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

[Dan Christensen]

On Friday, January 8, 2021 at 6:27:50 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]

Dan Christensen schrieb am Freitag, 8. Januar 2021 um 18:29:16 UTC+1:
> On Friday, January 8, 2021 at 6:27:50 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?

Great mathematicians have invented the limit. For N and the set E of positive even numbers this yields in correct mathematics:

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

There are twice as many natural numbers than positive even numbers.

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 12:27:50 UTC+1:
> There are less prime numbers P than natural numbers N

[Gus Gassmann]

Repeating something does not make it any less false. Your expression makes no sense. What is '/' when it is applied to sets? And how does it associate over '∩'?

[Dan Christensen]

Another idiocy for the ages! (See updated list below.)

In your algebra course, after you learn about variables, you will learn about functions and, in particular, about bijective functions. Then you will learn that, there is a bijective function f mapping N to E (1-to-1 and onto), such that f(x)=2x. And that its inverse is g(x)=x/2. Then you will understand that there are just as many natural numbers as even numbers.

Good luck with your algebra course, Mucke. Remember to do your homework or you will continue to fail as here.

***************************************************
More absurd quotes from Wolfgang Muckenheim (WM, aka Mucke):

"There are twice as many natural numbers than positive even numbers." **** NEW ****
--sci.math, 2021/01/09

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

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

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

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

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

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

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

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


Slipping ever more deeply into madness...

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

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

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

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

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

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

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

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

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

[WM]

Mostowski Collapse schrieb am Samstag, 9. Januar 2021 um 14:28:10 UTC+1:
> Because of bijection n |-> pn, we have P| = |N|.

There is no bijection between undefinable elements.

Try to define as many unit fractions as possible. Remove them from (0, 1]. Set theory asserts that Infinitely many unit fractions will remain. But if you remove them all, then nothing remains. That means all is more than all definable.

Regards, WM

[WM]

Thank you for the hint. I meant Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2

Regards, WM

[WM]

Dan Christensen schrieb am Samstag, 9. Januar 2021 um 20:31:30 UTC+1:
> On Saturday, January 9, 2021 at 4:43:11 AM UTC-5, WM wrote:

> > There are twice as many natural numbers than positive even numbers.
> >

> In your algebra course, after you learn about variables, you will learn about functions and, in particular, about bijective functions. Then you will learn that, there is a bijective function f mapping N to E (1-to-1 and onto), such that f(x)=2x.

This is taught by teachers (like myself in earlier times, cp. W. Mückenheim: "Mathematik für die ersten Semester", 4th ed., De Gruyter, Berlin (2015)) who have not yet discovered dark numbers.

Infinite sets comprise undefinable elements. There is no bijection between undefinable elements

Simple example: Try to define as many unit fractions as possible. Remove them from (0, 1]. Set theory asserts that Infinitely many unit fractions will remain. But if you remove them all, then nothing remains. That means all is more than all definable. The difference is called dark.

Regards, WM

[Dan Christensen]

On Saturday, January 9, 2021 at 5:05:29 PM UTC-5, WM wrote:
> Dan Christensen schrieb am Samstag, 9. Januar 2021 um 20:31:30 UTC+1:
> > On Saturday, January 9, 2021 at 4:43:11 AM UTC-5, WM wrote:
>
> > > There are twice as many natural numbers than positive even numbers.
> > >
> > In your algebra course, after you learn about variables, you will learn about functions and, in particular, about bijective functions. Then you will learn that, there is a bijective function f mapping N to E (1-to-1 and onto), such that f(x)=2x.
> This is taught by teachers (like myself in earlier times, cp. W. Mückenheim: "Mathematik für die ersten Semester", 4th ed., De Gruyter, Berlin (2015)) who have not yet discovered dark numbers.

We are STILL waiting for your proof of the existence of even one such "dark" number, Mücke.

>
> Infinite sets comprise undefinable elements. There is no bijection between undefinable elements
>

There are no "undefinable" natural numbers or even numbers, Mücke.

> Simple example: Try to define as many unit fractions as possible.

Here is the set of all positive unit fractions: {1/x : x in N+}. Nothing mysterious or "dark" about it, Mücke.

> Remove them from (0, 1].

The remainder would be (0, 1] \ {1/x : x in N+}. Nothing mysterious or "dark" about this either. Just basic high-school algebra. (BTE how it your course going? You don't seem to have made much progress. Are you not doing your homework, Mücke?

> Set theory asserts that Infinitely many unit fractions will remain.

Wrong again, Mücke. After removing every unit fraction from an interval, no unit fractions will remain there. They should cover this in your algebra course. Please remember to do your homework.


More absurd quotes from Wolfgang Muckenheim (WM):

"There are twice as many natural numbers than positive even numbers."

--sci.math 2021/01/09 **** NEW ****

[WM]

Dan Christensen schrieb am Samstag, 9. Januar 2021 um 23:35:53 UTC+1:

> We are STILL waiting for your proof of the existence of even one such "dark" number.

You will be waiting in vain unless you can solve the first homework given below.

> > Simple example: Try to define as many unit fractions as possible.
> Here is the set of all positive unit fractions: {1/x : x in N+}.

First Homework: Name a unit fraction such that every reader knows whether or not it is larger than 1/17.

Regards, WM

[Gus Gassmann]

Thought so. We are therefore back to your tired old false argument that this limit implies that there are "more" natural numbers than even naturals, without defining what "more" actually means.

Just to jog your memory (in case this still helps): E is a strict subset of N, but card(E) = card(N). card is not a continuous function.

[Dan Christensen]

Everyone here, except maybe you, Mucke, will know that 1/2 is larger than 1/17.

Now, back to your "Math is Fun" homework. No more distractions!

Dan

Download my DC Proof 2.0 software at http://www.dcproof.com

[WM]

Gus Gassmann schrieb am Sonntag, 10. Januar 2021 um 15:33:03 UTC+1:

> E is a strict subset of N, but card(E) = card(N). card is not a continuous function.

That shows that card is incompatible with mathematics. 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.

Cardinality is an at least outrageously unprecise measure. The same result for Q and N is not acceptable. But I can tell you the reason: In fact every mapping between infinite sets concerns only a potentially infinite subcollection. The very different remainders are dark and cannot participate.

Have you never wondered why all sets have same cardinality? (This includes all distinguishable real numbers. Note that undistiguishable elements cannot appear in a mapping. Note further that distinguishing infinite digit sequences can only happen at finite places. Hence only countably many distinctions can exist.)

Regards, WM

[WM]

Dan Christensen schrieb am Sonntag, 10. Januar 2021 um 17:42:26 UTC+1:
> On Sunday, January 10, 2021 at 5:46:04 AM UTC-5, WM wrote:
> > Dan Christensen schrieb am Samstag, 9. Januar 2021 um 23:35:53 UTC+1:
> >
> > > We are STILL waiting for your proof of the existence of even one such "dark" number.
> >
> > You will be waiting in vain unless you can solve the first homework given below.
> > > > Simple example: Try to define as many unit fractions as possible.
> > > Here is the set of all positive unit fractions: {1/x : x in N+}.
> > First Homework: Name a unit fraction such that every reader knows whether or not it is larger than 1/17.
> >

> 1/2 is larger than 1/17.

Very good. Now find out why this cannot be said of 1/n.

Regards, WM

[Python]

Crank Wolfgang Mueckenheim, aka WM wrote:
> Gus Gassmann schrieb am Sonntag, 10. Januar 2021 um 15:33:03 UTC+1:
>
>> E is a strict subset of N, but card(E) = card(N). card is not a continuous function.
>
> That shows that card is incompatible with mathematics.

Being non-continuous is incompatible with mathematics?

You are a utter idiot, Crank Wolfgang Mueckenheim, from Hochschule
Augsburg.

[WM]

Python schrieb am Sonntag, 10. Januar 2021 um 19:08:13 UTC+1:
> Crank Wolfgang Mueckenheim, aka WM wrote:
> > Gus Gassmann schrieb am Sonntag, 10. Januar 2021 um 15:33:03 UTC+1:
> >
> >> E is a strict subset of N, but card(E) = card(N). card is not a continuous function.
> >
> > That shows that card is incompatible with mathematics.
> Being non-continuous is incompatible with mathematics?

The function |N ∩ [0, n]| / |E ∩ [0, n]| counting the natural and even numbers in the given interval has the limit 2. This is not only mathematics but the result that every sensible thinker will obtain.

But I can tell you why cardinality is going so much astray: All infinite sets have mainly dark elements. The mapping with |N can only concern the first few definable elements. Therefore all have the same size. Or do you *really* believe that there are as many fractions as integers?

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 and have corrupted thousands of students.

Regards, WM

[FredJeffries]

On Sunday, January 10, 2021 at 10:26:38 AM UTC-8, WM wrote:
> Python schrieb am Sonntag, 10. Januar 2021 um 19:08:13 UTC+1:
> > Crank Wolfgang Mueckenheim, aka WM wrote:
> > > Gus Gassmann schrieb am Sonntag, 10. Januar 2021 um 15:33:03 UTC+1:
> > >
> > >> E is a strict subset of N, but card(E) = card(N). card is not a continuous function.
> > >
> > > That shows that card is incompatible with mathematics.
> > Being non-continuous is incompatible with mathematics?
> The function |N ∩ [0, n]| / |E ∩ [0, n]| counting the natural and even numbers in the given interval has the limit 2. This is not only mathematics but the result that every sensible thinker will obtain.

Thus we have the remarkable result that the combination of two 'imprecise and wrong and therefore useless' and 'perverse' 'measures' we arrive at THE absolutely 'precise' 'result that every sensible thinker will obtain'.

[Dan Christensen]

On Sunday, January 10, 2021 at 1:02:04 PM UTC-5, WM wrote:
> Dan Christensen schrieb am Sonntag, 10. Januar 2021 um 17:42:26 UTC+1:
> > On Sunday, January 10, 2021 at 5:46:04 AM UTC-5, WM wrote:
> > > Dan Christensen schrieb am Samstag, 9. Januar 2021 um 23:35:53 UTC+1:
> > >
> > > > We are STILL waiting for your proof of the existence of even one such "dark" number.
> > >
> > > You will be waiting in vain unless you can solve the first homework given below.
> > > > > Simple example: Try to define as many unit fractions as possible.
> > > > Here is the set of all positive unit fractions: {1/x : x in N+}.
> > > First Homework: Name a unit fraction such that every reader knows whether or not it is larger than 1/17.
> > >

> > Everyone here, except maybe you, Mucke, will know that 1/2 is larger than 1/17.

>
> Very good. Now find out why this cannot be said of 1/n.
>

I guess you still haven't got to the part about variables in your course. When you do, you SHOULD understand that 1/n > 1/17 iff n < 17 for all n in N+. None of your mysterious "dark numbers" required.

> > Now, back to your "Math is Fun" homework. No more distractions!

Still applies.

[Gus Gassmann]

On Sunday, 10 January 2021 at 14:02:04 UTC-4, WM wrote:
> Dan Christensen schrieb am Sonntag, 10. Januar 2021 um 17:42:26 UTC+1:

[...]

> > 1/2 is larger than 1/17.
> Very good. Now find out why this cannot be said of 1/n.

In this context I invite you again to compare the distance from Augsburg to Neustadt with the distance from Augsburg to Frankfurt.

[Gus Gassmann]

On Sunday, 10 January 2021 at 16:02:56 UTC-4, FredJeffries wrote:
> On Sunday, January 10, 2021 at 10:26:38 AM UTC-8, WM wrote:
> > Python schrieb am Sonntag, 10. Januar 2021 um 19:08:13 UTC+1:
> > > Crank Wolfgang Mueckenheim, aka WM wrote:
> > > > Gus Gassmann schrieb am Sonntag, 10. Januar 2021 um 15:33:03 UTC+1:
> > > >
> > > >> E is a strict subset of N, but card(E) = card(N). card is not a continuous function.
> > > >
> > > > That shows that card is incompatible with mathematics.
> > > Being non-continuous is incompatible with mathematics?
> > The function |N ∩ [0, n]| / |E ∩ [0, n]| counting the natural and even numbers in the given interval has the limit 2. This is not only mathematics but the result that every sensible thinker will obtain.
> Thus we have the remarkable result that the combination of two 'imprecise and wrong and therefore useless' and 'perverse' 'measures' we arrive at THE absolutely 'precise' 'result that every sensible thinker will obtain'.

Note: *Only* by combining two or more imprecise results can you *ever* hope to obtain precise results in Mueckemyth. Moreover, you cannot possibly be a sensible thinker if you do not agree.

[Jim Burns]

On 1/10/2021 1:00 PM, WM wrote:

> Have you never wondered why all sets have same cardinality?

The time has come,' the Walrus said,
To talk of many things:
Of shoes — and ships — and sealing-wax —
Of cabbages — and kings —
And why the sea is boiling hot —
And whether pigs have wings.'

[Khong Dong]

At any rate:

<quote>

The time has passed that we’ve gone astray from this knowledge boundary, believing wrongly -- if not complacently --that it’d be just a matter of time before we could prove the absolute truth or falsehood of any specific mathematical conjecture we come across, as if we were entitled to such reasoning privilege.

The time has come that we should finally go home, to our root of being finite, to humbly realize that, in so far as reasoning is an introspective attempt to answer the question of where we are --what we can possibly know --in the infinite realm of mathematical abstraction, our mortal deducing and knowing mathematical truths is only a matter of endowment - not entitlement.

