken,

"...but covers only the *potentially* infinite part of *individually*
*identifiable elements* of the paired sequences."

>
> The simplest proof is this: In the interval (0, 1/n] there remain, for every identified n, ℵ0 unit fractions which cannot be identified as individuls. Therefore they cannot be applied for enumerating purposes.
>
>>> The mathematical result contradict this. If for every n there are about twice as many integers, then there is no chance to egalize.
>> The twice as many is only true for finite sets, not infinite ones.
>>
>> Again, you cannot from the finite, extrapolate the infinite.
>
> That is just what Cantor did. Otherwise he could not have found any result about the infinite.
>
>>> There are twice as many fractions in (2, 4) as in (2, 3).
>> There isn't, they are exactly equal as many.
>
> Every schoolboy could tell you the truth. Mathematics derives precisely the limit 2 for every sequence intervals (0, n], whatever stepwidth may be chosen, when n --> oo. There cannot be the least doubt.

not when infinity is involved. 2 * oo = oo , k + oo = oo

>
>
> To deny this means to throw away the achievements of mathematics in order to keep the religious nonsense that has been explained as such above.

red herring. Learn your limits.

>
> You must claim that all unit fractions in (0, 1] must be individually definable and that analytical limits are wrong.

"all unit fractions.....must be individually definable" is red herring
as your definition of "definable" is totally unworkable.

>
>>> You have been taught that, but it is wrong.
>> It is true, because there exists a surjection N->Q and anther surjection Q->N, that means there is a bijection between them, hence they have the same cardinality.
>
> No, that is not a bijection between actually infinite sets. It concerns only definable parts.
>>
>>> Mathematics is universal.
>>
>> But not everything in mathematics applies to everything in it.
>
> There is no doubt that every interval (0, n] contains infinitely many fractions and only finitely many natnumbers. Since after every n there is no further natnumber, this ratio cannot be egalized.
>>
>>> lim_{n->oo} |N ∩ [0, n]| / |E∩ [0, n]| = 2 is no intuition.
>>
>> Nope but it does not however in any way shape or form imply anything about the relative cardinality between the two sets N and 2N.
>
> It does not concern cardinality, it concerns mathematical fact.

same mistake again.

>
>>> Of course set theorists will claim that everything happens simultaneously. But that was true, then it could be analyzed in the way that I showed above.
>> You cannot do these in a step by step fashion
>
> In mathematics every step can be investigated. There we do not need to apply tricks.

no tricks involved. it is an area you have not learned yet, or an area
you like to troll in.

>
> Regards, WM
>

[Sergio]

nope, you miss apply finite to infinite, a common mistake among those
new to math,

it is also obtained by the fact

2 * oo = oo and k + oo = oo

[Gus Gassmann]

Sadly, he is not "new to math", although he never progressed past the intro level. What he did not understand during his math courses in the 1970s, he never properly took on board, and pretty much everything he may have understood once he has forgotten or repressed. Bob Dylan's lyrics fit WM to a T: "his brain has been mismanaged with great skill". So who, oh who, is going to take away his license to kill (his students' minds).

[Python]

These days "license to kill" may mean "academic freedom", according
to Hochschule Augsburg social meadia team...

These days "inciting riots and election overturn" means "free of
speech", according to right wing medias in the U.S. ...

These days "raping his 14 years old son-in-law" means "teenager
romance", according to a french so-called "philosopher" as long
as an high rank analyst of French politics is involved...

[WM]

Gus Gassmann schrieb am Freitag, 15. Januar 2021 um 12:13:47 UTC+1:
> On Friday, 15 January 2021 at 06:20:06 UTC-4, WM wrote:
> > Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 23:53:55 UTC+1:
> > > On Thursday, 14 January 2021 at 18:13:02 UTC-4, WM wrote:
> >
> > > > Wrong, cardinality has nothing to do with "what is". In mathematics the contrary is true.
> > > You used a cardinality argument to "prove" that there are half as many even naturals as there are naturals.
> > Wrong. I use mathematics.
> > Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2
> > has nothing to do with cardinality.
> ??? What the fuck do you think these funny vertical bars represent

They represent the amount, the number of of numbers - counted one by one.

