Set Theory is DEAD!

[Adam Polak]

Dear Friends,

The Set Theory, creator of which is considered to be Professor Georg Cantor, currently adhered to by the vast majority of scientists, is an undoubtedly flawed theory, based on erroneous assumptions and, as a result, filled with errors and internal contradictions.

The wide "Analysis of mistakes in infinity study attempts" within set theory can be found here on YouTube:
https://www.youtube.com/watch?v=s23Cz8A0BKs

In the upcoming presentations, we will together take a colser look on numerous errors in set theory, we will identify Hilbert's Grand Hotel Paradox errors, easily solve the Continuum Hypothesis (allegedly undecidable), Russell's Paradox, the Paradox of the set of all sets, and we will confirm even more emphatically that the set theory can be seen only as erroneous and disproven.

A small sample below. A comparison that decisively, in an unquestionable manner, refutes the Cantor's Diagonal Argument as evidence of the inequality of the infinite set of real numbers relative to the infinite set of natural numbers.

A hotel with an infinite number of rooms.
There is a guest in each room.
As a result, you have two infinite sets:

An infinite SET OF ROOMS containing elements with the following symbols: R1, R2, R3, ...

An infinite SET OF GUEST containing elements with the following symbols: G1, G2, G3...

A new guest appears: NG1
The new guest is definitely not among the guests that are already in the hotel because he is different from them, his name is: ("NG" + its individual number ) , everyone present in the hotel is: ("G"+ individual number of each ).

If you claim that you can accommodate a new guest in room 1 and move everyone currently present in the hotel to rooms n+1
you can do exactly the same thing with a "new" real number supposedly created by diagonal method.

You assign "new" real numb to 1, and you shift all the real numbers previously in the right column of the diagonal matrix down by one: the one that was assigned to 1 is now assigned to 2, the one assigned to 2 is now assigned to 3, etc.

It is mutually contradictory to say that you can accommodate a new guest in Hilbert's hotel and at the same time to say that you cannot find a natural number as a pair for a "new" real number "created" by the diagonal method.

The set theory is clearly contradictory in many places.

Best Regards,
Adam Polak

[Ben Bacarisse]

Adam Polak <mt69...@gmail.com> writes:

> The Set Theory [...] Cantor [...] flawed theory [...] filled with
> errors [...]

> https://www.youtube.com/watch?v=s23Cz8A0BKs

That well-known Journal, YouTube!

So rather than publish a paper and gain the respect of mathematicians
round the world, you choose to post on YouTube and in this (other)
crank-filled corner of the Internet. If you are wrong, doing so is just
a wast of time, but if you right it monumentally stupid to post in the
only paces where the default assumption will be that you are a lunatic.

--
Ben.

[Adam Polak]

Ben, I know what you mean.
Note however, that from what you write it follows that:
TRUTH is what is published in a "well-known scientific journal" and if the same statement is published on YouTube, it is not TRUTH but is probably stupid.
As a result, it doesn't matter WHAT someone states,
what matters is WHO states it, and WHERE it is stayed.
The heart of the matter (the topic, the statement) is pushed to the margins.
Isn't this approach completely unscientific and even unwise?

Try to read and refute at least one element of what I write.
I assure you that you won't be able to.

If you refute one element, I will tell you why I am not writing to "well-known scientific journal"

Regards,
Adam

[Fritz Feldhase]

On Monday, November 6, 2023 at 12:01:03 PM UTC+1, Adam Polak wrote:

> A hotel with an infinite number of rooms.
> There is a guest in each room.
> As a result, you have two infinite sets:
>
> An infinite SET OF ROOMS containing elements with the following symbols: R1, R2, R3, ...
>

> An infinite SET OF GUESTS containing elements with the following symbols: G1, G2, G3...
>
> A new guest appears: G0 [<< changed for simplicity, FF]
>
> The new guest is [...] not among the guests that are already in the hotel [...].

>
> If you claim that you can accommodate a new guest in room 1 and move everyone currently present in the hotel to rooms n+1
> you can do exactly the same thing with a "new" real number supposedly created by diagonal method.

Sure. No one denies that.

> You assign "new" real number to 1, and you shift all the real numbers previously in the [list] down by one: the one that was assigned to 1 is now assigned to 2, the one assigned to 2 is now assigned to 3, etc.

Exactly.

> It is mutually contradictory to say that you can accommodate a new guest in Hilbert's hotel and at the same time to say that you cannot find a natural number as a pair for a "new" real number "created" by the diagonal method.

No one claims the latter (except you, it seems).

What we claim is that there is no "list" (of real numbers) which contains _all_ real numbers. (We can actually PROVE that in each and every "list" of real numbers at least one real number is missing.)

While on the other hand, we may very well think of a "hotel" (let's call it, say, /earth/) which accommodates _all_ living persons (at once).

[Adam Polak]

"No one claims the latter (except you, it seems)."

I'll take it as a joke. First, because I don't think you have the authority to speak for EVERYONE/"No one". Secondly, because I have recorded lectures by professors of mathematics, physics, logic and publications in which exactly what I wrote is clearly stated.
Never mind.

"... there is no "list" (of real numbers) which contains _all_ real numbers. (We can actually PROVE that in each and every "list" of real numbers at least one real number is missing.) "

It is completely obvious that the elements of any infinite set cannot be arranged into a "list" that would contain all (from a quantitative perspective) the elements of such an infinite set. Of course, this applies to the infinite set of real numbers, and of course this applies to the infinite set of natural numbers and of course this applies to any other infinite set. A set whose elements can be arranged into a "list"/series containing ALL the elements - > i.e. the first element, the last element and the elements in between would be a finite set. Read my presentation, I explain it in quite detail and in a way that cannot be questioned. Then ask a question if you don't understand something. Quote a passage! " " that you don't understand before asking.

[Fritz Feldhase]

On Monday, November 6, 2023 at 3:25:20 PM UTC+1, Adam Polak wrote:
>
> "... there is no "list" (of real numbers) which contains _all_ real numbers. (We can actually PROVE that in each and every "list" of real numbers at least one real number is missing.) "

Note, that I didn't say "finite list", idiot. We are talking about infinite sets [and hence "lists"] here, aren't we?

> It is completely obvious that the elements of any infinite set cannot be arranged into a "list" that would [bla bla bla]

Get a grip, idiot. By a /list/ of the elements of an infinite set A, I mean a bijection from IN onto A here.

Hence a "list" of, say, the natural numbers is possible. You may consider id: IN --> IN defined with id(n) = n for all n in IN. But a "list" of all reals is n o t possible.

THAT's what *we* are talking about and what Cantor proved.

___________________

Hint: You are talking nonsense in your vid. Here too.