</quote>

[WM]

Dan Christensen schrieb am Sonntag, 10. Januar 2021 um 22:35:22 UTC+1:
> On Sunday, January 10, 2021 at 1:02:04 PM UTC-5, WM wrote:

> > > > First Homework: Name a unit fraction such that every reader knows whether or not it is larger than 1/17.
> > > >

> > > Everyone here will know that 1/2 is larger than 1/17.

> >
> > Very good. Now find out why this cannot be said of 1/n.
> >

> When you do, you SHOULD understand that 1/n > 1/17 iff

You should understand that you have to answer the question with no if and no iff.

Here the question is again:
Is 1/n > 1/17?
The answers are plainly yes or no or unknown.

Regards, WM

[WM]

FredJeffries schrieb am Sonntag, 10. Januar 2021 um 21:02:56 UTC+1:
> On Sunday, January 10, 2021 at 10:26:38 AM UTC-8, WM wrote:

> > The function |N ∩ [0, n]| / |E ∩ [0, n]| counting the natural and even numbers in the given interval has the limit 2. This is not only mathematics but the result that every sensible thinker will obtain.
> Thus we have the remarkable result that the combination of two 'imprecise and wrong and therefore useless' and 'perverse' 'measures' we arrive at THE absolutely 'precise' 'result that every sensible thinker will obtain'.

You seem to have misunderstood mathematics. The limit 2 is an absolutely precise measure although not all terms of the function are existing. The limit is not obtained by checking all aleph_0 terms of the sequence, which is impossible, but by proving that, whatever term a_n for n > n_0 may be identified, it will deviate less from the limit than the chosen epsilon. For every epsilon that can be chosen there exists such an n_0.

Regards, WM

[FredJeffries]

Our Great and Powerful Professor thus demonstrates that that he does not understand what he has posted.

He finds 'an absolutely precise measure' of the limit of a sequence. He boasts that that 'whatever term a_n for n > n_0 may be identified', the value may be determined with enough precision that 'will deviate less from the limit than the chosen epsilon'.

He is able to attain these remarkable results in spite of the fact that he DEFINES sequence by combining two 'imprecise and wrong and therefore useless' and 'perverse' 'measures' , i.e. cardinality!

Daffynition: The function |N ∩ [0, n]| / |E ∩ [0, n]| counting the natural and even numbers in the given interval.

eod

[Ralf Bader]

According to your idiotic nonsense
Message-ID: <0051d8df-7bc7-448c...@googlegroups.com>
omly finitely many epsilon can be chosen. The stuff with limits does not
work in those circumstances. But I am sure that you are too stupid to
grasp this.

[Dan Christensen]

On Monday, January 11, 2021 at 12:37:42 PM UTC-5, WM wrote:
> Dan Christensen schrieb am Sonntag, 10. Januar 2021 um 22:35:22 UTC+1:
> > On Sunday, January 10, 2021 at 1:02:04 PM UTC-5, WM wrote:
>
> > > > > First Homework: Name a unit fraction such that every reader knows whether or not it is larger than 1/17.
> > > > >
> > > > Everyone here will know that 1/2 is larger than 1/17.
> > >
> > > Very good. Now find out why this cannot be said of 1/n.
> > >

> > I guess you still haven't got to the part about variables in your course. When you do, you SHOULD understand that 1/n > 1/17 iff n < 17 for all n in N+. None of your mysterious "dark numbers" required.

> You should understand that you have to answer the question with no if and no iff.
>
> Here the question is again:
> Is 1/n > 1/17?
> The answers are plainly yes or no or unknown.
>

It may or may not be true. As I said above, its truth value depends on the value of the variable n. Yes, I know, you have no idea what I am talking about and you won't until to you get to the part about variables in your course, Mucke. Until then, just DO YOUR HOMEWORK. You are looking like a complete idiot here.

[WM]

FredJeffries schrieb am Montag, 11. Januar 2021 um 19:30:02 UTC+1:
> On Monday, January 11, 2021 at 9:49:53 AM UTC-8, WM wrote:

> He is able to attain these remarkable results in spite of the fact that he DEFINES sequence by combining two 'imprecise and wrong and therefore useless' and 'perverse' 'measures' , i.e. cardinality!

I do not apply cardinality here at all.

>
> The function |N ∩ [0, n]| / |E ∩ [0, n]| counting the natural and even numbers in the given interval.

You are really unable to understand that for n > 1 every term of the potentially infinite sequence and its limit are precisely defined by the above formula? Here are some examples:
|N ∩ [0, 1]| / |E ∩ [0, 1]| = 1/0 = undefined
|N ∩ [0, 2]| / |E ∩ [0, 2]| = 2/1 = 2
|N ∩ [0, 3]| / |E ∩ [0, 3]| = 3/1 = 3
|N ∩ [0, 10]| / |E ∩ [0, 10]| = 10/5 = 2
and so on

Regards, WM

[Gus Gassmann]

Very good! But it does not mean what you seem to think (or at least what you want your students to swallow and regurgitate).

Try with the set S of squares of integers instead of E. Then lim{n->oo} [ |N ∩ [0, n]| / |S ∩ [0, n]| ] = 0. Notice anything?

[WM]

Dan Christensen schrieb am Dienstag, 12. Januar 2021 um 05:35:11 UTC+1:
> On Monday, January 11, 2021 at 12:37:42 PM UTC-5, WM wrote:
> > Dan Christensen schrieb am Sonntag, 10. Januar 2021 um 22:35:22 UTC+1:
> > > On Sunday, January 10, 2021 at 1:02:04 PM UTC-5, WM wrote:
> >
> > > > > > First Homework: Name a unit fraction such that every reader knows whether or not it is larger than 1/17.
> > > > > >
> > > > > Everyone here will know that 1/2 is larger than 1/17.
> > > >
> > > > Very good. Now find out why this cannot be said of 1/n.
> > > >
> > > I guess you still haven't got to the part about variables in your course. When you do, you SHOULD understand that 1/n > 1/17 iff n < 17 for all n in N+. None of your mysterious "dark numbers" required.
> > You should understand that you have to answer the question with no if and no iff.
> >
> > Here the question is again:
> > Is 1/n > 1/17?
> > The answers are plainly yes or no or unknown.
> >
> It may or may not be true.

That means unknown. Well, you should have learned a lot by this answer.

> As I said above, its truth value depends on the value of the variable n.

The variable has no value. Otherwise you would have answered yes or no.

> Yes, I know, you have no idea what I am talking about

I do not think so about you. I think you are a stubborn liar who never well stand by his mistakes.

> You are looking like a complete idiot here.

And you are an evil tongue.

But all not completely stupid readers will have grasped these facts now.

Regards, WM

[WM]

Ralf Bader schrieb am Montag, 11. Januar 2021 um 21:59:20 UTC+1:
> On 01/11/2021 06:49 PM, WM wrote:
> > FredJeffries schrieb am Sonntag, 10. Januar 2021 um 21:02:56 UTC+1:
> >> On Sunday, January 10, 2021 at 10:26:38 AM UTC-8, WM wrote:
> >
> >>> The function |N ∩ [0, n]| / |E ∩ [0, n]| counting the natural and
> >>> even numbers in the given interval has the limit 2. This is not
> >>> only mathematics but the result that every sensible thinker will
> >>> obtain.
> >> Thus we have the remarkable result that the combination of two
> >> 'imprecise and wrong and therefore useless' and 'perverse'
> >> 'measures' we arrive at THE absolutely 'precise' 'result that every
> >> sensible thinker will obtain'.
> >
> > You seem to have misunderstood mathematics. The limit 2 is an
> > absolutely precise measure although not all terms of the function are
> > existing. The limit is not obtained by checking all aleph_0 terms of
> > the sequence, which is impossible, but by proving that, whatever
> > term a_n for n > n_0 may be identified, it will deviate less from the
> > limit than the chosen epsilon. For every epsilon that can be chosen
> > there exists such an n_0.
> According to you

> omly finitely many epsilon can be chosen.

That is obvious because no-one can choose infinitely many numbers individually. But we use a general dependence of n_0 and eps. For instance the simple function f(n) = 1/n can be proven convergent by taking eps = 1/k . For every such epsilon we can find an n_0, for instance n_0 = k+1, to make sure the criterion is satisfied.

> The stuff with limits does not
> work in those circumstances.

It did and does. In above dependence we can insert every k that is not forever dark. More is not necessary (because we cannot be sure that dark numbers are existing at all). This kind of analysis has been introduced way before Cantor when only potential infinity existed. And it has worked well.

Regards, WM

[WM]

I notice that a single wrong result proves a theory wrong. And there are many wrong results of set theory: Take the natnumbers and the integers, or the natnumbers and the numbers divisible by 3, or divisible by 4, or ... , or the uncancelled fractions. These examples prove the cardinality measure wrong. More is not required.

Regards, WM

[Gus Gassmann]

On Tuesday, 12 January 2021 at 08:36:06 UTC-4, WM wrote:
> Gus Gassmann schrieb am Dienstag, 12. Januar 2021 um 13:05:06 UTC+1:
> > On Tuesday, 12 January 2021 at 07:30:32 UTC-4, WM wrote:
>
> > > > The function |N ∩ [0, n]| / |E ∩ [0, n]| counting the natural and even numbers in the given interval.
> > > You are really unable to understand that for n > 1 every term of the potentially infinite sequence and its limit are precisely defined by the above formula? Here are some examples:
> > > |N ∩ [0, 1]| / |E ∩ [0, 1]| = 1/0 = undefined
> > > |N ∩ [0, 2]| / |E ∩ [0, 2]| = 2/1 = 2
> > > |N ∩ [0, 3]| / |E ∩ [0, 3]| = 3/1 = 3
> > > |N ∩ [0, 10]| / |E ∩ [0, 10]| = 10/5 = 2
> > > and so on
> > Very good! But it does not mean what you seem to think (or at least what you want your students to swallow and regurgitate).
> >
> > Try with the set S of squares of integers instead of E. Then lim{n->oo} [ |N ∩ [0, n]| / |S ∩ [0, n]| ] = 0.
> > Notice anything?
> I notice that a single wrong result proves a theory wrong.

That's your theory, my good friend. Everybody else here understands that card is not a continuous function and hence cannot be used to infer limiting behaviour for its arguments. So now you acknowledge the faults in your theory yourself?

[Dan Christensen]

On Tuesday, January 12, 2021 at 7:09:15 AM UTC-5, WM wrote:
> Dan Christensen schrieb am Dienstag, 12. Januar 2021 um 05:35:11 UTC+1:
> > On Monday, January 11, 2021 at 12:37:42 PM UTC-5, WM wrote:
> > > Dan Christensen schrieb am Sonntag, 10. Januar 2021 um 22:35:22 UTC+1:
> > > > On Sunday, January 10, 2021 at 1:02:04 PM UTC-5, WM wrote:
> > >
> > > > > > > First Homework: Name a unit fraction such that every reader knows whether or not it is larger than 1/17.
> > > > > > >
> > > > > > Everyone here will know that 1/2 is larger than 1/17.
> > > > >
> > > > > Very good. Now find out why this cannot be said of 1/n.
> > > > >
> > > > I guess you still haven't got to the part about variables in your course. When you do, you SHOULD understand that 1/n > 1/17 iff n < 17 for all n in N+. None of your mysterious "dark numbers" required.
> > > You should understand that you have to answer the question with no if and no iff.
> > >
> > > Here the question is again:
> > > Is 1/n > 1/17?
> > > The answers are plainly yes or no or unknown.
> > >
> > It may or may not be true.

> That means unknown. Well, you should have learned a lot by this answer.

> > As I said above, its truth value depends on the value of the variable n.

> The variable has no value. Otherwise you would have answered yes or no.

It seems you have STILL not grasped the notion of a variable, Mucke. You really need to take some time out from spreading your lies and misinformation here, and educate yourself to at least high-school level math. It won't be easy for you, I know, but you really must DO YOUR HOMEWORK!

That link again: https://www.mathsisfun.com/algebra/introduction.html

[Ralf Bader]

On 01/12/2021 01:25 PM, WM wrote:
> Ralf Bader schrieb am Montag, 11. Januar 2021 um 21:59:20 UTC+1:
>> On 01/11/2021 06:49 PM, WM wrote:
>>> FredJeffries schrieb am Sonntag, 10. Januar 2021 um 21:02:56
>>> UTC+1:
>>>> On Sunday, January 10, 2021 at 10:26:38 AM UTC-8, WM wrote:
>>>
>>>>> The function |N ∩ [0, n]| / |E ∩ [0, n]| counting the natural
>>>>> and even numbers in the given interval has the limit 2. This
>>>>> is not only mathematics but the result that every sensible
>>>>> thinker will obtain.
>>>> Thus we have the remarkable result that the combination of two
>>>> 'imprecise and wrong and therefore useless' and 'perverse'
>>>> 'measures' we arrive at THE absolutely 'precise' 'result that
>>>> every sensible thinker will obtain'.
>>>
>>> You seem to have misunderstood mathematics. The limit 2 is an
>>> absolutely precise measure although not all terms of the function
>>> are existing. The limit is not obtained by checking all aleph_0
>>> terms of the sequence, which is impossible, but by proving that,
>>> whatever term a_n for n > n_0 may be identified, it will deviate
>>> less from the limit than the chosen epsilon. For every epsilon
>>> that can be chosen there exists such an n_0.
>> According to you omly finitely many epsilon can be chosen.