> > And I can explain why Cantor went wrong: His result of same cardinality of naturals, rationals, algebraics, etc. is simply caused by the fact that every “bijection” is not a surjection but covers only the potentially infinite part of individually identifiable elements of the paired sequences.
> Which is it? Potentially infinite or light and dark, with only finitely many light numbers? Believe it or not, it does make a difference

It does not make the least difference. Whether the dark numbers are existing or not is irrelevant since they cannot be addressed and used in mathematics.

Regards, WM

[WM]

zelos...@gmail.com schrieb am Freitag, 15. Januar 2021 um 13:29:25 UTC+1:
> >But wrong means nonsense. And Cantor went wrong, and I can explain why he went wrong: Cantor’s result of same cardinality of naturals, rationals, algebraics, etc. is simply caused by the fact that every “bijection” is not a surjection but covers only the potentially infinite part of individually identifiable elements of the paired sequences.
> This is entirely wrong. The bijections are surjective adn injective, that is what a fucking bijection means.

It means that but is impossible for infinite sets.

> >The simplest proof is this: In the interval (0, 1/n] there remain, for every identified n, ℵ0 unit fractions which cannot be identified as individuls. Therefore they cannot be applied for enumerating purposes.
> We can identify all of them

No, that would contradict the theorem that in all cases aleph_0 numbers beyond any identified number remain *not* identified.

Regards, WM

[WM]

Sergio schrieb am Freitag, 15. Januar 2021 um 16:19:33 UTC+1:
> On 1/15/2021 4:35 AM, WM wrote:

> > Every schoolboy could tell you the truth. Mathematics derives precisely the limit 2 for every sequence intervals (0, n], whatever stepwidth may be chosen, when n --> oo. There cannot be the least doubt.
> not when infinity is involved. 2 * oo = oo , k + oo = oo

Try to learn analysis. Analysis allows to calculate the lim 2n/n for n --> oo precisely.

> >
> > You must claim that all unit fractions in (0, 1] must be individually definable and that analytical limits are wrong.
> "all unit fractions.....must be individually definable" is red herring

It is what matheologians claim. Of course it is nonsense.

> as your definition of "definable" is totally unworkable.

Representation by digits is basic in mathematics and workable in every application of mathematics to sciences.

Regards, WM

[Gus Gassmann]

On Friday, 15 January 2021 at 18:02:01 UTC-4, WM wrote:
> Gus Gassmann schrieb am Freitag, 15. Januar 2021 um 12:13:47 UTC+1:
> > On Friday, 15 January 2021 at 06:20:06 UTC-4, WM wrote:
> > > Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 23:53:55 UTC+1:
> > > > On Thursday, 14 January 2021 at 18:13:02 UTC-4, WM wrote:
> > >
> > > > > Wrong, cardinality has nothing to do with "what is". In mathematics the contrary is true.
> > > > You used a cardinality argument to "prove" that there are half as many even naturals as there are naturals.
> > > Wrong. I use mathematics.
> > > Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2
> > > has nothing to do with cardinality.
> > ??? What the fuck do you think these funny vertical bars represent
> They represent the amount, the number of of numbers - counted one by one.

Use any euphemism you want; they still represent the cardinality of the sets in question.

> > > And I can explain why Cantor went wrong: His result of same cardinality of naturals, rationals, algebraics, etc. is simply caused by the fact that every “bijection” is not a surjection but covers only the potentially infinite part of individually identifiable elements of the paired sequences.
> > Which is it? Potentially infinite or light and dark, with only finitely many light numbers? Believe it or not, it does make a difference
> It does not make the least difference. Whether the dark numbers are existing or not is irrelevant since they cannot be addressed and used in mathematics.

That's not what everyone else understands by "potentially infinite", though, is it?