[Dan Christensen]

On Monday, November 6, 2023 at 6:01:03 AM UTC-5, Adam Polak wrote:
[snip]

> A hotel with an infinite number of rooms.
> There is a guest in each room.
> As a result, you have two infinite sets:
>
> An infinite SET OF ROOMS containing elements with the following symbols: R1, R2, R3, ...
>
> An infinite SET OF GUEST containing elements with the following symbols: G1, G2, G3...
>
> A new guest appears: NG1
> The new guest is definitely not among the guests that are already in the hotel because he is different from them, his name is: ("NG" + its individual number ) , everyone present in the hotel is: ("G"+ individual number of each ).
>
> If you claim that you can accommodate a new guest in room 1 and move everyone currently present in the hotel to rooms n+1
> you can do exactly the same thing with a "new" real number supposedly created by diagonal method.
>
> You assign "new" real numb to 1, and you shift all the real numbers previously in the right column of the diagonal matrix down by one: the one that was assigned to 1 is now assigned to 2, the one assigned to 2 is now assigned to 3, etc.
>
> It is mutually contradictory to say that you can accommodate a new guest in Hilbert's hotel and at the same time to say that you cannot find a natural number as a pair for a "new" real number "created" by the diagonal method.
>

Hilbert's Infinite Hotel should not be taken too literally. It is simply a humorous illustration that an infinite set like the set of natural numbers N = {1, 2, 3, ... } can be mapped bijectively to a proper subset of itself, namely {2, 3, 4, ... }. This property is the defining characteristic of ANY infinite set. In this case, the required bijection is f: N --> {2, 3, 4, ... } such that f(x)=x+1.

For a somewhat less mind-blowing development, you might consider my alternative approach. I start by defining what we mean by a finite set. Then an infinite set is just one that is not finite.

See my blog posting at https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/

There, I present informal and formal set-theoretic developments of a non-numeric definition of a finite set. Refute it if you think you can.,

Dan

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

[Ben Bacarisse]

Adam Polak <mt69...@gmail.com> writes:

> poniedziałek, 6 listopada 2023 o 12:47:47 UTC+1 Ben Bacarisse napisał(a):
>> Adam Polak <mt69...@gmail.com> writes:
>>
>> > The Set Theory [...] Cantor [...] flawed theory [...] filled with
>> > errors [...]
>>
>> > https://www.youtube.com/watch?v=s23Cz8A0BKs
>>
>> That well-known Journal, YouTube!
>>
>> So rather than publish a paper and gain the respect of mathematicians
>> round the world, you choose to post on YouTube and in this (other)
>> crank-filled corner of the Internet. If you are wrong, doing so is just
>> a wast of time, but if you right it monumentally stupid to post in the
>> only paces where the default assumption will be that you are a lunatic.
>

> Ben, I know what you mean.

No, you draw a totally unwarranted conclusion from what I said, namely
this:

> Note however, that from what you write it follows that:
> TRUTH is what is published in a "well-known scientific journal" and if
> the same statement is published on YouTube, it is not TRUTH but is
> probably stupid.

No, that does not follow from what I wrote. In fact, that is directly
contradicted by what I stated: you can be stating the truth in the wrong
place for it to be recognised as significant.

Why don't you want to have your work recognised as significant?
Mathematicians don't scour sci.math and YouTube looking for the one
person (you) who really /has/ refuted modern set theory in amongst the
hundreds of other posts claiming to do buy which don't. No, they read
the journals that are lying around in the common room, or that sit in
the library.

--
Ben.

[Adam Polak]

FINITE
A set S is finite when there exists an n ϵ N such that S has exactly n elements.

INFINITE
Consequently:
A set S is infinite when there does not exist an n ϵ N such that S has exactly n elements.

From the above, it follows that:
Infinity is not a number, particularly not a natural number. It is not the number of elements in an infinite set, nor is it the largest or last element in an infinite set, e.g. such as the infinite set of natural numbers N.

[Adam Polak]

Ben,
Honestly, I have very little time for issues other than those related to the substantive content, I am expanding the material and preparing further presentations. I also have no experience in dealing with science journals.

Do you want to help? to be the manager of such a publication?

I can assure you that it will be a very interesting and unforgettable experience for you. (The content of the presentation cannot be denied - and it refutes set theory with several proofs from several perspectives)
We can think overe some "success fee" for you.

[Chris M. Thomasson]

A finite set is a finite set. Now, an infinite set can be composed of
infinite finite sets...

{ 0 }, { 0, 1 }, { 0, 1, 2 }

Or, lets say a simple binary tree:


l[0] = 0
_______________________________
/ \
/ \
/ \
/ \
l[1] = 1 2
_______________________________
/ \ / \
/ \ / \
l[2] = 3 4 5 6
...............................

Comprised of levels { 0 }, { 1, 2 }, { 3, 4, 5, 6 }, ...


Each level is finite, say l[2] at { 3, 4, 5, 6 }, however, there are an
infinite number of levels in the tree... See? :^)

To get the parent node of say, 3 or 4:

ceil(3/2)-1 = 1
4/2-1 = 1

Say 5 and 6, they have a parent of 2:

ceil(5/2)-1 = 2
6/2-1 = 2

Oh lets try the children of 5, that would be in level l[3] at:

{ 7, 8, 9, 10, 11, 12, 13, 14 }

So, 11 and 12 would be:

ceil(11/2)-1 = 5
12/2-1 = 5


Lets try 1 and 2, with a parent of zero, the root of the tree:

ceil(1/2)-1 = 0
2/2-1 = 0


What is the parent of 15? Lets see:

ceil(15/2)-1 = 7

Humm, what is the parent of 16?

16/2-1 = 7

Yes!

[Dan Christensen]

On Monday, November 6, 2023 at 4:09:22 PM UTC-5, Adam Polak wrote:

> poniedziałek, 6 listopada 2023 o 17:37:41 UTC+1 Dan Christensen napisał(a):
> > On Monday, November 6, 2023 at 6:01:03 AM UTC-5, Adam Polak wrote:
> > [snip]
> > > A hotel with an infinite number of rooms.
> > > There is a guest in each room.
> > > As a result, you have two infinite sets:
> > >
> > > An infinite SET OF ROOMS containing elements with the following symbols: R1, R2, R3, ...
> > >
> > > An infinite SET OF GUEST containing elements with the following symbols: G1, G2, G3...
> > >
> > > A new guest appears: NG1
> > > The new guest is definitely not among the guests that are already in the hotel because he is different from them, his name is: ("NG" + its individual number ) , everyone present in the hotel is: ("G"+ individual number of each ).
> > >
> > > If you claim that you can accommodate a new guest in room 1 and move everyone currently present in the hotel to rooms n+1
> > > you can do exactly the same thing with a "new" real number supposedly created by diagonal method.
> > >
> > > You assign "new" real numb to 1, and you shift all the real numbers previously in the right column of the diagonal matrix down by one: the one that was assigned to 1 is now assigned to 2, the one assigned to 2 is now assigned to 3, etc.
> > >
> > > It is mutually contradictory to say that you can accommodate a new guest in Hilbert's hotel and at the same time to say that you cannot find a natural number as a pair for a "new" real number "created" by the diagonal method.
> > >
> > Hilbert's Infinite Hotel should not be taken too literally. It is simply a humorous illustration that an infinite set like the set of natural numbers N = {1, 2, 3, ... } can be mapped bijectively to a proper subset of itself, namely {2, 3, 4, ... }. This property is the defining characteristic of ANY infinite set. In this case, the required bijection is f: N --> {2, 3, 4, ... } such that f(x)=x+1.
> >
> > For a somewhat less mind-blowing development, you might consider my alternative approach. I start by defining what we mean by a finite set. Then an infinite set is just one that is not finite.
> >
> > See my blog posting at https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
> >
> > There, I present informal and formal set-theoretic developments of a non-numeric definition of a finite set. Refute it if you think you can.,
> >

> FINITE
> A set S is finite when there exists an n ϵ N such that S has exactly n elements.
>

It turns out that it is not necessary to assume the existence of an infinite set to define a finite set. A set X can be said to be finite if every injective (1-1) function on X must also be surjective (onto). At the above link, I develop this notion starting from an informal thought experiment.