I did not write this, quotefaker. Now for some more of your idiotic
nonsense:

> That is obvious because no-one can choose infinitely many numbers
> individually. But we use a general dependence of n_0 and eps. For
> instance the simple function f(n) = 1/n can be proven convergent by
> taking eps = 1/k . For every such epsilon we can find an n_0, for
> instance n_0 = k+1, to make sure the criterion is satisfied.

>> The stuff with limits does not work in those circumstances.
>
> It did and does. In above dependence we can insert every k that is
> not forever dark. More is not necessary (because we cannot be sure
> that dark numbers are existing at all). This kind of analysis has
> been introduced way before Cantor when only potential infinity
> existed. And it has worked well.

So it is nothing but your whim according to which things have to be
"individually chosen" and so on, or a "general dependence" may be used.
What you are telling here is just shit.

[Jim Burns]

Mathematics deals in _descriptions_

It is a small truth, but a mighty one, that a description is
_absolutely true_ of what it describes. Anything for which
a description is false is something which it does not describe.

----
Consider the sequence 0,1,2,3,4,... of what seems to be
the natural numbers.

We can attempt to describe the things that are in 0,1,2,3,4,...
We can make statements about these things so obvious that
_all reasonable people will agree_ to them. Still, as high as
that standard is, that's probably not absolute.

Here are some _very reasonable_ claims about a thing k
in 0,1,2,3,4,...

For a thing k, one and only thing k+1 follows immediately.
k+1 does not follow immediately any thing other than k.

For a thing k, there is a finite linear sequence 0,1,...,k.
Every sub-sequence of a finite linear sequence is a
finite linear sequence.
Every finite linear sequence either contains a first entry and
a last entry, or it is the empty sequence.

We are able to reason from these claims to some further claim
using only _truth-preserving_ rules. By "truth-preserving",
I mean that, if the claims we start with are true, then the
claim we end with cannot be false. (This deserves more attention.)

One example of such a further claim is that there is no pair
of things (intended to be natural numbers) which are in the ratio
of sqrt(2). Historically, this is not obviously true. In fact,
I am told that the cult of Pythagorus considered it obviously
false.

----
We only know it as _very reasonable_ that each natural number k
has its k+1 and its 0,1,...,k.

*However* we know it as _absolutely true_ that each thing k
described as having its k+1 and its 0,1,...,k
has its k+1 and its 0,1,...,k.
And, because we use only truth-preserving rules, we also know it
as _absolutely true_ that there is no pair of _these things_
in the ratio of sqrt(2), even though it flies in the face of
what _seems_ obvious to us.

----
Here we are with two descriptions.

The things in 0,1,2,3,4,...

The things, each k of which has its k+1 and its 0,1,...,k.

Although we might have originally intended the second to be
a description of the first, the second is a much more useful
description, without that vague '...' hand-waving toward
infinity. What we see happening is the second description
supplanting the first description as _what we mean_ by
"natural numbers".

Maybe, once upon a time, "each k having its k+1 and its
0,1,...,k" was a _very reasonable_ description of the natural
numbers in 0,1,2,3,4,... However, it seem to me that, today,
it is more accurate to swap their places and say that
"0,1,2,3,4..." is a _very reasonable_ description of the
natural numbers, each k with its k+1 and its 0,1,...,k.

With that newer (though still pretty old) understanding of
how to _describe_ a natural number, it is _absolutely true_
that there is no pair of natural numbers in the ratio of
sqrt(2).

[Khong Dong]

To be precise: Mathematics deals in _interpretation_ via well-formed-formulas.

>
> It is a small truth, but a mighty one, that a description is
> _absolutely true_ of what it describes. Anything for which
> a description is false is something which it does not describe.

So, of the concept "natural numbers", is the arithmetic interpretation of cGC a true or false "description" -- to you and mathematicians -- as we speak?

And if you and no mathematician can answer, what makes you think cGC doesn't describe (an already proven) truth-relativity?

[Khong Dong]

To be succinct, what makes you think the concept "natural numbers" doesn't _intrinsically_ harbor -- require -- a notion of absolute truth-undecidability, which incidentally Gödel tumbled upon but failed to understand the fact?

[WM]

Gus Gassmann schrieb am Dienstag, 12. Januar 2021 um 19:08:07 UTC+1:
> On Tuesday, 12 January 2021 at 08:36:06 UTC-4, WM wrote:
> > Gus Gassmann schrieb am Dienstag, 12. Januar 2021 um 13:05:06 UTC+1:
> > > On Tuesday, 12 January 2021 at 07:30:32 UTC-4, WM wrote:
> >
> > > > > The function |N ∩ [0, n]| / |E ∩ [0, n]| counting the natural and even numbers in the given interval.
> > > > You are really unable to understand that for n > 1 every term of the potentially infinite sequence and its limit are precisely defined by the above formula? Here are some examples:
> > > > |N ∩ [0, 1]| / |E ∩ [0, 1]| = 1/0 = undefined
> > > > |N ∩ [0, 2]| / |E ∩ [0, 2]| = 2/1 = 2
> > > > |N ∩ [0, 3]| / |E ∩ [0, 3]| = 3/1 = 3
> > > > |N ∩ [0, 10]| / |E ∩ [0, 10]| = 10/5 = 2
> > > > and so on
> > > Very good! But it does not mean what you seem to think (or at least what you want your students to swallow and regurgitate).
> > >
> > > Try with the set S of squares of integers instead of E. Then lim{n->oo} [ |N ∩ [0, n]| / |S ∩ [0, n]| ] = 0.
> > > Notice anything?
> > I notice that a single wrong result proves a theory wrong.
> That's your theory, my good friend.

That's a fact acknowledged by every academic who deserves this name.

> Everybody else here understands that card is not a continuous function and hence cannot be used to infer limiting behaviour for its arguments.

It is irrelevant whether cardinality is a contininuous function. It is irrelevant what cardinality is at all. Relevant is only that its result is wrong when compared with the result of mathematics. This result is the limit of the sequence 2/1, 4/2, 6/3, ... --> 2.

Regards, WM

[WM]

Ralf Bader schrieb am Dienstag, 12. Januar 2021 um 21:26:43 UTC+1:
> On 01/12/2021 01:25 PM, WM wrote:

> >> According to you omly finitely many epsilon can be chosen.
> I did not write this

You did, including the typing error.

> > It did and does. In above dependence we can insert every k that is
> > not forever dark. More is not necessary (because we cannot be sure
> > that dark numbers are existing at all). This kind of analysis has
> > been introduced way before Cantor when only potential infinity
> > existed. And it has worked well.
> So it is nothing but your whim according to which things have to be
> "individually chosen" and so on

It is simply correct mathematics.

Regards, WM

[Jim Burns]

On 1/12/2021 7:37 PM, Khong Dong wrote:
> On Tuesday, 12 January 2021 at 14:22:58 UTC-7,
> Jim Burns wrote:

>>> The time has passed that we’ve gone astray from this
>>> knowledge boundary, believing wrongly -- if not complacently
>>> --that it’d be just a matter of time before we could prove
>>> the absolute truth or falsehood of any specific mathematical
>>> conjecture we come across, as if we were entitled to such
>>> reasoning privilege.
>>>
>>> The time has come that we should finally go home, to our root
>>> of being finite, to humbly realize that, in so far as reasoning
>>> is an introspective attempt to answer the question of where we
>>> are --what we can possibly know --in the infinite realm of
>>> mathematical abstraction, our mortal deducing and knowing
>>> mathematical truths is only a matter of endowment -
>>> not entitlement.

>> Mathematics deals in _descriptions_
>
> To be precise: Mathematics deals in _interpretation_
> via well-formed-formulas.

The finger pointing at the moon is not the moon itself.

There are some things we can know about the infinite realm of
mathematical abstraction, even though we ourselves are not
infinite. This is something like pointing at the moon
without being the moon. Which is also something we can do.

>> It is a small truth, but a mighty one, that a description is
>> _absolutely true_ of what it describes. Anything for which
>> a description is false is something which it does not describe.
>
> So, of the concept "natural numbers", is the arithmetic
> interpretation of cGC a true or false "description" --
> to you and mathematicians -- as we speak?

You seem to think that I have claimed somewhere that I know
everything about the natural numbers. I haven't.

I claim that we know _some true things_ about the
natural numbers.

For example, here is a small but mighty truth:
Each natural number is a natural number.

We can describe what we mean by "natural number".
More known truths.

( For example...
( A natural number k is followed by its successor k+1.
( The natural numbers that precede k form a finite linear
( sequence 0,1,...,k.

We can reason from previously known truths by
truth-preserving steps to even more known truths.

( For example...
( If P and P -> Q are previously know truths,
( then Q is a know truth.

I do not currently know either cGC or ~cGC.
From what I hear, cGC is not a currently-known truth,
though it may become known to be true in the future.

That would be an odd question coming from anyone else.
My answer mimics the form of an admission, but my
"admission" does not affect anything I've said.

My claim is that there are things we know about the
natural numbers.
~(2 < 2) is one of those things.
cGC is not one of those things.

> And if you and no mathematician can answer, what makes you
> think cGC doesn't describe (an already proven)
> truth-relativity?

To be precise: _you have not proven_ that cGC has different
truth values for different models of the natural numbers.
If I recall correctly, you present two structures of the
_language_ of arithmetic. In one cGC is true, in one cGC
is false. However, the two structures are not _models_ of
the natural numbers. They are irrelevant to the question.

_Other people_ have proven that a wide range of extensions
of the concept of natural numbers are either inconsistent
or have true-but-unprovable statements.

_You_ haven't proved anything like this. Or proved anything
at all, that I have ever seen.

Both these things are true:
There are some things we can prove about the natural numbers.
There are some things we cannot prove about the natural numbers.

[Khong Dong]

On Wednesday, 13 January 2021 at 11:32:51 UTC-7, Jim Burns wrote:
> On 1/12/2021 7:37 PM, Khong Dong wrote:

> > So, of the concept "natural numbers", is the arithmetic
> > interpretation of cGC a true or false "description" --
> > to you and mathematicians -- as we speak?

> You seem to think that I have claimed somewhere that I know
> everything about the natural numbers. I haven't.

It's just a question, Jim.

> Both these things are true:
> There are some things we can prove about the natural numbers.

Right. Like "Two plus two is four".

> There are some things we cannot prove about the natural numbers.

For example ... ?

[Gus Gassmann]

On Wednesday, 13 January 2021 at 13:26:28 UTC-4, WM wrote:
> Gus Gassmann schrieb am Dienstag, 12. Januar 2021 um 19:08:07 UTC+1:
> > On Tuesday, 12 January 2021 at 08:36:06 UTC-4, WM wrote:
> > > Gus Gassmann schrieb am Dienstag, 12. Januar 2021 um 13:05:06 UTC+1:
> > > > On Tuesday, 12 January 2021 at 07:30:32 UTC-4, WM wrote:
> > >
> > > > > > The function |N ∩ [0, n]| / |E ∩ [0, n]| counting the natural and even numbers in the given interval.
> > > > > You are really unable to understand that for n > 1 every term of the potentially infinite sequence and its limit are precisely defined by the above formula? Here are some examples:
> > > > > |N ∩ [0, 1]| / |E ∩ [0, 1]| = 1/0 = undefined
> > > > > |N ∩ [0, 2]| / |E ∩ [0, 2]| = 2/1 = 2
> > > > > |N ∩ [0, 3]| / |E ∩ [0, 3]| = 3/1 = 3
> > > > > |N ∩ [0, 10]| / |E ∩ [0, 10]| = 10/5 = 2
> > > > > and so on
> > > > Very good! But it does not mean what you seem to think (or at least what you want your students to swallow and regurgitate).
> > > >
> > > > Try with the set S of squares of integers instead of E. Then lim{n->oo} [ |N ∩ [0, n]| / |S ∩ [0, n]| ] = 0.
> > > > Notice anything?
> > > I notice that a single wrong result proves a theory wrong.
> > That's your theory, my good friend.
> That's a fact acknowledged by every academic who deserves this name.

I repeat myself. Reading comprehension is not yor strong suit.

> > Everybody else here understands that card is not a continuous function and hence cannot be used to infer limiting behaviour for its arguments.
> It is irrelevant whether cardinality is a contininuous function. It is irrelevant what cardinality is at all. Relevant is only that its result is wrong when compared with the result of mathematics. This result is the limit of the sequence 2/1, 4/2, 6/3, ... --> 2.

Sure. And it tells you dick all about the comparison of card(N) and card(E). And that is the consequence of the discontinuity. My, you are in fine form today.

[WM]

It tells me, and it should tell everybody, that cardinality is not a correct measure.

Regards, WM

[WM]

Jim Burns schrieb am Mittwoch, 13. Januar 2021 um 19:32:51 UTC+1:

> There are some things we can prove about the natural numbers.
> There are some things we cannot prove about the natural numbers.

The formet comprises the fact that there are twice as many integers as even integers. A measure which deviates from this result is wrong, at least it is useless.

Regards, WM

[Gus Gassmann]

On Thursday, 14 January 2021 at 06:54:30 UTC-4, WM wrote:
> Gus Gassmann schrieb am Mittwoch, 13. Januar 2021 um 20:22:54 UTC+1:
> > On Wednesday, 13 January 2021 at 13:26:28 UTC-4, WM wrote:
>
> > > > Everybody else here understands that card is not a continuous function and hence cannot be used to infer limiting behaviour for its arguments.
> > > It is irrelevant whether cardinality is a contininuous function. It is irrelevant what cardinality is at all. Relevant is only that its result is wrong when compared with the result of mathematics. This result is the limit of the sequence 2/1, 4/2, 6/3, ... --> 2.