[Sergio]

On 1/15/2021 4:09 PM, WM wrote:
> Sergio schrieb am Freitag, 15. Januar 2021 um 16:19:33 UTC+1:
>> On 1/15/2021 4:35 AM, WM wrote:
>
>>> Every schoolboy could tell you the truth. Mathematics derives precisely the limit 2 for every sequence intervals (0, n], whatever stepwidth may be chosen, when n --> oo. There cannot be the least doubt.
>> not when infinity is involved. 2 * oo = oo , k + oo = oo
>
> Try to learn analysis. Analysis allows to calculate the lim 2n/n for n --> oo precisely.

wrong. that is not analysis either

"lim 2n/n for n --> oo"

2n/n = 2, for any n

>>>
>>> You must claim that all unit fractions in (0, 1] must be individually definable and that analytical limits are wrong.
>> "all unit fractions.....must be individually definable" is red herring
>
> It is what matheologians claim. Of course it is nonsense.

so we all agree it is nonsense. It is also silly.

>
>> as your definition of "definable" is totally unworkable.
>
> Representation by digits is basic in mathematics and workable in every application of mathematics to sciences.

and that has nothing to do with your daffynition of "definable"

>
> Regards, WM
>

[Sergio]

the theorem is wrong.

[Sergio]

On 1/15/2021 6:42 PM, Sergio wrote:
> On 1/15/2021 4:09 PM, WM wrote:
>> Sergio schrieb am Freitag, 15. Januar 2021 um 16:19:33 UTC+1:
>>> On 1/15/2021 4:35 AM, WM wrote:
>>
>>>> Every schoolboy could tell you the truth. Mathematics derives precisely the limit 2 for every sequence intervals (0, n], whatever stepwidth may be chosen, when n --> oo. There cannot be the least doubt.
>>> not when infinity is involved. 2 * oo = oo , k + oo = oo
>>
>> Try to learn analysis. Analysis allows to calculate the lim 2n/n for n --> oo precisely.
>
> wrong. that is not analysis either
>
> "lim 2n/n for n --> oo"
>
> 2n/n = 2, for any n

but when n = oo

2 * oo = oo , 2 + oo = oo

[Abelardo Ippolito]

Sergio wrote:

>> No, that would contradict the theorem that in all cases aleph_0 numbers
>> beyond any identified number remain *not* identified. Regards, WM
>
> the theorem is wrong.

If capitalism is not based on a social humanitarian faith system,
capitalism is nothing more than a hunting license. A license to hunt.

Also, capitalism is a predatory ideology, which preys on legitimated
economic activity, through usury, primarily. The bankers ruined your
country, now they demand a reset and preserve their privileges. You, to
stay the slave.

[Chris M. Thomasson]

Ding! :^)

[WM]

Gus Gassmann schrieb am Samstag, 16. Januar 2021 um 00:24:30 UTC+1:
> On Friday, 15 January 2021 at 18:02:01 UTC-4, WM wrote:
> > Gus Gassmann schrieb am Freitag, 15. Januar 2021 um 12:13:47 UTC+1:
> > > On Friday, 15 January 2021 at 06:20:06 UTC-4, WM wrote:
> > > > Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 23:53:55 UTC+1:
> > > > > On Thursday, 14 January 2021 at 18:13:02 UTC-4, WM wrote:
> > > >
> > > > > > Wrong, cardinality has nothing to do with "what is". In mathematics the contrary is true.
> > > > > You used a cardinality argument to "prove" that there are half as many even naturals as there are naturals.
> > > > Wrong. I use mathematics.
> > > > Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2
> > > > has nothing to do with cardinality.
> > > ??? What the fuck do you think these funny vertical bars represent
> > They represent the amount, the number of of numbers - counted one by one.
> Use any euphemism you want; they still represent the cardinality of the sets in question.

Of course for finite sets the cardinality is same as what mathematics supplies. But for the limit it differs. I use the analycal limit here but not cardinality.