> INFINITE
> Consequently:
> A set S is infinite when there does not exist an n ϵ N such that S has exactly n elements.
>

A set X can be said to be infinite if is not finite, i.e. there exists a function on X that is both injective and NOT surjective. (From Dedekind.)

> From the above, it follows that:
> Infinity is not a number, particularly not a natural number.

I don't define "infinity," I define "infinite set."

[Fritz Feldhase]

On Monday, November 6, 2023 at 10:34:12 PM UTC+1, Adam Polak wrote:
>
> The content of the presentation cannot be denied - and

So you didn't realize that I had to *correct* the nonense you wrote here, idiot?

Or just want to ignore the correction (which is quite typical for cranks)?

[Fritz Feldhase]

On Monday, November 6, 2023 at 10:09:22 PM UTC+1, Adam Polak wrote:

> Infinity is not a number, particularly not a

AGAIN, no one (except possibly you) claimed that it is, you silly idiot.

[Hint: With "no one", I mean no one in the present context; as well as no (distinguished) set theoriest.]

Are you sure that your name isn't Don Quixote?

[Fritz Feldhase]

On Monday, November 6, 2023 at 10:39:54 PM UTC+1, Chris M. Thomasson wrote:

> A finite set is a finite set. Now, an infinite set can be composed of
> infinite finite sets...
>
> { 0 }, { 0, 1 }, { 0, 1, 2 }

I guess you meant (to write)

{ 0 }, { 0, 1 }, { 0, 1, 2 }, ...

here. Right?

So you were referring to the infinte set

{{ 0 }, { 0, 1 }, { 0, 1, 2 }, ... }

here, right?

Do we have a case here where the blind tries to lead the lame (or the other way round)? Sorta?

[Fritz Feldhase]

On Monday, November 6, 2023 at 3:25:20 PM UTC+1, Adam Polak wrote:

> It is completely obvious that

you don't know what you are talking about.

> the elements of any infinite set cannot be arranged into a [finite] "list" that would contain all [...] the elements of such an infinite set.

Obviously.

> A set whose elements can be arranged into a [finite] "list"/sequence containing ALL the elements - > i.e. the first element, the last element and the elements in between would be a finite set.

Indeed.

But since we are considering infinite sets we are talking about infinite lists her too.

Formally, we may define such a list as a function from IN onto the set we are considering. In this case "lists" usually are called (infinite) sequences in mathematics.

See: https://de.wikipedia.org/wiki/Folge_(Mathematik)#Formale_Definition

Some such "lists"/"sequences":

(1, 2, 3, 4, ...) - the sequence of all natural numbers

(2, 4, 6, 8...) - the sequence of all even numbers

(2, 3, 5, 7...) - the sequence of all prime numbers

etc.

Note that these lists/sequences do not have a "last element/term".

[Fritz Feldhase]

On Monday, November 6, 2023 at 3:41:55 PM UTC+1, Fritz Feldhase wrote:

Correction:

> By a /list/ of the elements of an infinite set A, I mean a __function___ from IN onto A here.

Sorry about that.

So the sequence (1, 1, 2, 2, 3, 3, 4, 4, ...) is a "list of all natural numbers" in my book, too.

[Fritz Feldhase]

Or rather:

> (1, 2, 3, 4, ...) - _a_ sequence of all natural numbers
>
> (2, 4, 6, 8...) - _a_ sequence of all even numbers
>
> (2, 3, 5, 7...) - _a_ sequence of all prime numbers

[Ben Bacarisse]

Adam Polak <mt69...@gmail.com> writes:

> Honestly, I have very little time for issues other than those related
> to the substantive content,

I doubt that. I expect you have lots of spare time. And don't fuss
about not answering my question. I'm sure you post here for the same
reasons everyone else who has "refuted Cantor" posts here.

> Do you want to help? to be the manager of such a publication?

Gosh, why would I do that? As soon as you clarify your refutation,
someone else will publish it to get the credit. And if it's one of the
other cranks, they will say you copied the points they made decades ago
if you try to claim credit!

More likely, you won't address the points put to you (you haven't so
far) but you will have fun chatting about it. Enjoy.

--
Ben.

[Mathin3D]

It has bee 6 days since Halloween and the nutcases are still out!

Do you need contact numbers for some good psychiatrists?

On Monday, November 6, 2023 at 4:09:22 PM UTC-5, Adam Polak wrote:

[Mathin3D]

I have not seen this dude around here before. He heading the John Gabriel direction in the mental sanity department. Start collecting snippets. LOL

[Dan joyce]

On Monday, November 6, 2023 at 6:01:03 AM UTC-5, Adam Polak wrote:

Just look at my post of Oct 2023 -- Hilbert original idea of an infinite hotel was manipulated.
This is explained in great detail on why these rooms in my infinite pyramid hotel --->oo,
In other words these rooms go on forever. Where does infinite set theory fit in here?
Showing this 2d pyramid being built in real time with explanations along the way.
Not a peep from any of the ZFC guys refuting my thoughts.
My thought process on this could be wrong and if so please enlighten me

Dan

[Fritz Feldhase]

On Monday, November 6, 2023 at 3:25:20 PM UTC+1, Adam Polak wrote:

"It is mutually contradictory to say that you can accommodate a new guest in Hilbert's hotel and at the same time to say that you cannot find a natural number as a pair for a "new" real number "created by the diagonal method."

> > No one claims the latter (except you, it seems). [FF]

Hint: With "no one", I mean no one in the present context; as well as no (distinguished) set theoriest [as far as I can tell].

> [...] I have recorded lectures by professors of mathematics, physics [...] in which exactly what I wrote is clearly stated.

Please _ignore physicists_ in this context. Mathematicans though should know what they are talking about. Do you have some quotes?

Of course, we may/can always add the new number "created by the diagonal method" to the list of real numbers we are considering.

But that's not the point here (concerning Cantor's proof). The point is:

> "there is no sequence/"list" (of real numbers) which contains _all_ real numbers. (After all, we can PROVE that in each and every sequence/"list" of real numbers at least one real number is missing.)"

On the other hand, for example, (1, 2, 3, 4, ...) is a _complete_ sequence of all natural numbers (none is missing).

[Fritz Feldhase]

On Tuesday, November 7, 2023 at 1:21:22 PM UTC+1, Ben Bacarisse wrote:

> [...] and if it's one of the other cranks, they will say you copied the points they made decades ago

> if you try to claim credit!

Ironically the only reference he mentions is WM's cranky nonsense manuscript:

| Interesting sources and documents related to Set Theory:
|
| "Transfinity - A Source Book" by Wolfgang Mückenheim
| https://www.hs-augsburg.de/~mueckenh/...

Seems this guy fell for it. And it shows!