I am just noticing that that is not even the correct sequence. The correct sequence should be 1/0, 2/1, 3/1, 4/2, ...

> > Sure. And it tells you dick all about the comparison of card(N) and card(E).
> It tells me, and it should tell everybody, that cardinality is not a correct measure.

And yet you use an "incorrect" measure to derive the statement that there are twice as many natural numbers as there are even natural numbers. Does the use of an "incorrect" measure no longer invalidate the inference? (Ex falso quodlibet, and all that.)

And tell me again: What do you make of the fact that lim{n->oo} |S ∩ [0, n]| / |N ∩ [0, n]| = 0, where S is the set of squared naturals? Actually, tell me for the first time, because you ignored the question earlier.

[WM]

Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 12:51:54 UTC+1:
> On Thursday, 14 January 2021 at 06:54:30 UTC-4, WM wrote:
> > Gus Gassmann schrieb am Mittwoch, 13. Januar 2021 um 20:22:54 UTC+1:
> > > On Wednesday, 13 January 2021 at 13:26:28 UTC-4, WM wrote:
> >
> > > > > Everybody else here understands that card is not a continuous function and hence cannot be used to infer limiting behaviour for its arguments.
> > > > It is irrelevant whether cardinality is a contininuous function. It is irrelevant what cardinality is at all. Relevant is only that its result is wrong when compared with the result of mathematics. This result is the limit of the sequence 2/1, 4/2, 6/3, ... --> 2.
> I am just noticing that that is not even the correct sequence. The correct sequence should be 1/0, 2/1, 3/1, 4/2, ...

That depends on the sequence. If you take the sequence that I yesterday used in my lecture, http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT, p. 91 then I am correct.

If you take another sequence, then you can also find many more terms than you have quoted.

> > > Sure. And it tells you dick all about the comparison of card(N) and card(E).
> > It tells me, and it should tell everybody, that cardinality is not a correct measure.
> And yet you use an "incorrect" measure to derive the statement that there are twice as many natural numbers as there are even natural numbers.

I use the correct measure. The limit is 2. That is independent of the step width. I used stepwidth 2, you used 1, another one may use 0.1 or 0.01. The limit will not change. It is 2.

> Does the use of an "incorrect" measure no longer invalidate the inference?

It does. That's why set theory is invalid.

> And tell me again: What do you make of the fact that lim{n->oo} |S ∩ [0, n]| / |N ∩ [0, n]| = 0, where S is the set of squared naturals? Actually, tell me for the first time, because you ignored the question earlier.

I am not obliged to solve this problem. I am only obliged to show my students and other people of comparable intelligence and open-mindedness that set theory is in contradiction with mathematics. But that's not a tragedy, because fortunately “the actual infinite is not required for the mathematics of the physical world.” [S. Feferman, "In the light of logic", Oxford Univ. Press (1998)]. Of course my students know that.

Regards, WM

[FredJeffries]

On Thursday, January 14, 2021 at 4:25:08 AM UTC-8, WM wrote:

> I am not obliged to solve this problem.

This is surely the purest statement of the true scientific attitude.

Every scientist should have it emblazoned on his office door.

Someone should produce posters with cute kittens and puppies espousing it.

[Jim Burns]

On 1/13/2021 1:58 PM, Khong Dong wrote:
> On Wednesday, 13 January 2021 at 11:32:51 UTC-7,
> Jim Burns wrote:
>> On 1/12/2021 7:37 PM, Khong Dong wrote:

>>> So, of the concept "natural numbers", is the arithmetic
>>> interpretation of cGC a true or false "description" --
>>> to you and mathematicians -- as we speak?
>
>> You seem to think that I have claimed somewhere that I know
>> everything about the natural numbers. I haven't.
>
> It's just a question, Jim.

You have, in the past, claimed that you prove that cGC
is undecidable. You have not proved that.

You have also, in the past, drawn conclusions that would not
be justified, even if you had proved that cGC was undecidable,
by over-generalizing the (alleged) undecidedness of cGC.

Should I have waited for you to repeat the incorrect claims
you've made in the past before telling you they're incorrect?

>> Both these things are true:
>> There are some things we can prove about the natural numbers.
>
> Right. Like "Two plus two is four".

Or like ~(2 < 2).

>> There are some things we cannot prove about the
>> natural numbers.
>
> For example ... ?

My point is that NOT proving one thing about the natural numbers
won't change that we HAVE proved a different thing about the
natural numbers.

Do you agree that the proof of ~(2 < 2) stays valid
with or without an example of a true-but-unprovable
statement? If you like, we can review the proof of ~(2 < 2).

Let's circle back to cGC. You:

>>> So, of the concept "natural numbers", is the arithmetic
>>> interpretation of cGC a true or false "description" --
>>> to you and mathematicians -- as we speak?

I don't know. And it doesn't matter that I don't know
so far as it concerns anything I've said to you.

[Gus Gassmann]

On Thursday, 14 January 2021 at 08:25:08 UTC-4, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 12:51:54 UTC+1:
> > On Thursday, 14 January 2021 at 06:54:30 UTC-4, WM wrote:
> > > Gus Gassmann schrieb am Mittwoch, 13. Januar 2021 um 20:22:54 UTC+1:
> > > > On Wednesday, 13 January 2021 at 13:26:28 UTC-4, WM wrote:
> > >
> > > > > > Everybody else here understands that card is not a continuous function and hence cannot be used to infer limiting behaviour for its arguments.
> > > > > It is irrelevant whether cardinality is a contininuous function. It is irrelevant what cardinality is at all. Relevant is only that its result is wrong when compared with the result of mathematics. This result is the limit of the sequence 2/1, 4/2, 6/3, ... --> 2.
> > I am just noticing that that is not even the correct sequence. The correct sequence should be 1/0, 2/1, 3/1, 4/2, ...
> That depends on the sequence. If you take the sequence that I yesterday used in my lecture, http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT, p. 91 then I am correct.

Bullshit! You don't even know how to read your own expression, which was Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2. Even that required me to point out that "card" was needed to make sense of the expression. Nothing in this suggests that n can only be even. And no, I was not at your lecture yesterday (thankfully!), so how the fuck should I know or care what you said then?

[Jim Burns]

By the same argument, there are no square integers.
This clarifies what you (WM) mean by "useful" and "useless".

----
There are some things we can prove, and there are
some things we cannot prove.

We can prove that every thing in the sequence 0,1,2,...
is a thing in the sequence 0,1,2,...

We can describe an indefinite thing-in-the-sequence k.
Thing k is followed immediately by thing k+1.
Thing k is connected to 0 by a finite linear sequence 0,1,...,k.

Having seen the two descriptions,
-- thing k in sequence 0,1,2,...
-- thing k with k+1 and 0,1,...,k
we can decide that the second is the better description of

what we mean by "natural number".

If what we mean by "natural number" is
"thing with k+1 and 0,1,...,k"
then we can prove that every natural number k is
a thing with k+1 and 0,1,...,k.

And we can prove quite a lot about natural numbers from that.

[Khong Dong]

On Thursday, 14 January 2021 at 09:58:26 UTC-7, Jim Burns wrote:
> On 1/13/2021 1:58 PM, Khong Dong wrote:
> > On Wednesday, 13 January 2021 at 11:32:51 UTC-7,
> > Jim Burns wrote:
> >> On 1/12/2021 7:37 PM, Khong Dong wrote:
>
> >>> So, of the concept "natural numbers", is the arithmetic
> >>> interpretation of cGC a true or false "description" --
> >>> to you and mathematicians -- as we speak?
> >
> >> You seem to think that I have claimed somewhere that I know
> >> everything about the natural numbers. I haven't.
> >
> > It's just a question, Jim.
> You have, in the past, claimed that you prove that cGC
> is undecidable. You have not proved that.

I did in my quantum_Mathematics.pdf: you and the others (Rupert, Peter) failed to understand even just an introduction part like Section 2.2.

[Ralf Bader]

Let S comprise all and only those natural numbers which have only the
digits 0 and 1 in their decimal expansion. Then your limit is 0 again.
On the other hand, if those decimal expansions are viewed as binary
expansions, then all natural numbers are represented. So are there as
many binary expansions as there are natural numbers as there are decimal
expansions, or are the binary expansions some kind of infinitesimal
fraction of the decimal expansions? And no set theory involved.

I am sure that you either refuse to see the problem, or explain that you
are not obliged to solve it, or that you "solve" it by babbling some
nonsense about "dark numbers". Because for mathematics you are too dumb
and stupid.

[WM]

FredJeffries schrieb am Donnerstag, 14. Januar 2021 um 17:39:40 UTC+1:
> On Thursday, January 14, 2021 at 4:25:08 AM UTC-8, WM wrote:
>
> > I am not obliged to solve this problem.
> This is surely the purest statement of the true scientific attitude.
>
> Every scientist should have it emblazoned on his office door.

Every set theorists has it secretly in his desk: There are nonmeasurable sets.

Regards, WM

[WM]

Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 18:51:16 UTC+1:
>

> Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2. Even that required me to point out that "card" was needed to make sense of the expression.

No, that is a wrong assertion. I simply count the natural numbers and the even natural numbers in the intervals
(0, 2], (0, 4], (0, 6], ...
or in the intervals
(0, 1.1], (0, 0.2], (0, 0.3], ...
or in any other sequence of intervals increasing from 0 to infinity.
For every such sequence I can obtain a limit, and this limit, surprise, is the same in all cases.

> Nothing in this suggests that n can only be even.

That is really not required. I did not say that your sequence was wrong. But you mistakenly claimed mine wrong.

>>> I am just noticing that that is not even the correct sequence. The correct sequence should be 1/0, 2/1, 3/1, 4/2, ...

>> That depends on the sequence. If you take the sequence that I yesterday used in my lecture,

> And no, I was not at your lecture yesterday

But you could see a very suggestive sequence of pictures here: http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT, p. 91.

> so how the fuck should I know or care what you said then?

The pictures are self-explaining if you use the presentation modus.

Regards, WM

[Peter]

Could you say where quantum_Mathematics.pdf is? No doubt you've already
done so, but I've lost it.

--
When, once, reference was made to a statesman almost universally
recognized as one of the villains of this century, in order to
induce him to a negative judgment, he replied: "My situation is
so different from his, that it is not for me to pass judgment".
Ernst Specker on Paul Bernays

[WM]

Jim Burns schrieb am Donnerstag, 14. Januar 2021 um 20:29:15 UTC+1:
> On 1/14/2021 5:56 AM, WM wrote:
> > Jim Burns schrieb
> > am Mittwoch, 13. Januar 2021 um 19:32:51 UTC+1:
>
> >> There are some things we can prove about the
> >> natural numbers.
> >> There are some things we cannot prove about the
> >> natural numbers.
> >

> > The former comprises the fact that there are twice as many

> > integers as even integers. A measure which deviates from
> > this result is wrong, at least it is useless.
> By the same argument, there are no square integers.

No-one is obliged to find the share of square integers. True is that it falls below every positive fraction. But if we had to decide whether their share is 1 or 0, then the latter comes closer to reality than the former.

> This clarifies what you (WM) mean by "useful" and "useless".

To determine the share 0 is certainly more useful than the statement that every natural number is a square.

>
> ----
> There are some things we can prove, and there are
> some things we cannot prove.
>
> We can prove that every thing in the sequence 0,1,2,...
> is a thing in the sequence 0,1,2,...
>
> We can describe an indefinite thing-in-the-sequence k.
> Thing k is followed immediately by thing k+1.
> Thing k is connected to 0 by a finite linear sequence 0,1,...,k.

That makes k defined. But we have the theorem that every defined k is followed by infinitely many undefined natnumbers among which there are infinitely many undefinable natnumbers which re required to satisfy the theorem.

>
> Having seen the two descriptions,
> -- thing k in sequence 0,1,2,...
> -- thing k with k+1 and 0,1,...,k
> we can decide that the second is the better description of
> what we mean by "natural number".
> If what we mean by "natural number" is
> "thing with k+1 and 0,1,...,k"
> then we can prove that every natural number k is
> a thing with k+1 and 0,1,...,k.
>
> And we can prove quite a lot about natural numbers from that.

In particular we can prove that, if Cantor was right and omega - k = omega, every k is followed by infinitely many undefined natnumbers among which there are infinitely many undefinable natnumbers which are required to satisfy the theorem for every k. If they could be used up, then the theorem was contradicted.

But the dark numbers supply a wonderful explanation why all sets have same cardinality: Only comparatively few elements of the two sequences can be paired. Most are dark and not in the bijection. Notice that all natnumbers, all even natnumbers, all fractions, all algebraic numbers, all definable real numbers, all prime numbers, all natnumbers divisible by 10^10^10^100000000000000000 and many, maqny further sets are in "bijection".

I think that I have found a wonderful explanation for this provably wrong result.

Regards, WM

[WM]

Ralf Bader schrieb am Freitag, 15. Januar 2021 um 09:12:20 UTC+1:
>
> Let S comprise all and only those natural numbers which have only the
> digits 0 and 1 in their decimal expansion. Then your limit is 0 again.

Yes. But note that I do not claim that my approach is suitable to solve all problems of relative sizes of infinite sets, in particular because either only pot. inf sets exist where no fixed proportions are definable or dark parts exist where the relative share is also impossible to fix unless we can assume very simple relations like "every odd number is followed by an even number and vice versa".

> On the other hand, if those decimal expansions are viewed as binary
> expansions, then all natural numbers are represented.

Yes. Then we have inserted between them a larger and infinitely increasing set of additional elements.