> > > > And I can explain why Cantor went wrong: His result of same cardinality of naturals, rationals, algebraics, etc. is simply caused by the fact that every “bijection” is not a surjection but covers only the potentially infinite part of individually identifiable elements of the paired sequences.
> > > Which is it? Potentially infinite or light and dark, with only finitely many light numbers? Believe it or not, it does make a difference
> > It does not make the least difference. Whether the dark numbers are existing or not is irrelevant since they cannot be addressed and used in mathematics.
> That's not what everyone else understands by "potentially infinite", though, is it?

The dark numbers have no influence. Everybody should understand by potential infinity the collection of identified numbers. Either the dark complement of |N exists or there is nothing dark and consequently no completed set |N.

Regards, WM

[WM]

> the theorem is wrong.

Show a counter example. Identify a number with less than aleph_0 not identified successors.

Regards, WM

[Sergio]

You need to provide a formal Proof of your Theorem first.

That is how it is done in Mathematics.

[FredJeffries]

On Friday, January 15, 2021 at 2:02:01 PM UTC-8, WM wrote:
> Gus Gassmann schrieb am Freitag, 15. Januar 2021 um 12:13:47 UTC+1:
> > On Friday, 15 January 2021 at 06:20:06 UTC-4, WM wrote:
> > > Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 23:53:55 UTC+1:
> > > > On Thursday, 14 January 2021 at 18:13:02 UTC-4, WM wrote:
> > >
> > > > > Wrong, cardinality has nothing to do with "what is". In mathematics the contrary is true.
> > > > You used a cardinality argument to "prove" that there are half as many even naturals as there are naturals.
> > > Wrong. I use mathematics.
> > > Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2
> > > has nothing to do with cardinality.
> > ??? What the fuck do you think these funny vertical bars represent
> They represent the amount, the number of of numbers - counted one by one.

Here we have an exquisite example of the mathematics-by-magic that prevails in our Professor's kingdom.

No understanding of concepts is required -- indeed, it is a positive hindrance.

Only correct enunciation of the spell is required. The professor pronounces that |E ∩ [0, n]| is not a 'cardinality', therefore it is NOT a 'cardinality' no matter how much its daffynition resembles cardinality. Two objects that share the same name, by the magic Law of Names, ARE the same. The same object passing under two different names IS two different objects.

There is an obvious carryover to his classroom where students are required to correctly pronounce the spells they have been taught without any understanding of the concepts involved.

[FredJeffries]

On Friday, January 15, 2021 at 2:02:01 PM UTC-8, WM wrote:
> Gus Gassmann schrieb am Freitag, 15. Januar 2021 um 12:13:47 UTC+1:
> > On Friday, 15 January 2021 at 06:20:06 UTC-4, WM wrote:
> > > Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 23:53:55 UTC+1:
> > > > On Thursday, 14 January 2021 at 18:13:02 UTC-4, WM wrote:
> > >
> > > > > Wrong, cardinality has nothing to do with "what is". In mathematics the contrary is true.
> > > > You used a cardinality argument to "prove" that there are half as many even naturals as there are naturals.
> > > Wrong. I use mathematics.
> > > Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2
> > > has nothing to do with cardinality.
> > ??? What the fuck do you think these funny vertical bars represent
> They represent the amount, the number of of numbers - counted one by one.

One can only wonder what poor wretch our Professor forced to 'count' the even numbers less than a billion 'one by one'

[WM]

Sergio schrieb am Samstag, 16. Januar 2021 um 19:20:43 UTC+1:
> On 1/16/2021 10:54 AM, WM wrote:
> > Sergio schrieb am Samstag, 16. Januar 2021 um 01:43:18 UTC+1:
> >> On 1/15/2021 4:04 PM, WM wrote:
> >
> >>>> We can identify all of them
> >>>
> >>> No, that would contradict the theorem that in all cases aleph_0 numbers beyond any identified number remain *not* identified.
> >
> >> the theorem is wrong.
> >
> > Show a counter example. Identify a number with less than aleph_0 not identified successors.
> >

> You need to provide a formal Proof of your Theorem first.
>
> That is how it is done in Mathematics

and that is why matheology could rise. Facts are shown by the inability of anyone to identify unit fractions in the interval (0, 1/n) such that less than aleph_0 unit fractions remain.