[Chris M. Thomasson]

On 11/6/2023 2:16 PM, Fritz Feldhase wrote:
> On Monday, November 6, 2023 at 10:39:54 PM UTC+1, Chris M. Thomasson wrote:
>
>> A finite set is a finite set. Now, an infinite set can be composed of
>> infinite finite sets...
>>
>> { 0 }, { 0, 1 }, { 0, 1, 2 }
>
> I guess you meant (to write)
>
> { 0 }, { 0, 1 }, { 0, 1, 2 }, ...

Ding! Indeed.

>
> here. Right?
>
> So you were referring to the infinte set
>
> {{ 0 }, { 0, 1 }, { 0, 1, 2 }, ... }

Right:

>
> here, right?
>
> Do we have a case here where the blind tries to lead the lame (or the other way round)? Sorta?

I was trying to build up to a case of an n-ary tree with 2-ary as an
example:

[Fritz Feldhase]

On Wednesday, November 8, 2023 at 12:30:35 AM UTC+1, Chris M. Thomasson wrote:

> I was trying to build up to a case of an n-ary tree with 2-ary as an example: [...]

N/p. Seems that you have a typical programmer's view concerning mathematics. :-P

Trees are interesting objects, no doubt.

[Fritz Feldhase]

On Wednesday, November 8, 2023 at 12:36:05 AM UTC+1, Fritz Feldhase wrote:
> On Wednesday, November 8, 2023 at 12:30:35 AM UTC+1, Chris M. Thomasson wrote:
> >
> > I was trying to build up to a case of an n-ary tree with 2-ary as an example: [...]
> >
> N/p. Seems that you have a typical programmer's view concerning mathematics. :-P

Btw. I'd recommend the folling book to you - you might like it very much:

Graham, Knuth, Patashnik: Concrete Mathematics: A Foundation for Computer Science

"Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems." The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study."

The bright guy you are, you might take advantage of it.

[mitchr...@gmail.com]

It was dead at its beginning. What is a set with nothing in it?
What can math make out of it?

[Fritz Feldhase]

On Wednesday, November 8, 2023 at 2:49:56 AM UTC+1, mitchr...@gmail.com wrote:
>
> What is a set with nothing in it?

An empty set?

[FromTheRafters]

The empty set.

[Python]

We could do memes from this :-)

[FromTheRafters]

Python presented the following explanation :

I asked the grocer for an empty box, he asked "An empty box of what?"

[Adam Polak]

these doubts are explained in my presentation (link in the first message) page 23 and around :

"
Set theory, in its various formulations, avoids presenting a single coherent definition of the concept of a SET.

Defining this concept based on the notion of an ELEMENT of a set reveals that the so-called "empty set" does not satisfy the definition of a set and should not be considered as part of a coherent set theory.
On the other hand, defining the concept of a SET in a way that allows for the omission of having elements as a condition for the existence of a set makes it extremely difficult, if not impossible, to maintain the illusion that set theory explains the phenomenon of creating "something out of nothing„.

In practice, the concept of a SET is usually described through the concept of the elements of the set. At the same time, by force, by axiom, set theory introduces a concept of empty set containing no elements, even though it does not fulfill the description /definition of a set in terms of having elements.

There should be no doubt that regardless of whether a SET is understood as a collection of mushrooms or as a basket into which mushrooms have been collected, there can be no talk of any creatio ex nihilo in any case.
"

I have many tasks and limited time, so I may not visit and reply here often
BR, Adam

[Adam Polak]

Ben, Thank you for this answer.
Coincidentally, it well illustrates the problem with Set Theory aspiring to be a correct description of REALITY in terms of the relationship between infinite sets and their elements.
In fact, it is just an unsubstantiated and erroneous DREAM on the subject.
Exactly like your dream/imagination of what it's like i "real" with my free time, etc.

Regards, Adam

[Fritz Feldhase]

On Wednesday, November 8, 2023 at 1:11:37 PM UTC+1, Adam Polak wrote:

> the problem with Set Theory aspiring to be a correct description of REALITY in terms of <bla>

<Holy shit!>

The aim of set theory certainly isn't "to be a correct description of REALITY" whatsoever, knucklehead.

[Fritz Feldhase]

On Wednesday, November 8, 2023 at 1:03:25 PM UTC+1, Adam Polak wrote:

> Set theory, in its various formulations, avoids presenting a single coherent definition of the concept of a SET.

Indeed! So what?

> [allowing] for [an empty] set makes it extremely difficult, if not impossible, to maintain the illusion that set theory explains the phenomenon of creating "something out of nothing'.

Huh?! Who told you that "set theory [tries to] explain[.] the phenomenon of creating 'something out of nothing'"?!

That's just nonsense.

Hint: It doesn't.

> by axiom[s], set theory [allows for] a[n] empty set containing no elements

Indeed! So what?

There's no written or unwritten law that forbids that.

> There should be no doubt that regardless of whether a SET is understood as a collection of mushrooms or as a basket into which mushrooms have been collected,

The latter is the "appropriate interpretation". Hint: A set just containing one element is not identical with this element: {a} =/= a. Moreover, the former would certainly not allow for an empty set (I'd say). Please do not mix up mereology with set theory.

> there can be no talk of any creatio ex nihilo in any case.

Again, no one is claiming that (except you, it seems).

(Sure, some physicist may have come up with such a phrase. I already told you to forget about physicists in the present context.)

[Fritz Feldhase]

On Wednesday, November 8, 2023 at 2:09:15 PM UTC+1, Fritz Feldhase wrote:

> Again, no one is claiming that (except you, it seems).
>
> (Sure, some physicist may have come up with such a phrase. I already told you to forget about physicists in the present context.)

Hint: Mückenheim is a physicist.

So it isn't a particularly good idea to base ones views on the nonsense he wrote concerning "set theory".

[Timothy Golden]

On Monday, November 6, 2023 at 7:15:55 AM UTC-5, Adam Polak wrote:

> poniedziałek, 6 listopada 2023 o 12:47:47 UTC+1 Ben Bacarisse napisał(a):
> > Adam Polak <mt69...@gmail.com> writes:
> >

> > > The Set Theory [...] Cantor [...] flawed theory [...] filled with
> > > errors [...]
> >
> > > https://www.youtube.com/watch?v=s23Cz8A0BKs
> >
> > That well-known Journal, YouTube!
> >
> > So rather than publish a paper and gain the respect of mathematicians
> > round the world, you choose to post on YouTube and in this (other)
> > crank-filled corner of the Internet. If you are wrong, doing so is just
> > a wast of time, but if you right it monumentally stupid to post in the
> > only paces where the default assumption will be that you are a lunatic.
> >
> > --
> > Ben.
> Ben, I know what you mean.
> Note however, that from what you write it follows that:
> TRUTH is what is published in a "well-known scientific journal" and if the same statement is published on YouTube, it is not TRUTH but is probably stupid.
> As a result, it doesn't matter WHAT someone states,
> what matters is WHO states it, and WHERE it is stayed.
> The heart of the matter (the topic, the statement) is pushed to the margins.
> Isn't this approach completely unscientific and even unwise?
>
> Try to read and refute at least one element of what I write.
> I assure you that you won't be able to.
>
> If you refute one element, I will tell you why I am not writing to "well-known scientific journal"
>
> Regards,
> Adam

Um, I find an integrity conflict back at set and function as two fundamental concepts. When your set can be defined with a function, and your function requires a set, then a transgression has been committed. This is a compiler level error, and Cantor may have been one of the compilers, but so was Peano with his successor function.
You can certainly shrug this off as a human, and the anthropic principle will live within our systems forever. For some of us this condition is unacceptable. Simplicity was to be found another way, and computing hardware as a limited yet operable system exposes the chase, and perhaps the product and division really best pose the puzzle. In finality could it be that the product is taken, and only then? Well, says you: "I don't mind taking sums of products, nor products of sums." And yet whether the product is itself is a grandiose sum has been left adrift. Then too, the integral, and the sigma notation, so often entering the laws that we recover. Don't mathematicians regard their trade as being ultimately at the bottom of it all?