> So are there as
> many binary expansions as there are natural numbers as there are decimal
> expansions, or are the binary expansions some kind of infinitesimal
> fraction of the decimal expansions? And no set theory involved.
>

Set theory is involved as soon as you assume that the sets are complete.

For potentially infinite sets the relative shares of square numbers or decimal representations with only digits 0 and 1 are decreasing and converging to zero. That is certainly a better result than Cantor's, where all natnumbers, all even natnumbers, all fractions, all algebraic numbers, all definable real numbers, all prime numbers, all natnumbers divisible by 10^10^10^100000000000000000 and many, many further sets are equinumerous.

In short: I have no tool to measure the relative sizes of arbitrary infinite sets, but I have the tool to measure the sizes of even and odd and all natnumbers because the results are constant for all sufficiently large intervals and would not change for dark numbers, because the successor of different sign is simply a property of every natural number.

Regards, WM

[Gus Gassmann]

On Friday, 15 January 2021 at 07:35:42 UTC-4, WM wrote:
> Gus Gassmann schrieb am Donnerstag, 14. Januar 2021 um 18:51:16 UTC+1:
> >
>
> > Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2. Even that required me to point out that "card" was needed to make sense of the expression.
> No, that is a wrong assertion. I simply count the natural numbers and the even natural numbers in the intervals
> (0, 2], (0, 4], (0, 6], ...
> or in the intervals
> (0, 1.1], (0, 0.2], (0, 0.3], ...
> or in any other sequence of intervals increasing from 0 to infinity.
> For every such sequence I can obtain a limit, and this limit, surprise, is the same in all cases.
> > Nothing in this suggests that n can only be even.
> That is really not required. I did not say that your sequence was wrong. But you mistakenly claimed mine wrong.

No mistake about it, just sloppy notation on your part. Par for the course. Equally predictable that you can't admit your shortcomings.

[WM]

Gus Gassmann schrieb am Freitag, 15. Januar 2021 um 14:04:39 UTC+1:
> On Friday, 15 January 2021 at 07:35:42 UTC-4, WM wrote:

> > That is really not required. I did not say that your sequence was wrong. But you mistakenly claimed mine wrong.
> No mistake about it, just sloppy notation on your part.

No, there is lack of understanding on your part. You said:

> > >>> I am just noticing that that is not even the correct sequence. The correct sequence should be 1/0, 2/1, 3/1, 4/2, ...

That is nonsense. There is no "correct sequence". There are many equally correct sequences.

Regards, WM

[Ralf Bader]

On 01/15/2021 01:48 PM, WM wrote:

idiotic nonsense

[Jim Burns]

On 1/15/2021 6:56 AM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 14. Januar 2021 um 20:29:15 UTC+1:
>> On 1/14/2021 5:56 AM, WM wrote:
>>> Jim Burns schrieb
>>> am Mittwoch, 13. Januar 2021 um 19:32:51 UTC+1:

>>>> There are some things we can prove about the
>>>> natural numbers.
>>>> There are some things we cannot prove about the
>>>> natural numbers.
>>>
>>> The former comprises the fact that there are twice as many
>>> integers as even integers. A measure which deviates from
>>> this result is wrong, at least it is useless.
>>
>> By the same argument, there are no square integers.
>
> No-one is obliged to find the share of square integers.

No-one is obliged to respect your argument.

That's the *point* of "no square integers".
Your argument does not always give a correct answer.
Apparently, that doesn't bother you.

Mathematics gives a description, then it reasons from
the description to new statements about what's described.

The only reasoning steps permitted are those which, when
starting from true statements, cannot end on false statements.
_Truth-preserving_ steps.

Every statement that is part of a description _must be_
true of what it describes. Because _description_

Every statement that is reasoned to from a description
*also* _must be_ true of what the description describes.
Because _truth-preserving_

This is a central feature of mathematical reasoning:
its results _must be_ true of what is described.
This obliges rational beings to respect those results.

( Note that its results _must be_ true of infinitely-many
( unexamined natural numbers, and they _must be_ true
( even if they contradict our intuition. The "errors"
( you imagine in "matheology" are its greatest triumphs.

No-one is obliged to respect your argument, if it sometimes
gives incorrect results. As it does. In at least this way,
your argument is different from mathematics.

[Khong Dong]

On Friday, 15 January 2021 at 04:49:20 UTC-7, Peter wrote:
> Khong Dong wrote:
> > On Thursday, 14 January 2021 at 09:58:26 UTC-7, Jim Burns wrote:
> >> On 1/13/2021 1:58 PM, Khong Dong wrote:
> >>> On Wednesday, 13 January 2021 at 11:32:51 UTC-7,
> >>> Jim Burns wrote:
> >>>> On 1/12/2021 7:37 PM, Khong Dong wrote:
> >>
> >>>>> So, of the concept "natural numbers", is the arithmetic
> >>>>> interpretation of cGC a true or false "description" --
> >>>>> to you and mathematicians -- as we speak?
> >>>
> >>>> You seem to think that I have claimed somewhere that I know
> >>>> everything about the natural numbers. I haven't.
> >>>
> >>> It's just a question, Jim.
> >> You have, in the past, claimed that you prove that cGC
> >> is undecidable. You have not proved that.
> >
> > I did in my quantum_Mathematics.pdf: you and the others (Rupert, Peter) failed to understand even just an introduction part like Section 2.2.
> >
> Could you say where quantum_Mathematics.pdf is? No doubt you've already
> done so, but I've lost it.

https://drive.google.com/file/d/1a8CvDCwv3Q_x1sgRiQGfZHOZqr1tug6F/view?usp=sharing

[WM]

Jim Burns schrieb am Freitag, 15. Januar 2021 um 20:41:50 UTC+1:
> On 1/15/2021 6:56 AM, WM wrote:
> > Jim Burns schrieb
> > am Donnerstag, 14. Januar 2021 um 20:29:15 UTC+1:
> >> On 1/14/2021 5:56 AM, WM wrote:
> >>> Jim Burns schrieb
> >>> am Mittwoch, 13. Januar 2021 um 19:32:51 UTC+1:
>
> >>>> There are some things we can prove about the
> >>>> natural numbers.
> >>>> There are some things we cannot prove about the
> >>>> natural numbers.
> >>>
> >>> The former comprises the fact that there are twice as many
> >>> integers as even integers. A measure which deviates from
> >>> this result is wrong, at least it is useless.
> >>
> >> By the same argument, there are no square integers.
> >
> > No-one is obliged to find the share of square integers.
> No-one is obliged to respect your argument.

If you don't respect mathematics, you may continue to adhere to matheology.
If you respect mathematics, then you are obliged to agree that the sequence
2/1, 4/2, 6/3, ... converges against limit 2.

>
> Your argument does not always give a correct answer.
> Apparently, that doesn't bother you.

In order to disprove a theory, one counterexample suffices.

>
> Mathematics gives a description, then it reasons from
> the description to new statements about what's described.
>
> The only reasoning steps permitted are those which, when
> starting from true statements, cannot end on false statements.
> _Truth-preserving_ steps.

This is the sequence above and its analytical limit.

>
> Every statement that is part of a description _must be_
> true of what it describes. Because _description_

Cantor's statement of equinumerosity of |N and |E is wrong. Note that bijection implies equinumerosity according to its definition.

>
> ( Note that its results _must be_ true of infinitely-many
> ( unexamined natural numbers, and they _must be_ true
> ( even if they contradict our intuition.

But they cannot be true if they contradict mathematics.

>
> No-one is obliged to respect your argument, if it sometimes
> gives incorrect results.

My argument does not give incorrect results. My theory is restricted to simple cases. Because of the existence of dark numbers its range of validity is restricted to simple properties which are satisfied even by dark numbers, like change of parity from even to odd to even ...

Regards, WM

[Dan Christensen]

On Saturday, January 9, 2021 at 5:05:29 PM UTC-5, WM wrote:
> Dan Christensen schrieb am Samstag, 9. Januar 2021 um 20:31:30 UTC+1:
> > On Saturday, January 9, 2021 at 4:43:11 AM UTC-5, WM wrote:
>
> > > There are twice as many natural numbers than positive even numbers.
> > >
> > In your algebra course, after you learn about variables, you will learn about functions and, in particular, about bijective functions. Then you will learn that, there is a bijective function f mapping N to E (1-to-1 and onto), such that f(x)=2x.
> This is taught by teachers (like myself in earlier times, cp. W. Mückenheim: "Mathematik für die ersten Semester", 4th ed., De Gruyter, Berlin (2015)) who have not yet discovered dark numbers.
>
> Infinite sets comprise undefinable elements. There is no bijection between undefinable elements
>

If the function f: N --> E such that f(x)=2x is a bijection (as above). Do you deny it?

We have:

f(0)=0
f(1)=2
f(2)=4
f(3)=6
...

The inverse of f is g: E --> N such that g(x)=x/2. We have:

g(0)=0
g(2)=1
g(4)=2
g(6)=3
...

Do you deny it?

> Simple example: Try to define as many unit fractions as possible. Remove them from (0, 1]. Set theory asserts that Infinitely many unit fractions will remain. But if you remove them all, then nothing remains. That means all is more than all definable. The difference is called dark.
>

As we have already seen here, your "dark numbers" seem to be a result of your ignorance of basic high-school math, specifically, the notion of a variable.

*********************************************************************

More absurd quotes from Wolfgang Muckenheim (WM):

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

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

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

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

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

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

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

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


Slipping ever more deeply into madness...

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

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

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

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

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

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

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

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

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


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

[Jim Burns]

On 1/16/2021 8:02 AM, WM wrote:

> My argument does not give incorrect results.

> My theory is restricted to simple cases....

You restrict your theory to simple cases in order to avoid
answering for incorrect results like "no square numbers".

If anyone were serious about using your theory, including you,
they would need to include the clause
| This will work, unless it doesn't work.

This doesn't _save_ your theory, as you seem to think.
It tosses it in the garbage can.
What use is a theory that works unless it doesn't?

Note that it is _your own hand_ with your "simple cases only"
that tosses your theory in the garbage can.

> Because of the existence of dark numbers its range of
> validity is restricted to simple properties which are
> satisfied even by dark numbers, like change of parity
> from even to odd to even

So successors are simple enough for you.

What about addition, if we define it from successors?
x + 0 = x
x + Sy = S(x + y)

If addition is simple enough, what about multiplication,
if we define it from successors and multiplication?
x*0 = 0
x*Sy = x*y + x

But, if multiplication is simple enough,
how can squares be too complex?

( I am reminded of an Albert Einstein quote:
( | Everything should be made as simple as possible,
( | but no simpler.

Here's a simple property:
having a finite linear sequence of predecessors.

For each k, there is a finite linear sequence 0,1,...,k

But it gives answers you don't like.
This, then, is what you mean by "simple enough":
it gives answers you like.
Anything you don't like "contradicts mathematics".

https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect
Dunning–Kruger effect

[WM]

Jim Burns schrieb am Samstag, 16. Januar 2021 um 19:18:29 UTC+1:
> On 1/16/2021 8:02 AM, WM wrote:
>
> > My argument does not give incorrect results.
> > My theory is restricted to simple cases....
>
> You restrict your theory to simple cases in order to avoid
> answering for incorrect results like "no square numbers".

I restrict my arguing (it is not my theory, but classical mathematics) to simple cases because so I can provide reliable results.

>
> If anyone were serious about using your theory, including you,

Every mathematician should be serious about the fact that there are twice as many natnumbers as even natnumbers. This is the limit 2 supplied by mathematics for every interval-width (0, k_n). The limit 1 however can be excluded.

> they would need to include the clause
> | This will work, unless it doesn't work.

Attach this label to set theory: This theory will never work unless by improbable accident.

>
> This doesn't _save_ your theory, as you seem to think.
> It tosses it in the garbage can.
> What use is a theory that works unless it doesn't?

Not every sequence has a limit in analysis. Nevertheless mathematicians accept limits that can be proved to exist by the well-known criteria.

>
> Note that it is _your own hand_ with your "simple cases only"
> that tosses your theory in the garbage can.

Sorry, that is not "my theory", it is simply an applications of classical mathematics.

> > Because of the existence of dark numbers its range of
> > validity is restricted to simple properties which are
> > satisfied even by dark numbers, like change of parity
> > from even to odd to even
> So successors are simple enough for you.

If dark natnumbers exist, then they have the properties of natnumbers. That includes neighbours of different parity. But even if dark numbers do not exist, the limit 2 is the only correct result.

>
> What about addition, if we define it from successors?
> x + 0 = x
> x + Sy = S(x + y)

It works for every definable x and y.

>
> If addition is simple enough, what about multiplication,
> if we define it from successors and multiplication?
> x*0 = 0
> x*Sy = x*y + x
>
> But, if multiplication is simple enough,
> how can squares be too complex?

All that works for every definable x and y.

>
> ( I am reminded of an Albert Einstein quote:
> ( | Everything should be made as simple as possible,
> ( | but no simpler.
>
> Here's a simple property:
> having a finite linear sequence of predecessors.

That works for every definable natnumber.

>
> For each k, there is a finite linear sequence 0,1,...,k
>
> But it gives answers you don't like.

No it is true for every definable k. But every definable k is followed by aleph_0 natnumbers, not all of which can be definable, because if all were definable, then they could become defined and then none would remain as successors.

> This, then, is what you mean by "simple enough":
> it gives answers you like.
> Anything you don't like "contradicts mathematics".