Regards, WM

[WM]

Cantor: man kann eine solche Menge ... nach einem bestimmten leicht zu definierenden Gesetze in die Form einer einfach unendlichen Reihe mit dem allgemeinen Gliede w_n, wo n ein positiver unbeschränkter ganzzahliger Index ist, bringen, so daß jedes Glied oder Element der Menge an einer bestimmten Stelle n dieser Reihe steht und auch umgekehrt jedes Glied w_n der Reihe ein Element der gedachten Mannigfaltigkeit ist.

Regards, WM

[Gus Gassmann]

On Saturday, 16 January 2021 at 12:51:08 UTC-4, WM wrote:
> Gus Gassmann schrieb am Samstag, 16. Januar 2021 um 00:24:30 UTC+1:
> > On Friday, 15 January 2021 at 18:02:01 UTC-4, WM wrote:
> > > Gus Gassmann schrieb am Freitag, 15. Januar 2021 um 12:13:47 UTC+1:
> > > > On Friday, 15 January 2021 at 06:20:06 UTC-4, WM wrote:
> > > > > Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 23:53:55 UTC+1:
> > > > > > On Thursday, 14 January 2021 at 18:13:02 UTC-4, WM wrote:
> > > > >
> > > > > > > Wrong, cardinality has nothing to do with "what is". In mathematics the contrary is true.
> > > > > > You used a cardinality argument to "prove" that there are half as many even naturals as there are naturals.
> > > > > Wrong. I use mathematics.
> > > > > Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2
> > > > > has nothing to do with cardinality.
> > > > ??? What the fuck do you think these funny vertical bars represent
> > > They represent the amount, the number of of numbers - counted one by one.
> > Use any euphemism you want; they still represent the cardinality of the sets in question.
> Of course for finite sets the cardinality is same as what mathematics supplies. But for the limit it differs. I use the analycal limit here but not cardinality.

... which you then turn into an unwarranted insinuation about the infinite sets E and N. That is proof by intimidation. When you are then confronted with two clear examples where your approach leads to nonsense and contradiction, you mumble about your method not being universally applicable. You are despicable charlatan and a cheat.

[zelos...@gmail.com]

>They represent the amount, the number of of numbers - counted one by one.

WHen it comes to set, |A| means THE CARDINALITY

>It means that but is impossible for infinite sets.

Nope, f(x)=x for f:N->N is entirely surjective

>No, that would contradict the theorem that in all cases aleph_0 numbers beyond any identified number remain *not* identified.

There is no such theorem because your idea of "identified" has no meaning in mathematics.

>Try to learn analysis. Analysis allows to calculate the lim 2n/n for n --> oo precisely.

Set theory and analysis are not the fucking same you imbecile.

>Representation by digits is basic in mathematics and workable in every application of mathematics to sciences

Representation is however irrelevant in mathematics.

[WM]

Gus Gassmann schrieb am Sonntag, 17. Januar 2021 um 23:37:13 UTC+1:
> On Saturday, 16 January 2021 at 12:51:08 UTC-4, WM wrote:

> > Of course for finite sets the cardinality is same as what mathematics supplies. But for the limit it differs. I use the analycal limit here but not cardinality.

> ... which you then turn into an unwarranted insinuation about the infinite sets E and N.

That is the mathematical limit: Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2.

> When you are then confronted with two clear examples where your approach leads to nonsense and contradiction,

Nonsense? Let P denote the set of prime numbers, then |N ∩ [0, n]| / |P∩ [0, n]| decreases below every positive eps. That means Lim_{n--> oo} |N ∩ [0, n]| / |P ∩ [0, n]| = 0. It does not mean that there are no prime numbers. Note that Lim_{n--> oo} 1/n = 0 although there is no term zero in the sequence.

Regards, WM