[Timothy Golden]

On Monday, November 6, 2023 at 4:09:22 PM UTC-5, Adam Polak wrote:
> FINITE
> A set S is finite when there exists an n ϵ N such that S has exactly n elements.
>

> INFINITE
> Consequently:
> A set S is infinite when there does not exist an n ϵ N such that S has exactly n elements.

It is really modulo behaved elements that satisfy here. All others give me the creeps.

>
> From the above, it follows that:

> Infinity is not a number, particularly not a natural number. It is not the number of elements in an infinite set, nor is it the largest or last element in an infinite set, e.g. such as the infinite set of natural numbers N.

> poniedziałek, 6 listopada 2023 o 17:37:41 UTC+1 Dan Christensen napisał(a):

> > On Monday, November 6, 2023 at 6:01:03 AM UTC-5, Adam Polak wrote:
> > [snip]

> > > A hotel with an infinite number of rooms.
> > > There is a guest in each room.
> > > As a result, you have two infinite sets:
> > >
> > > An infinite SET OF ROOMS containing elements with the following symbols: R1, R2, R3, ...
> > >
> > > An infinite SET OF GUEST containing elements with the following symbols: G1, G2, G3...
> > >
> > > A new guest appears: NG1
> > > The new guest is definitely not among the guests that are already in the hotel because he is different from them, his name is: ("NG" + its individual number ) , everyone present in the hotel is: ("G"+ individual number of each ).
> > >
> > > If you claim that you can accommodate a new guest in room 1 and move everyone currently present in the hotel to rooms n+1
> > > you can do exactly the same thing with a "new" real number supposedly created by diagonal method.
> > >
> > > You assign "new" real numb to 1, and you shift all the real numbers previously in the right column of the diagonal matrix down by one: the one that was assigned to 1 is now assigned to 2, the one assigned to 2 is now assigned to 3, etc.
> > >

> > > It is mutually contradictory to say that you can accommodate a new guest in Hilbert's hotel and at the same time to say that you cannot find a natural number as a pair for a "new" real number "created" by the diagonal method.
> > >

> > Hilbert's Infinite Hotel should not be taken too literally. It is simply a humorous illustration that an infinite set like the set of natural numbers N = {1, 2, 3, ... } can be mapped bijectively to a proper subset of itself, namely {2, 3, 4, ... }. This property is the defining characteristic othenf ANY infinite set. In this case, the required bijection is f: N --> {2, 3, 4, ... } such that f(x)=x+1.

> >
> > For a somewhat less mind-blowing development, you might consider my alternative approach. I start by defining what we mean by a finite set. Then an infinite set is just one that is not finite.
> >
> > See my blog posting at https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
> >
> > There, I present informal and formal set-theoretic developments of a non-numeric definition of a finite set. Refute it if you think you can.,
> >

[Ross Finlayson]

Hmm, "SET THEORY, is dead", I like it, Adam.

It's fair of you to say "here's a theory where I've axiomatized away my model of
a trust in set theory, I must make myself another way to trust", the set theory,
then you constructively bring up what you want, while then pointing out a
crankish argument in set theory, that shows something you've hobbled yourself, from.

Powers of 2?

What you get is ordering, numbering, and counting, and when numbering invoves counting.

Set theory models these sufficiently all their "regular" way, "well-founded", for example,
the regular set theory.

You can make inconsistent models of set theory and show how they're inconsistent.
It's not considered constructivist, say, insofar as formal rigor and "can't not trust it".

So, you want to square away your Aleph numbers, cardinals, and the Omega-many ordinals.
The Aleph, is the counting infinity, while the Omega, is moreso the numbering infinity
and the ordering infinity, in ordinals.

The counting infinities the Aleph numbers, their arithmetic builds the orders of the spaces,
above each constructive, regular, ordinary, ..., theory of words like sets, here elementary
objects.

That's one reason why cardinals and ordinals are different, different infinities.

Anyways usually insofar as any mistake you write here someone will point it out to you.

Anyways what results I enjoyed this for some time, currently looking at my own slates,
I sort of organize analysis in continuum mechanics.

"Infinitely-many", ....

So, what you want to do, I think to really get an understanding of the cardinal and ordinal
numbers, and, the cardinal and ordinal infinities, is give yourself axioms for example "inverse",
but for example "counting" or whatever other results "infinity" axioms, then figuring out
where their sameness and differences, do or strongly do or don't or strongly don't, hold,
what do.

"It's a continuum mechanics, ...", just saying, Adam, that if you're looking for a theory that
really digs up set theory, I made one with both cardinals and ordinals and their infinities
what otherwise sometimes aren't "extra" enough to be real.

I've even gone so far as to stand up letting a simplest mathematical infinity, back into
the philosophy, of the theory, the one that science is missing.

So, when I suggest, "Powers of 2?", I suggest that you're thoroughly familiar with them,
all the powers of 2, then for the two infinities you call "w", omega, and "2^w", 2 to the omega,
which as a number, is an ordinal, but also results when writing ordinals regular-ly, is the
space "2^w", each of the infinite sequences of zeros and ones, with a beginning.

It's set theory, ....

[Timothy Golden]

Did you ever consider whether the top down form could apply rather than the bottom up approach that mathematics seems to formalize?
Some will insist that this top is infinity somehow, but according to cosmology as I understand it they may be claiming a true finite albeit very large value. To what degree then we work down from this large thing which is essentially unknown to us other than to suppose it, possibly even with peculiarities built into its own actual factors, for instance, dealing symmetries and so forth, and what, so three is one of its factors? Is that it, Ross? Isn't there a one in three chance that this would occur?
And whether odd or even: you pick which by what means? Hmmm.... I think our reality is quite odd.

[Adam Polak]

"This is a compiler level error, and Cantor may have been one of the compilers, but so was Peano with his successor function."

with the difference that Cantor made a mistake and Peano not (not in the definition of natural numbers at least)

"You can certainly shrug this off as a human, and the anthropic principle will live within our systems forever. For some of us this condition is unacceptable."

In a way I agree. (not only WHAT, but also who, when, why, for what purpose, etc... states labeling it as a "scientific statement")

[Python]

Adam Polak wrote:
> [not even original nonsense]

> I have many tasks and limited time, so I may not visit and reply here often
> BR, Adam

Don't worry, there a lot of cranks of your kind down here, we won't
miss you.

[FromTheRafters]

Adam Polak expressed precisely :

Status confirmed.

[Phil Carmody]

That wound you up so much you went to the local cafe, and asked for
a strong coffee - no milk. The cafe owner replied "sorry, we're out
of milk, but I can make you one with no cream".