Contrary to you I can prove my case by the most convincing proof. It is 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]

Dan Christensen schrieb am Samstag, 16. Januar 2021 um 16:30:43 UTC+1:
> On Saturday, January 9, 2021 at 5:05:29 PM UTC-5, WM wrote:

> > Infinite sets comprise undefinable elements. There is no bijection between undefinable elements
> >
> If the function f: N --> E such that f(x)=2x is a bijection (as above). Do you deny it?

It holds for all definable elements, but not for dark elements.

I can prove the existence of dark elements by the most convincing proof. It is the inability of anyone and any program to identify unit fractions in the interval (0, 1/n) such that less than aleph_0 unit fractions remain not identified.

Regards, WM

[Dan Christensen]

On Sunday, January 17, 2021 at 5:58:13 AM UTC-5, WM wrote:
> Dan Christensen schrieb am Samstag, 16. Januar 2021 um 16:30:43 UTC+1:
> > On Saturday, January 9, 2021 at 5:05:29 PM UTC-5, WM wrote:
>
> > > Infinite sets comprise undefinable elements. There is no bijection between undefinable elements
> > >
> > If the function f: N --> E such that f(x)=2x is a bijection (as above). Do you deny it?
> It holds for all definable elements, but not for dark elements.
>

Wrong again, Mucke. No matter how you classify them, we have this bijection mapping EVERY natural number x to an even number 2x. And its inverse maps every even number y to a natural number y/2. Deal with it!

> I can prove the existence of dark elements by the most convincing proof. It is the inability of anyone and any program to identify unit fractions in the interval (0, 1/n) such that less than aleph_0 unit fractions remain not identified.
>

I have done so repeatedly. The set of unit fractions in (0, 1/n) for ALL n in N is just:

{1/(n+1), 1/(n+2), 1/(n+3), ... }

Or equivalently

{1/(n+x) : x in N+}

where each element can be uniquely identified by the value of x.

Yes, I know, you probably STILL don't understand what a variable is, but everyone here who has completed elementary school will know. Maybe you can take a night-school class to catch up.

********************************************************************


More absurd quotes from Wolfgang Muckenheim (WM, aka Mucke):

"There are twice as many natural numbers than positive even numbers."

--sci.math 2021/01/09

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

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

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

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

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

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

"No set is countable, not even |N."

[Jim Burns]

On 1/17/2021 5:53 AM, WM wrote:
> Jim Burns schrieb
> am Samstag, 16. Januar 2021 um 19:18:29 UTC+1:
>> On 1/16/2021 8:02 AM, WM wrote:

>>> My argument does not give incorrect results.
>>> My theory is restricted to simple cases....
>>
>> You restrict your theory to simple cases in order to avoid
>> answering for incorrect results like "no square numbers".
>
> I restrict my arguing
> (it is not my theory, but classical mathematics)

In mathematics, if truth-preserving inferences disagree with
intuition, intuition is set aside.

In your theory, if truth-preserving inferences disagree with
intuition, the inferences are set aside.

This is a pretty important difference.

> to simple cases because so I can provide reliable results.

We aren't disagreeing on this point.
We differ on whether bad results, when ignored, don't matter.

>>> Because of the existence of dark numbers its range of
>>> validity is restricted to simple properties which are
>>> satisfied even by dark numbers, like change of parity
>>> from even to odd to even

...

>> Here's a simple property:
>> having a finite linear sequence of predecessors.
>
> That works for every definable natnumber.

That works for every thing *having a finite linear sequence*
*of predecessors* It doesn't matter what you call it.

I'll call it a finlinsop. A finlinsop is a thing *having*
*a finite linear sequence of predecessors*

Look at 0,1,2,3,4,...
_As far as we can see_ it is composed of finlinsops.
_Everything we ever use_ in 0,1,2,3,4... is a finlinsop.
Therefore, we restrict our attention to the finlinsops in
0,1,2,3,4,...

We can reason about *all* the infinitely-many finlinsops.
We do not define each of them individually, or look at
each of them individually, or interact in any way with
each of them individually.

But we know that each finlinsop is a finlinsop, and thus
it has a finite linear sequence of predecessors.

And we know some of the consequences of having a finite linear
sequence of predecessors. We know those consequences even if
they disagree with our intuition. Even if you don't like it.

When we talk about natural numbers, we are talking about
finlinsops. It doesn't matter what you have to say about it.
We know what we are talking about.

>> For each k, there is a finite linear sequence 0,1,...,k
>>
>> But it gives answers you don't like.
>
> No it is true for every definable k.
> But every definable k is followed by aleph_0 natnumbers,
> not all of which can be definable,

No, they all can.

> because if all were definable,
> then they could become defined

Yes, they can.

> and then none would remain as successors.

No, and it has nothing to do with your dark numbers.

Each finlinsop natural is followed by aleph_0 finlinsop
naturals. Throwing non-finlinsop numbers into the mix won't
produce a finlinsop natural *without* aleph_0 finlinsop
naturals after it.

Piling stones on top of stones will not produce
floating stones.

[Khong Dong]

On Sunday, 17 January 2021 at 12:12:49 UTC-7, Jim Burns wrote:
> On 1/17/2021 5:53 AM, WM wrote:
> > Jim Burns schrieb
> > am Samstag, 16. Januar 2021 um 19:18:29 UTC+1:
> >> On 1/16/2021 8:02 AM, WM wrote:
>
> >>> My argument does not give incorrect results.
> >>> My theory is restricted to simple cases....
> >>
> >> You restrict your theory to simple cases in order to avoid
> >> answering for incorrect results like "no square numbers".
> >
> > I restrict my arguing
> > (it is not my theory, but classical mathematics)

> In mathematics, if truth-preserving inferences disagree with
> intuition, intuition is set aside.

So you -- and we all -- should set aside Gödel's intuition of "natural numbers" since there can be _no_ truth-preserving inference for at last one arithmetic sentence, e.g., cGC, to begin with.

[Khong Dong]

Iow, what is Gödel's notion of "natural numbers", when the notion itself would have no clue as to whether or not it'd like to preserve infinitely many arithmetic truths?

[Jim Burns]

On 1/17/2021 2:12 PM, Jim Burns wrote:
> On 1/17/2021 5:53 AM, WM wrote:
>> Jim Burns schrieb
>> am Samstag, 16. Januar 2021 um 19:18:29 UTC+1:
>>> On 1/16/2021 8:02 AM, WM wrote:

>>>> My argument does not give incorrect results.
>>>> My theory is restricted to simple cases....
>>>
>>> You restrict your theory to simple cases in order to avoid
>>> answering for incorrect results like "no square numbers".
>>
>> I restrict my arguing
>> (it is not my theory, but classical mathematics)
>
> In mathematics, if truth-preserving inferences disagree with
> intuition, intuition is set aside.

|

| He was 40 yeares old before he looked on Geometry; which
| happened accidentally. Being in a Gentleman's Library,
| Euclid's Elements lay open, and 'twas the 47 El. Libri 1
| [Pythagoras' Theorem]. He read the proposition. By G-,
| sayd he (he would now and then sweare an emphaticall Oath
| by way of emphasis) this is impossible! So he reads the
| Demonstration of it, which referred him back to such a
| Proposition; which proposition he read. That referred him
| back to another, which he also read. Et sic deinceps
| [and so on] that at last he was demonstratively convinced
| of that trueth. This made him in love with Geometry .
|
"Of Thomas Hobbes, in 1629" -- John Aubrey

> In your theory, if truth-preserving inferences disagree with
> intuition, the inferences are set aside.

I'll have to settle for a paraphrase. You (more or less):
"I stopped reading at your first error"
where "error" includes disagreeing with your intuition.

> This is a pretty important difference.

The little story is NOT a matter of Hobbes CHOOSING to decide
that Euclid must be correct, perhaps because Euclid is very old
and famous. It is a matter of Euclid winning the argument,
across a gap of a few thousand years.

Not everyone is willing to admit that they have lost,
when they have lost. Thomas Hobbes apparently had that
talent naturally. Acquiring that talent is an important
part of what is taught in mathematics classes.

Mathematics is the language of physics.
Physics, if it's any good, has some results that are
counter-intuitive.
Mathematics is capable of arguing for results that counter
intuition.
This is not a proof, exactly, but these facts strike me
as far from coincidental.

[Jim Burns]

On 1/17/2021 2:46 PM, Khong Dong wrote:

> Iow, what is Gödel's notion of "natural numbers",

Goedel says (with the notation updated),

[I 1]
~(S(x1) = 0)

[I 2]
S(x1) = S(y1) -> x1 = y1

[I 3]
( x2(0) & forall x1, x2(x1) -> x2(x1) ) -> forall x1, x2(x1)

Any notion of "natural numbers" that agrees with
axioms I, parts 1,2,3 is sufficiently similar to Goedel's
notion for his argument to apply.

> when the notion itself would have no clue as to
> whether or not it'd like to preserve infinitely many
> arithmetic truths?

This is some pretty intense anthropomorphization going on here.

According to my best guess at what you mean,
what the notion would like has no effect on the
preservation of truth. It is a thing that is so.

Mathematics deals in descriptions.
Certain inferences (which we describe) preserve truth.
It is so, for them.
There are arguments by which we can see that it is so.

[Khong Dong]

On Sunday, 17 January 2021 at 13:43:01 UTC-7, Jim Burns wrote:
> On 1/17/2021 2:46 PM, Khong Dong wrote:
>
> > Iow, what is Gödel's notion of "natural numbers",
> Goedel says (with the notation updated),
>
> [I 1]
> ~(S(x1) = 0)
>
> [I 2]
> S(x1) = S(y1) -> x1 = y1
>
> [I 3]
> ( x2(0) & forall x1, x2(x1) -> x2(x1) ) -> forall x1, x2(x1)

So how do all that formalization, inferences "preserve" -- your word -- the _arithmetic truth_ that exists between cGC and its negation?

[WM]

Jim Burns schrieb am Sonntag, 17. Januar 2021 um 21:18:22 UTC+1:

> Mathematics is the language of physics.

Set theory is not the language of physics. There are more fractions in (1, oo) than in (0, 1) because there are as many in (2, 3) as in (3, 4).

If you were not severly blinded by set theory, you would immediately see it.

> Physics, if it's any good, has some results that are
> counter-intuitive.

That is not an excuse for believing that mathematics is wrong.

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

is indispensable.

> Mathematics is capable of arguing for results that counter
> intuition.

That is not an excuse to believe obvious shit like the bankrupt of McDuck. Moreover this bankrupt depends on the choice of issued dollars. In science the choice of labels will never change the result. In my recent exam not any single student accepted that nonsense.

> This is not a proof, exactly, but these facts strike me
> as far from coincidental.

Put Cantor instead of Euclid and Burns instead of Hobbes.

Regards, WM

[WM]

Jim Burns schrieb am Sonntag, 17. Januar 2021 um 20:12:49 UTC+1:
> On 1/17/2021 5:53 AM, WM wrote:

> In mathematics, if truth-preserving inferences disagree with
> intuition, intuition is set aside.

Therefore we use well-defined tools, for instance:

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

> In your theory, if truth-preserving inferences disagree with
> intuition, the inferences are set aside.

There is no intuition but hard scientific facts that every bijection will remain a bijection under any permutation of the involved sets.

For finite sets a bijection shows equinumerosity and remains a bijection under every permutation.
For infinite sets the latter fails. Why should thw former remain?

>
> > to simple cases because so I can provide reliable results.
> We aren't disagreeing on this point.

But you claim that a completely different result would not show a contradiction.

> We differ on whether bad results, when ignored, don't matter.

What bad results? With increasing interval (0, n) the relative abundance of square numbers or prime numbers decreases beyond every positive epsilon.

> >> Here's a simple property:
> >> having a finite linear sequence of predecessors.
> >
> > That works for every definable natnumber.
> That works for every thing *having a finite linear sequence*
> *of predecessors* It doesn't matter what you call it.

Fine. But I can call it definable natnumber.

>
> We can reason about *all* the infinitely-many finlinsops.

Hardly. There is no completion. It is a potentially infinite collection.

>
> > But every definable k is followed by aleph_0 natnumbers,
> > not all of which can be definable,
> No, they all can.

Ask any intelligent person not yet infected by matheology whether they can identify all unit fractions in (0, 1/n) such that only (0, 0) remains unidentified.

Regards, WM

[Peter]

Thank you.

[Carmello Fumero-Diaz]

Dan Christensen <Dan_Chr...@sympatico.ca> wrote:

>> I can prove the existence of dark elements by the most convincing
>> proof. It is the inability of anyone and any program to identify unit
>> fractions in the interval (0, 1/n) such that less than aleph_0 unit
>> fractions remain not identified.
>
> I have done so repeatedly. The set of unit fractions in (0, 1/n) for ALL
> n in N is just: {1/(n+1), 1/(n+2), 1/(n+3), ... }

Not exactly, you have to have a minimum of *double accuracy* available in
order to count those 10.2299999954326 exactly.

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[Jim Burns]

On 1/17/2021 4:58 PM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 17. Januar 2021 um 21:18:22 UTC+1:

>> Mathematics is the language of physics.
>
> Set theory is not the language of physics.

Bullshit.

What _you_ (WM) mean by "set theory" is _precisely_
the language of physics.

Physics is in the business of reasoning about _what we_
_have not seen_ Does the conservation of energy also hold
in the core of the Sun? Physics says "Yes, even though we
have not seen the core of the Sun."

You (WM) oppose this. You demand that we _see_ numbers if
we are to reason about them. What you oppose, what you call
"set theory" (really, first order logic) is _precisely_ what
physics uses to speak of the conservation of energy
_everywhere_ seen, unseen, never yet seen, or never to be seen.