Phil
--
We are no longer hunters and nomads. No longer awed and frightened, as we have
gained some understanding of the world in which we live. As such, we can cast
aside childish remnants from the dawn of our civilization.
-- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/

[Chris M. Thomasson]

On 11/9/2023 4:55 PM, Phil Carmody wrote:
> FromTheRafters <F...@nomail.afraid.org> writes:
>> Python presented the following explanation :
>>> Le 08/11/2023 à 05:29, FromTheRafters a écrit :
>>>> Fritz Feldhase wrote on 11/7/2023 :
>>>>> On Wednesday, November 8, 2023 at 2:49:56 AM UTC+1,
>>>>> mitchr...@gmail.com wrote:
>>>>>>
>>>>>> What is a set with nothing in it?
>>>>>
>>>>> An empty set?
>>>>
>>>> The empty set.
>>>
>>> We could do memes from this :-)
>>
>> I asked the grocer for an empty box, he asked "An empty box of what?"

Fractalize it, Say, an empty box with smaller empty boxes in them. What
boxes are empty?

>
> That wound you up so much you went to the local cafe, and asked for
> a strong coffee - no milk. The cafe owner replied "sorry, we're out
> of milk, but I can make you one with no cream".

LOL!

[Fritz Feldhase]

No joke, in Russellian Type Theory each type has its own "empty set".

[Fritz Feldhase]

On Thursday, November 9, 2023 at 4:27:52 AM UTC+1, Python wrote:

> Adam Polak wrote:
> >
> > I have many tasks and limited time, so I may not visit and reply here often
> >

> Don't worry, there a lot of cranks of your kind down here, we won't miss you.

Ironically, the only reference this buffoon mentions is WM's cranky nonsense manuscript:

[Fritz Feldhase]

On Thursday, November 9, 2023 at 4:27:53 AM UTC+1, FromTheRafters wrote:
> Adam Polak expressed precisely :

> >
> > Ben, Thank you for this answer.
> > Coincidentally, it well illustrates the problem with Set Theory aspiring to
> > be a correct description of REALITY in terms of the relationship between
> > infinite sets and their elements. In fact, it is just an unsubstantiated and
> > erroneous DREAM on the subject. Exactly like your dream/imagination of what
> > it's like i "real" with my free time, etc.
> >
> > Regards, Adam
> >
> Status confirmed.

Yeah, a genius!

[Adam Polak]

Thank you for your comment Ross. Very interesting.
e.g.:

"So, you want to square away your Aleph numbers, cardinals, and the Omega-many ordinals.
The Aleph, is the counting infinity, while the Omega, is moreso the numbering infinity and the ordering infinity, in ordinals.
The counting infinities the Aleph numbers, their arithmetic builds the orders of the spaces, above each constructive, regular, ordinary, ..., theory of words like sets, here elementary objects.
That's one reason why cardinals and ordinals are different, different infinities."

"I sort of organize analysis in continuum mechanics."

I'll start from the end. I can assure you, Ross, and provide numerous independent, but logically related pieces of evidence that everything what quantitatively is represented by any infinite set, including: Alephs, cardinals, and Omegas, fits in, "happens" within the quantity represented by the infinite set of natural numbers.

The theory of infinite sets by Cantor, aspires to be a correct description of the reality in the realm of infinite sets, the relationships between infinite sets, and their elements. In reality, however, it is simply a FALLACIOUS description. It is a dream of grandeur dreamt by the mind, or perhaps the ego of Cantor (initially), and now by the minds/egos of the overwhelming majority of the scientific community. It is a beautiful dream in which a finite human comprehends infinity, e.g. infinity of natural numbers, and probably due to its beauty, it has been so infectious and widely spread. In the end, it is, unfortunately, only a waking dream with very little in common with reality, especially in terms of the true nature of infinity.

From the history of science, we know many sometimes very spectacular cases when "great" theories, accepted by "everyone," turned out to be greatly flawed or at least imperfect at some point. The same is about to happen soon with Cantor's Set Theory. I'm looking for a volunteer, preferably someone who is a "expert" in set theory professionally, someone who professionally deals with it, BUT! who allows for even the slightest possibility that there might be something wrong with set theory, just like with anything created by humans. (a person without this "BUT" will not wake up, he is simply fanatically attached to the illusion he is stuck in)

I will undertake to guide such a person through detox and awaken from the dream of cardinalities greater than N, to the reality, which I assure is no less, but more intriguing..

BR, Adam

[FredJeffries]

On Thursday, November 9, 2023 at 9:10:23 PM UTC-8, Adam Polak wrote:
>
> The theory of infinite sets by Cantor, aspires to be a correct description of the reality in the realm of infinite sets, the relationships between infinite sets, and their elements. In reality, however, it is simply a FALLACIOUS description. It is a dream of grandeur dreamt by the mind, or perhaps the ego of Cantor (initially), and now by the minds/egos of the overwhelming majority of the scientific community. It is a beautiful dream in which a finite human comprehends infinity, e.g. infinity of natural numbers, and probably due to its beauty, it has been so infectious and widely spread. In the end, it is, unfortunately, only a waking dream with very little in common with reality, especially in terms of the true nature of infinity.
>

In fact, Cantor took great pains to disassociate and distinguish the infinities both of theology and metaphysics from his mere mathematical transfinite.

<quote translated>
The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extra worldly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical magnitude, number or order type. In the latter two relations, where it obviously reveals itself as limited and capable for further proliferation and hence familiar to the finite, I call it Transfinitum and strongly contrast it with the absolute.
</quote>

<quote translated>
I have never assumed a “Genus Supremum” of the actual infinite. Quite on the contrary I have proved that there can be no such “Genus Supremum” of the actual infinite. What lies beyond all that is finite and transfinite is not a “Genus”; it is the unique, completely individual unity, in which everything is, which comprises everything, the ‘Absolute’, for human intelligence unfathomable, also that not subject to mathematics, unmeasurable, the “ens simplicissimum”, the “Actus purissimus”, which is by many called “God”.
</quote>

https://en.wikipedia.org/wiki/Absolute_Infinite

https://pointatinfinityblog.wordpress.com/2017/06/12/cantor-and-the-absolute-universal-structures-iv/

[Ross Finlayson]

Sure, trot out the oldest nag at the working farm.

That, "the infinite", is, "uncountable", is reduced to a footnote of Kant is his last Judgment on the Sublime,
Or his last Critique the Critique on Judgment, what's connected is sublime and it's continuous.

So, all the rest of the canon is kept then this also struck through as "an infinity". I.e., what philosophy
says logic says is only phenomenological and infinity is only noumenal. It is, Kant says Des Cartes says,
or says for him, that Duns Scotus, says. "There is a mathematical infinity."

Which Platonists say, .... On a good day.

There's still the rest of the Absolute and "G-d's Infinity", universal G-d, monist G-d, "All Infinity".
It's just there's let in the usual notion of sublime as greater that lets in continuity and
infinite-divisibility, all together, "only mathematically".

So, formally that's just a distinction rule with whatever finite, bounded model and all things.
"Infinity: there is one."

Then Kant in his "On the Old Saw", "Ethics", "Government", .... No, this is the "technical",
philosophy, logic, it's technical, humanity is arbitrarily an abstract concept, "philosophy".
So, in Kant's world, it's just pointed down "sublime" and the "subliminal", infinitely-divisible,
continuous, "Kant's DesCartes' Euclid's geometry's points including at infinity".

Then, foundations, I tell you, I'm glad I leafed all through a text of Courant, my differential
results, under "Cantor Space and Square Cantor Space, a non-Cartesian function", put
the most direct sorts applications, for differential analysis, into usual laws of large numbers
in the infinitely-divisible and continuum analysis. That: "the sweep function is its own anti-derivative",
is about the most remarkable fact to function theory, that "the exponential function is its own anti-derivative".

Not a real function, ..., that there is one.

Not-a-real-function, ....

So, for the differential analysis, there is this idee fixe for the fixed point, the identity dimension,
it is two Cartesian dimension, then x = y together, and in the infinite identity dimension x = y= z
and so on, relating two dimensions to indicate the level dimension, and having upper and lower
reflections, in the quadrant in the half-plane.

This being the envelope of a usual ramp, f(x) = x, involves a sort of style of differential analysis,
bounding to the corners under the hyperbola, families of those, that though fill the space and
area, like Fourier-style analysis has the windowing and the boxing as it were or the intervals,
this has results differentially from the origin and identity, out perspective what reflects bounds.

So, that's called "implicits" these days then also Courant introduces quite a few methods,
and justifications including "counterexamples in demonstration of convergence".
(That result convergence, ....) Here there's a completion established under the symmetries
of the two Cartesian dimensions and the identity dimension, either one and both, or
then simply the origin and identity, and envelope, "completions", that result similarly
to a Fourier-style analysis and its justifications windowing and boxing and thus resulting
transforms after the orthogonal and the kernels, "completions", is under hyperbolas,
resulting a sort wave-integration, the coefficients of the elements of the terms of the
infinite and the telescoping, series.

That though is defined algebraically itself, "foundations", yet of course such useful notions
as "the origin" and "angle", define all sorts forms, proportions, and geometry.

Analytically, ....

These days a generally approach of "completions" and "almost everywhere, equal",
(that completions are everywhere, and almost everywhere isn't), is most usual
"approximation, which is close" and "error, which is off".


Then you only need three definitions of continuity, any one of which are neat and small.
Given most objects are properties resulting, you don't have to attach your ontological
commitment to whatever makes it so, while of course there is what it is, as so.

So, renewed observance of what was expected a formal system, helps show not
just the common line, but all the so formal systems, what results a wealth common
results about usually laws of numbers: laws of large numbers and laws of small numbers,
in laws of numbers, here simply the types of infinities in the infinitely-divisible have their
own completions, and as large numbers of those sorts, laws of large numbers, and
"extra", actually "beyond", necessarily, any finite. I.e., together they're as so,
common systems of infinite divisibility.



Any regular language has a model in ordinary set theory.
They can't all use it at once, ..., the model of all ordinary set theory.


It's a set theory, it's a continuum mechanics, "foundations" has to be both.

[Ross Finlayson]

On Thursday, November 9, 2023 at 9:10:23 PM UTC-8, Adam Polak wrote:

Well, that should be rather clear in a direct sort of way,
that ordinary set theory is inconsistent.

It's not inconsistent at all insofar as it's regular, but there
are theorems resulting from quantification, that result
extra elements, "extra-ordinary", whether the ordinary is
extra or it's extra, the ordinary, not inconsistent at all,
then though that "as a model of a 'closed' regular no-infinite-
descending-chain well-founded theory, it is inconsistent,
set theory".

Ordinary set theory is inconsistent about infinity only insofar
as to be consistent again as it was consistent from and as it
is consistent, toward, what is ordering or numbering what results
counting in what's the most usual sort basis of a counting argument,
assigning counts upon numbers, those being all ordinary and finite
and usual, then all the ideas about their number and order,
qualitatively/quantitatively, that way.

So, what comes around from "set theory is inconsistent, because
infinite is infinite and quantification in the infinite", it only comes around
again, stronger, that it's "extra-consistent" and "infinity equals infinity plus one"
still retains "and also is greater than it".

Then, all that's just called "sublime", extra and greater, making all the thinkers
already set up, have to make it so, also, what they can accept.

[WM]

On 10.11.2023 06:10, Adam Polak wrote:

> From the history of science, we know many sometimes very spectacular
cases when "great" theories, accepted by "everyone," turned out to be
greatly flawed or at least imperfect at some point. The same is about to
happen soon with Cantor's Set Theory.

Cantor's mistake is this dilemma:

(1) He assumes that infinity is actual. This assumption need not be true
but it appears credible, in my opinion, when we look at the natural
numbers or at the fractions or at the points of the real line. However
we can use, define, or manipulate as individuals only a potentially
infinite collection, for instance
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| remains infinite.

(2) Cantor uses only the potentially infinite collections for his
bijections (because more is not possible). His result is that all
collections have the same number of elements that can be paired
(bypassing his mistaken result with real numbers here).

If infinity is actual however, then it is fixed. Then {0, 1, 2, 3, ...}
has exactly one element more than {1, 2, 3, ...}. Hilbert's hotel
suffers from the same mistake.

But since these "bijections" have resulted in so drastically
"counterintuitive", i.e. false results, a religious belief necessarily
has spread among the disciples of this matheology which will be
impossible to exorcise. No argument will have an effect.

Regards WM

[FromTheRafters]

It happens that WM formulated :

Then your work is done here, congratulations.

[WM]

On 11.11.2023 10:10, FromTheRafters wrote:
> It happens that WM formulated :
>

>> But since these "bijections" have resulted in so drastically
>> "counterintuitive", i.e. false results, a religious belief necessarily
>> has spread among the disciples of this matheology which will be
>> impossible to exorcise. No argument will have an effect.
>
> Then your work is done here, congratulations.

Are here only matheologians around? There are certainly many readers who
can become informed about the real situation.

Regards, WM

[Fritz Feldhase]

On Saturday, November 11, 2023 at 9:39:09 AM UTC+1, WM wrote:

> {0, 1, 2, ...} has exactly one element more than {1, 2, 3, ...}.

Look dumbo, I once asked you if {0, 1, 2, ...} and {{0}, {1}, {2}, ...} are equinumerous.

Your answer was YES. After all, {{0}, {1}, {2}, ...} is just {{n} : n e {0, 1, 2, ...}}. In other words, n |-> {n} is a bijection from {0, 1, 2, ...} onto {{0}, {1}, {2}, ...}.

This fact does not depend on any specific definition of the natural numbers.

Now in the context of Zermelo's ORIGINAL set theory (Z). The natural numbers are defined the following way: 0 = {}, 1 = {0}, 2 = {1}, 3 = {2}, in general: {n} = n+1 (for n e IN, n =/= 0).

THIS means that {1, 2, 3, ...} just *is* {{0}, {1}, {2}, ...} in the context of Z.

Hence you AGREE that {0, 1, 2, ...} and {1, 2, 3, ...} are equinumerous.

But this CONTRADICTS your claim: "{0, 1, 2, ...} has exactly one element more than {1, 2, 3, ...}".

Mückenheim, you are just too dumb for any form of mathematics.

MOST of your claims are SELF-CONTRADICTORY.