Take that away, and what do you have left? A catalog of
experimental results. Stamp-collecting.

Say that we want to build a bridge. If we make the girders this
thick and the foundation that deep, will it stay up? You would
insist on seeing it built before we make a "prediction".

Physical theories are generalizations over some collection --
of points of space-time, particles, girders. They use
_indefinite references_ (variables) to points, girders, etc
to describe each of them, seen or unseen. If a generalization
doesn't correctly describe what we _see_ (evidence),
we consider it falsified for what we _don't see_ (predictions).

Without "set theory" (first order logic) and indefinite
references, physics is useless.

> There are more fractions in (1, oo) than in (0, 1)

For each p/q in (1, inf) there is a unique q/p in (0, 1).
And vice versa.

> because there are as many in (2, 3) as in (3, 4).

There are also as many in (2, 3) as in (2, 4).
And as in (2, 5/2)

Here is the part of your proof you consider unnecessary
to write out:
| Cherry-pick results that seem to suggest the pre-selected
| result. Ignore the rest.
| Any flaws that are pointed out are unruly students, and
| must be dealt with as such. Because academic freedom.

[WM]

Jim Burns schrieb am Montag, 18. Januar 2021 um 19:14:53 UTC+1:
> On 1/17/2021 4:58 PM, WM wrote:
> > Jim Burns schrieb
> > am Sonntag, 17. Januar 2021 um 21:18:22 UTC+1:
>
> >> Mathematics is the language of physics.
> >
> > Set theory is not the language of physics.
> Bullshit.
>
> What _you_ (WM) mean by "set theory" is _precisely_
> the language of physics.

Really? It depends on the choice of labels, what result is occuring?

>
> Physics is in the business of reasoning about _what we_
> _have not seen_ Does the conservation of energy also hold
> in the core of the Sun? Physics says "Yes, even though we
> have not seen the core of the Sun."

Scrooge McDuck daily receives 1000000000000000 $ and issues only 1 $. Set theory yields the limit: bankrupt. BUT ONLY IF HE ISSUES THE OLDEST DOLLARS. Otherwise he will not get bankrupt. Can really any intelligent person recommend this matheologial nonsense for physics?

>
> You (WM) oppose this. You demand that we _see_ numbers if
> we are to reason about them.

The interior of the sun exists whether we see it or not.
A number sa an abstraction of reality does not exist if no-one abstracts it.
You should try to understand this big difference.

> What you oppose, what you call
> "set theory" (really, first order logic) is _precisely_ what
> physics uses to speak of the conservation of energy
> _everywhere_ seen, unseen, never yet seen, or never to be seen.

That is nonsense. Physics uses *logic* for its deductions but no first- or second-order logic. That is simply rubbish.

> Without "set theory" (first order logic) and indefinite
> references, physics is useless.

Obviously you don't know either.

> > There are more fractions in (1, oo) than in (0, 1)
> For each p/q in (1, inf) there is a unique q/p in (0, 1).
> And vice versa.

But each definable p/q is a member of potentially infinite set.

> > because there are as many in (2, 3) as in (3, 4).
> There are also as many in (2, 3) as in (2, 4).
> And as in (2, 5/2)

No. That is nonsense. Using mathematics we have a tool to get a precise result.

Your approach yield always the same result because you biject only a potentially infinite initial segment. Proof by the missing endsegments which are required to empty the set in steps of one natumber per endsegment.

>
> Here is the part of your proof you consider unnecessary
> to write out:
> | Cherry-pick results that seem to suggest the pre-selected
> | result. Ignore the rest.

I don't. 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 the probability to guess a certain number is zero. That does not imply that the number does not exist.

Regards, WM

[Gus Gassmann]

On Monday, 18 January 2021 at 15:52:39 UTC-4, WM wrote:
>[...] 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.

Look, I let you get away with this mistake once, but the second time, it's on you. Lim_{n--> oo} |N ∩ [0, n]| / |P ∩ [0, n]| does not exist. Lim_{n--> oo} |P ∩ [0, n]| / |N ∩ [0, n]| = 0.

> It does not mean that there are no prime numbers.

What then, does it mean to you? After all, *for you* Lim_{n--> oo} |N ∩ [0, n]| / |P ∩ [0, n]| = 2 implies that there are twice as many natural numbers as there are even numbers. What do *you* derive from the fact that Lim_{n--> oo} |P ∩ [0, n]| / |N ∩ [0, n]| = 0?

> Note that the probability to guess a certain number is zero. That does not imply that the number does not exist.

Do you understand that this requires that the cardinality of the set from which you select is infinite? *actually* infinite???

What a BLOODY idiot you are, Muckenheim!

[Dan Christensen]

On Monday, January 18, 2021 at 12:34:20 PM UTC-5, Carmello Fumero-Diaz wrote:

> Dan Christensen <> wrote:
>
> >> I can prove the existence of dark elements by the most convincing
> >> proof. It is the inability of anyone and any program to identify unit
> >> fractions in the interval (0, 1/n) such that less than aleph_0 unit
> >> fractions remain not identified.
> >
> > I have done so repeatedly. The set of unit fractions in (0, 1/n) for ALL
> > n in N is just: {1/(n+1), 1/(n+2), 1/(n+3), ... }
> Not exactly, you have to have a minimum of *double accuracy* available in
> order to count those 10.2299999954326 exactly.
>
> Hacked emails allegedly detail how EU drug regulator was pressured to
> approve Pfizer jab despite ‘problems’ with the vaccine

You and your pal Trump are giving conspiracy nuts a bad name.

Dan

[WM]

Gus Gassmann schrieb am Montag, 18. Januar 2021 um 22:16:04 UTC+1:
> On Monday, 18 January 2021 at 15:52:39 UTC-4, WM wrote:
> >[...] 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.

> Lim_{n--> oo} |N ∩ [0, n]| / |P ∩ [0, n]| does not exist.

There is an improper limit oo.

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

That is true.

> > It does not mean that there are no prime numbers.
> What then, does it mean to you?

It means that the fraction with increasing n grows above every positive value or falls below every positive epsilon.

After all, *for you* Lim_{n--> oo} |N ∩ [0, n]| / |P ∩ [0, n]| = 2 implies that there are twice as many natural numbers as there are even numbers.

No, P is not related to even numbers, on the contrary, it contains only one even number.

But Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2 shows twice as many natural numbers as even numbers not only *for me* but for every mathematician who has not become brain damaged.

> What do *you* derive from the fact that Lim_{n--> oo} |P ∩ [0, n]| / |N ∩ [0, n]| = 0?

The limit 2 means that the quotient is bounded by every +/- eps for sufficiently large n.

The quotient 0 means that the quotient falls below every positive value for sufficient n.

> > Note that the probability to guess a certain number is zero. That does not imply that the number does not exist.

> Do you understand that this requires that the cardinality of the set from which you select is infinite?

The cardinality is nonsense and cannot be required for mathematical arguments.

> *actually* infinite???
>
The limit 0 does not imply that the set is actually infinite. If you try to select a number, you will always select from a finite set.

Regards, WM

[Jim Burns]

On 1/18/2021 2:52 PM, WM wrote:
> Jim Burns schrieb
> am Montag, 18. Januar 2021 um 19:14:53 UTC+1:
>> On 1/17/2021 4:58 PM, WM wrote:
>>> Jim Burns schrieb
>>> am Sonntag, 17. Januar 2021 um 21:18:22 UTC+1:

>>>> Mathematics is the language of physics.
>>>
>>> Set theory is not the language of physics.
>>
>> Bullshit.
>> What _you_ (WM) mean by "set theory" is _precisely_
>> the language of physics.
>
> Really? It depends on the choice of labels,
> what result is occuring?

First order logic, which you don't know is labelled first order
logic, is what physics uses to reason about the unobserved-so-far
portion of our universe (this is called prediction), whatever
label you attach to it.

>> Physics is in the business of reasoning about _what we_
>> _have not seen_ Does the conservation of energy also hold
>> in the core of the Sun? Physics says "Yes, even though we
>> have not seen the core of the Sun."
>
> Scrooge McDuck daily receives 1000000000000000 $ and
> issues only 1 $. Set theory yields the limit: bankrupt.
> BUT ONLY IF HE ISSUES THE OLDEST DOLLARS. Otherwise he
> will not get bankrupt.

Each natural number can be matched with each multiple
of 1e15. This is why we say the infinitely many McDuck dollars
taken one at a time is precisely as many as the infinitely
many McDuck dollars taken 1e15 at a time.

You (WM) would like this to be complicated, but it's not.
Take the decimal numeral for a natrual number. Append
fifteen 0's to its right end. You have the numeral for
one and only one multiple of 1e15.
Take the decimal numeral for a multiple of 1e15. Remove
the fifteen 0'sfrom its right end. you have one and only
one numeral for a natural.
Match.

This is not how finite sets behave, but the set of naturals
is not finite.

> Can really any intelligent person recommend this
> matheologial nonsense for physics?

If we want to describe an (indefinite) thing in 0,1,2,3,4,...
then it is followed immediately by another thing and,
except for 0, it is preceded immediately by another thing.

Each thing in 0,1,2,3,4,... matches to a corresponding thing
in 1,2,3,4,5,... which is a proper subset of 0,1,2,3,4,...
This is what your Scrooge McDuck story comes down to,
matching with a proper subset.

Whether intelligent person should recommend using something
like 0,1,2,3,4,... to describe some physics depends upon
what's being described. If it's an infinite sequence that
needs to be described, then an intelligent person should
recommend something that can be matched with a proper subset.
Because that's what it is.

Your intuition disagrees with truth-preserving inferences
here. Truth-preserving inferences are more reliable.

>> You (WM) oppose this. You demand that we _see_ numbers if
>> we are to reason about them.
>
> The interior of the sun exists whether we see it or not.

Does it exist? How do you know? Have you seen it?

The interior of the Sun is a theoretical construct.
Very close to *all* of what we *think* the universe is
has never been observed by us and will never be observed
by us. The line you want to draw is not where you think it is.

My point is, though, that _we do not see_ the interior of
the Sun. *Nonetheless* we reason about it. Our logic works
just fine on the unseen

> A number sa an abstraction of reality does not exist
> if no-one abstracts it.
> You should try to understand this big difference.

It is a difference that makes no difference.
Numbers exist in the same sense (or do not exist,the same)
before and after they are "abstracted".

We describe a thing -- maybe a natural number is have
a finite linear sequence of predecessors --
We reason from the description to new statements.
The new statements are true of everything described.

Being "abstracted" plays no part.

>> What you oppose, what you call
>> "set theory" (really, first order logic) is _precisely_ what
>> physics uses to speak of the conservation of energy
>> _everywhere_ seen, unseen, never yet seen, or never to be seen.
>
> That is nonsense.
> Physics uses *logic* for its deductions but
> no first- or second-order logic. That is simply rubbish.

?
Maybe someone sometime mentioned to you that
*physics uses variables*

If you don't believe me, ask your colleagues.
PLEASE do that, and tell us how they react.

Don't overlook reminding your colleagues of your academic
freedom to lecture on things you're ignorant of.

>> Here is the part of your proof you consider unnecessary
>> to write out:
>> | Cherry-pick results that seem to suggest the pre-selected
>> | result. Ignore the rest.
>
> I don't. 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.

Neither does the other mean that there are twice as many
naturals as evens.

[Khong Dong]

On Monday, 18 January 2021 at 20:36:06 UTC-7, Jim Burns wrote:
> On 1/18/2021 2:52 PM, WM wrote:
> > Jim Burns schrieb
> > am Montag, 18. Januar 2021 um 19:14:53 UTC+1:
> >> On 1/17/2021 4:58 PM, WM wrote:
> >>> Jim Burns schrieb
> >>> am Sonntag, 17. Januar 2021 um 21:18:22 UTC+1:
>
> >>>> Mathematics is the language of physics.
> >>>
> >>> Set theory is not the language of physics.
> >>
> >> Bullshit.

But see below, Jim.

> >> What _you_ (WM) mean by "set theory" is _precisely_
> >> the language of physics.
> >
> > Really? It depends on the choice of labels,
> > what result is occuring?

> First order logic, which you don't know is labelled first order
> logic, is what physics uses to reason about the unobserved-so-far
> portion of our universe

Wow! So you've, as one with a Degree in Physics, really gone to the unobserved-so-far portion of a blackhole and seen some "First order logic" provability going on there? Any rate, outside the unobserved-so-far portion of that blackhole, _other_ physicists are still using Second Order Logic (e.g., Pi, e, Riemann Hypothesis) to gain their meager physics knowledge -- compared to you!

Good grief!

[WM]

Jim Burns schrieb am Dienstag, 19. Januar 2021 um 04:36:06 UTC+1:
> On 1/18/2021 2:52 PM, WM wrote:

> >> What _you_ (WM) mean by "set theory" is _precisely_
> >> the language of physics.
> >
> > Really? It depends on the choice of labels,
> > what result is occuring?
> First order logic,

*Logic* is applied in physics. *Logic* will never supply but contradict nonsense like McDuck, Tristram Shandy, or same number of farctions in (0, 1) and (0, 2).

> > Scrooge McDuck daily receives 1000000000000000 $ and
> > issues only 1 $. Set theory yields the limit: bankrupt.
> > BUT ONLY IF HE ISSUES THE OLDEST DOLLARS. Otherwise he
> > will not get bankrupt.

> Each natural number can be matched with each multiple
> of 1e15.

It appears so to stupids who are duped by deceivers.
You forget or do not even realize that mathematics proves a different result.

> This is why we say the infinitely many McDuck dollars
> taken one at a time is precisely as many as the infinitely
> many McDuck dollars taken 1e15 at a time.

But you don't realize that you say nonsense. Every child could understand that in (0, 1) there are less fractions than in (0, 2) because the former are a proper subset of the latter. No chance to prove equinumerosity. No chance to violate mathematics.

>
> This is not how finite sets behave, but the set of naturals
> is not finite.

The set of identified and mathematically existing naturals is finite, namely potentially infinite.

> Your intuition disagrees with truth-preserving inferences
> here.

No, you are only unable to understand formal mathematics:

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

and therefore you insult mathematicians applying it.

> > The interior of the sun exists whether we see it or not.
> Does it exist? How do you know? Have you seen it?

I need not see it. That is the advantage of reality.

>
> The interior of the Sun is a theoretical construct.
> Very close to *all* of what we *think* the universe is
> has never been observed by us and will never be observed
> by us.

Reality exists without having been observed. Ideas do not exist without havung been thought. That's the difference.

> > A number as an abstraction of reality does not exist

> > if no-one abstracts it.
> > You should try to understand this big difference.
> It is a difference that makes no difference.
> Numbers exist in the same sense (or do not exist,the same)
> before and after they are "abstracted".

Ideas do not exist without havung been thought.

> *physics uses variables*

and applies logic, but there is no reason to specify different kinds of logic. Do you know a reason why second-order logic is applied? There is none.

> >
> > I don't. 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} |P ∩ [0, n]| / |N ∩ [0, n]| = 0. [Corrected]

> > It does not mean that there are no prime numbers.

No, it means that the share of prime numbers decreases below every positive constant. It is same with the limit 0 of the sequence 1/n. It does not mean that the value zero is obtained, but the size of 1/n decreases below every positive constant. Same result. Same meaning. Both is mathematically correct.

> Neither does the other mean that there are twice as many
> naturals as evens.

Mathematics does show this for every finite interval (with increasing precision). But after all finite intervals, there where no further numbers exist, the share of even numbers will be blown up? How should that happen?

Regards, WM

[Python]

Crank Wolfgang Mueckenheim, aka WM wrote:
...

> But you don't realize that you say nonsense.
> Every child could understand that in (0, 1)
> there are less fractions than in (0, 2) because
> the former are a proper subset of the latter.

Oh, come on, Crank! This was noticed from Aristotle's time that
you can biject between some collections and some of their proper
subsets.

It appeared contradictory, but why would it be? Just because you,
a wannabee failed scientist in shameful Germany failed Academy, do
not like it? Give us a break.

THEN, it was taken as the *positive* definition of infinity by
several of your brilliant colleagues, less stubborn and idiot
than YOU.

It ended up as being a very useful and clever move. Everything
is far more simple and consistent that way, and it leads to NO
contradiction.

As Stanislaw Ulam once said :

"The infinite we shall do right away. The finite may take a little
longer."

> Reality exists without having been observed. Ideas
> do not exist without havung been thought. That's the difference.

Good as the whole collection of, say, natural number has been
thought by billions of people for ages, so it exists according to
your own words.

I've been recently teaching a refreshing course for CS students
about mathematics. I covered set theory and some specific concept
you do have issues with like infinite cardinalities and set theoretical
limits.

Guess what? Students didn't have the issues you do have. I even
mentioned, as a side note, *your* infamous alleged "work", providing a
link to your "book". Those who checked was not convince by your
sophistries, fallacies and lies.

This said a lot that *we*, as teachers, do not act like *you* :

Contrarily to you, who do not have the elementary decency and
courage to show Ben Bacarisse paper to your students, we do not
have any problem at all showing your "work" to students and
colleagues. Quite the contrary actually. The more we'll do that,
the quicker you'll be fired from Hochschule Augsburg and will be
asked to take responsibility for your crimes.

[Peter]

Khong Dong wrote:
> [...] _other_ physicists are still using Second Order Logic (e.g.,

> Pi, e, Riemann Hypothesis) to gain their meager physics knowledge --
> compared to you!

Please give an example of a physicist using second order logic. Also,
what is the connection between (these physicists and their) second order
logic and Pi, e, Riemann Hypothesis?

It is interesting (FSVO) that the Riemann hypothesis can be encoded in
FO arithmetic.

>
> Good grief!

[Gus Gassmann]

On Monday, 18 January 2021 at 17:59:15 UTC-4, WM wrote:
> Gus Gassmann schrieb am Montag, 18. Januar 2021 um 22:16:04 UTC+1:
> > On Monday, 18 January 2021 at 15:52:39 UTC-4, WM wrote:

[...]

> > > Note that the probability to guess a certain number is zero. (*)

> > > That does not imply that the number does not exist.
>
> > Do you understand that this requires that the cardinality of the set from which you select is infinite?

> [...]
> > *actually* infinite???
> [...]
This is a side issue, but it again shows up how your thinking is getting muddier and more confused by the day. Your senility is really getting the better of you, and you don't even realize it. If you select from a finite set of, say, cardinality C, then --- assuming a uniform distribution --- every member of that set is chosen with probability 1/C > 0. Your statement at the top (marked with an asterisk) is demonstrably false.

[Gus Gassmann]

On Monday, 18 January 2021 at 17:59:15 UTC-4, WM wrote:

> Gus Gassmann schrieb am Montag, 18. Januar 2021 um 22:16:04 UTC+1:
> > On Monday, 18 January 2021 at 15:52:39 UTC-4, WM wrote:
> > >[...] 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.
> > Lim_{n--> oo} |N ∩ [0, n]| / |P ∩ [0, n]| does not exist.
> There is an improper limit oo.
> >
> > Lim_{n--> oo} |P ∩ [0, n]| / |N ∩ [0, n]| = 0.
> That is true.
> > > It does not mean that there are no prime numbers.
> > What then, does it mean to you?
> It means that the fraction with increasing n grows above every positive value or falls below every positive epsilon.
> After all, *for you* Lim_{n--> oo} |N ∩ [0, n]| / |P ∩ [0, n]| = 2 implies that there are twice as many natural numbers as there are even numbers.
> No, P is not related to even numbers, on the contrary, it contains only one even number.

Yes, that was a typo. I meant to write Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2. You do conclude from this, however, that there are twice as many natural numbers as even natural numbers. However, you cannot conclude that there are 0 as many primes as there are natural numbers. In addition, you have not dealt in the least with the fact that there is a bijection between N and E (just as there are bijections between N ∩ [0, n] and E ∩ [0, 2n]). So your "result" is pretty much worthless.

> But Lim_{n--> oo} |N ∩ [0, n]| / |E ∩ [0, n]| = 2 shows twice as many natural numbers as even numbers not only *for me* but for every mathematician who has not become brain damaged.

Well, now you seem to be vacillating between an ultrafinitistic world where neither N nor E are infinite and a world like that of the surreal numbers, where there is omega/2, which is different from omega, and there are infinitesimals, and so on. Neither is ZFC, and you are welcome to use one of them on even-numbered days of the month and the other on odd-numbered days, but please indicate which side you are arguing in each thread.

[Jim Burns]

On 1/19/2021 5:51 AM, WM wrote:
> Jim Burns schrieb
> am Dienstag, 19. Januar 2021 um 04:36:06 UTC+1:
>> On 1/18/2021 2:52 PM, WM wrote:

>>> Physics uses *logic* for its deductions but
>>> no first- or second-order logic. That is simply rubbish.

>> *physics uses variables*
>
> and applies logic, but there is no reason to specify
> different kinds of logic. Do you know a reason why
> second-order logic is applied?
> There is none.

What chance is there that you (WM) know what you're talking
about, that you even know what second-order logic IS?
There is none.

This is what "academic freedom" means at Hochschule Augsburg.

[Jim Burns]

On 1/19/2021 5:51 AM, WM wrote:

> Jim Burns schrieb
> am Dienstag, 19. Januar 2021 um 04:36:06 UTC+1:
>> On 1/18/2021 2:52 PM, WM wrote:

>>> Scrooge McDuck daily receives 1000000000000000 $ and
>>> issues only 1 $. Set theory yields the limit: bankrupt.
>>> BUT ONLY IF HE ISSUES THE OLDEST DOLLARS. Otherwise he
>>> will not get bankrupt.
>
>> Each natural number can be matched with each multiple
>> of 1e15.
>
> It appears so to stupids who are duped by deceivers.

_Everyone knows_ that a decimal numeral can have fifteen 0s
appended to its right end.
_Everyone knows_ that a decimal numeral with fifteen 0s on
its right end can have the fifteen 0s removed.

Each natural has its own multiple of 1e15, and vice versa.

> You forget or do not even realize that [what WM calls]

> mathematics proves a different result.

You think that "mathematics" is a magic word.
You think that calling what you say "mathematics" _makes it_
free of error.
Your argument is _broken_ It doesn't matter what you call it,
it's still broken.

[Jim Burns]

On 1/19/2021 5:51 AM, WM wrote:

> Jim Burns schrieb
> am Dienstag, 19. Januar 2021 um 04:36:06 UTC+1:
>> On 1/18/2021 2:52 PM, WM wrote:

>>> The interior of the sun exists whether we see it or not.
>>
>> Does it exist? How do you know? Have you seen it?
>
> I need not see it. That is the advantage of reality.
>
>> The interior of the Sun is a theoretical construct.
>> Very close to *all* of what we *think* the universe is
>> has never been observed by us and will never be observed
>> by us.
>
> Reality exists without having been observed.
> Ideas do not exist without havung been thought.
> That's the difference.

We can reason about reality without observing it.

We can reason about what we describe, abstract or concrete,
without observing it.

That's the similarity.

We can, in some cases, be very-very-very _confident_ that
(at least some of) what a description is true of is reality.

We can be _absolutely certain_ that whatever statement follows
by truth-preserving steps from a description is true of
whatever the description is true of.

That's the difference.

>>> A number as an abstraction of reality does not exist
>>> if no-one abstracts it.
>>> You should try to understand this big difference.
>>
>> It is a difference that makes no difference.
>> Numbers exist in the same sense (or do not exist,the same)
>> before and after they are "abstracted".
>
> Ideas do not exist without havung been thought.

Descriptions are true statements about what they describe.

If you start with true statements about a thing, and proceed
_only_ by truth-preserving steps, then you _cannot_ arrive at
a false statement about that thing.

If your intuition disagrees
(it's almost guaranteed to disagree, sometimes, for anyone),
then your intuition is wrong.

[FredJeffries]

On Tuesday, January 19, 2021 at 2:51:24 AM UTC-8, WM wrote:

> Reality exists without having been observed. Ideas do not exist without havung been thought.

Therefore, ideas are not part of reality.

[WM]

FredJeffries schrieb am Mittwoch, 20. Januar 2021 um 03:12:08 UTC+1:
> On Tuesday, January 19, 2021 at 2:51:24 AM UTC-8, WM wrote:
>

> > Reality exists without having been observed. Ideas do not exist without having been thought.

> Therefore, ideas are not part of reality.

Ideas are part of reality, but not had ideas are not part of reality and their contents does not "exist" in any sense of the word, (i.e., either really or ideally). More precisely, any mention, or purported mention, of this contents is, literally, meaningless.

Regards, WM

[WM]

Jim Burns schrieb am Dienstag, 19. Januar 2021 um 18:59:57 UTC+1:
> On 1/19/2021 5:51 AM, WM wrote:
> > Jim Burns schrieb
> > am Dienstag, 19. Januar 2021 um 04:36:06 UTC+1:
> >> On 1/18/2021 2:52 PM, WM wrote:
>
> >>> Scrooge McDuck daily receives 1000000000000000 $ and
> >>> issues only 1 $. Set theory yields the limit: bankrupt.
> >>> BUT ONLY IF HE ISSUES THE OLDEST DOLLARS. Otherwise he
> >>> will not get bankrupt.
> >
> >> Each natural number can be matched with each multiple
> >> of 1e15.
> >
> > It appears so to stupids who are duped by deceivers.
> _Everyone knows_ that a decimal numeral can have fifteen 0s
> appended to its right end.
> _Everyone knows_ that a decimal numeral with fifteen 0s on
> its right end can have the fifteen 0s removed.
>
> Each natural has its own multiple of 1e15, and vice versa.

Alas that does not show equicardinality. That only shows that the initial segments of defined numbers have same cardinality. It does not show that there are as many natnumbers as ne15. This is easy to refute: The latter are a subset of the former. I can show elements that are not in the latter but in the former. Therefore equinumerosity is refuted.

>

> You think that "mathematics" is a magic word.

I use it as an important tool to refute nonsense.

> You think that calling what you say "mathematics" _makes it_
> free of error.

This is mathematics: Each of the infinitely many intervals (n, n+2] contains two natural numbers and only one even natural number. In order to catch up there must exist many even numbers outside of the domain covered by my intervals.

> Your argument is _broken_ It doesn't matter what you call it,
> it's still broken.

That is the typical uttering of religious faith. No matter whatever, you know the truth. I however use maths.

Regards, WM

[WM]

Jim Burns schrieb am Dienstag, 19. Januar 2021 um 20:08:46 UTC+1:

> We can be _absolutely certain_ that whatever statement follows
> by truth-preserving steps from a description is true of
> whatever the description is true of.

For instance that each of the infinitely many intervals (n, n+2] contains two natural numbers and only one even natural number.

> That's the difference.

What is the difference here?

Regards, WM