[Ross Finlayson]

Sure, I really think that intuition, abstraction itself, is the greatest precept of mathematics,
so as a constructivist, there's really so much to be said that it's pointed out that so-and-so,
said, "all mathematics" or "geometry" or "the logic", provides a subject and a classification,
that the intuitionist and intuitionistic surrounds even make for that "axiomatization" is as so,
"what if" and "any what if", that's though "theory", again of course, is both "the theory that's
reality's" and "all the rest of any of the theories", that a usual axiom is there are two halves
of the universe of theories: one is all reality and the other is none, yet at least, some (flights
of reality).

This way at least "theory is never wrong", while at the same time, with truth up front and throughout,
"must be a better theory".

This is so fundamental that it's couched in all philosophy and an entire, more or less,
entire East-West canon, and Western Canon, "reason", to say so much in so much time,
of what results a study, say of "foundations" as "classical foundations".

So, these days it's again "paleo-classical", that most terms what are super and extra,
like energy and entelechia, are new again, if though read from the "originals" or 2500
years ago, "paleo-classical modern inclusive extra-classical", so, what happens is that
the "post-modern, paleo-classical", approach, is digging up all sorts the recorded library,
of "authority", in science, reminding that the discussions concerned are same, and able
to gently, or generously, interpret the paleo-classical, as in some intent ideal:
that according to the canon, what's intended is even better than what's received,
"Perfect Plato: Never Wrong", and "What Plato might've said: if you defined what
he knew and thought", or, "Reading Plato: as if it always has to make sense".

It's easier in an imperfect world if "imperfect", it's perfect.


Then bounds and limits, it's most always usual that a symbolic calculator doesn't much
need the same space of arithmetic and language as all its expressions, ....

So, it's an intuitionist constructivist mathematics, and in the elementary, what's
central is central and what's primary is primary, elements in axiomatics, what's fundamental,
that mathematics also has a fundamental model, to the real.

"It's a continuum mechanics, ..., it's a set theory, ..., it's a gauge theory, ...", according
to mathematical science, that's about what it is.

(...A current working theory that current working theory altogether is sufficient, one model,
"that there is one".)

Then what I wrote is called an "apologetics", for foundations, so it wouldn't be algebraically
void or otherwise "un-conscientious" all the deliberation why "set theory", is viewed through
this generous lens, "never was wrong, since it was fixed", up into "paradoxes of set theory"
and "paradoxes of classical logic with ambiguity of the double-negative", "explanations of
continuum mechanics by quantum mechanics", "total field effects in universal gauge theory",
for my own interest I just wanted a neat table of formalisms, that in effect coordinate with
symbolry and other formalisms, providing fundamental theorems and lemmas, "non-paradoxes
of the extra-ordinary in set theory and extra-founded in ordinal theory", where there are
only sets in set theory and only ordinals in ordinal theory.

If there's one variable we all share it's time.

So, for set theory, Set Theory, there's Georg Cantor's et alia's Mengenlehre, set theory,
(set book or set reading), then it's not that difficult or so many pages, to write out
all the support for reasoning according to ZF and ZFC set theory, providing the usual
support and opinion to real analysis, and the open topology of the complete ordered field
or the real numbers algebra's complete ordered field and Dedekind completeness providing
Cauchy completeness, Eudoxus/Dedekind/Cauchy, what you do is you entirely separate the
"generous reading and perfect lens of the received principles the associated forms",
from, "that each restriction of comprehension establishes an incompleteness".

I.e., whether sets and classes are the same or different, "in a theory those being the objects
and there is only one relation, elt, but the reflexive or inverse part of the relation is contains,
or either way as the sole indicator in the theory relation", sets and classes are no different,
until up above axiomatics, results that at least an entirely different theory results,
when the relation is reflexive besides inverse, i.e. containing itself, reflexive, or together contained
in a greater, union and pairing.

Then the other way leads right to Cantor's and Russell's paradox what for relation,
show that there are other ways to resolve these paradoxes entirely theories of their own.

Then "that surely being intuitionistic the what must be constructivist", is also "that
being what also makes upon what rests the entire theory".

Trust in theory, ....


A working theory is usually whatever its elements "are". Trust in theory then "is that
they are", things are as they are, then trust in science has "or last least aren't not",
knowing "I don't know everything".

So, people with a lot of time to study it call all of mathematics, science, physics, and
so on one thing all the way to reason and philosophy, then that the entire industry,
is "foundations".

These days there's "axiomatic, descriptive set theory", which is fundamental that sets
are elementary and primary and central and in suitable organizations of type, the contents
of sets, that it is "descriptive" what the set "is", as its contents from the language of set
theory, are its own, "set", or class, of constants unique to the description.

Sets are only sets in set theory, there are only sets in set theory. All the rest attachment
is under model theory, or proof theory equivalently, then "descriptive set theory", is the
usual notion, of any sort use of sets in mathematics, in a sense, where the constants are
one model. So, the idea of that "the language of sets, must be a class, simply to indicate
this overall organization of description, and models", helps explain why "thus pretty much
all of mathematics has a descriptive set theory, vis-a-vis, fundamental set theory".

[Adam Polak]

Dear Ross,
Many words, for which I thank you very much. Some are quite valuable. I apologize if I'm not interpreting them correctly, but it seems a bit like an (perhaps not entirely conscious) attempt to "talk out/nock out" the problem, blur it in words, drown it in words. The problem EXISTS, it is significant, and it won't sink.

Many statements (aspiring to the role of a proper description of reality) contained in set theory or arising from it are simply UNTRUE, a description that is FLAWED.

Example: The claim that any of infinite sets should be perceived quantitatively as more numerous than the infinite set of natural numbers is a huge mistake. (It is easy to demonstrate, and I will do it in several ways, and in a louder manner - limiting presence here to get some spare time).

Of course, in the working phase, various theories can be formulated and explored as potential alternative descriptions of reality. However, if one of these theories turns out to be erroneous in many places, describing an imaginary reality it generated itself, it ceases to be a scientific theory, ceases to be a part of science. It becomes science-fiction, and in some cases, pure fantasy.

I know that engaging with science-fiction or outright fantasy can be very enjoyable and, in a way, very satisfying. Fiction offers far greater possibilities than science. However, consciously playing with fantasy while pretending to be a scientist is simply extremely immoral; it is deceit. It essentially degrades the function of a scientist to that of a fraud.

On the other hand, unconsciously indulging in fantasy under the guise of running scientific work undermines intellectual competence and essentially reduces such a "scientist" to the role of a fool.

I apologize for these blunt words. I don't mean anyone specific, but many should contemplate this.

The only possible result of mixing milk with a puddle is more puddle.

BR,
Adam

[Ross Finlayson]

Well, you might learn about Skolem, and about how he shows there's a countably infinite model
of things.

I understand that some people's idea of the integers, zero to infinity, scale exactly, to zero to one.
Of course, not everybody, and indeed that the powerset theorem is a thing gets into how there
can be a space of 0's and 1's, and, what's "ordinary" and "extra-ordinary".

There's a bunch to study to figure out that these days' classical logic is at contest with logic
of refutations and relevance, what most would consider as common sense, because what's
called classical pretty much isn't and de Morgan has more going on for resolving duals, than Boole.

But, it's it's not really so much of a bunch: it's not really that much to get a grasp on what goes
into various logics what result they address all the objects of logical discourse, about same.

Then, whether "there doesn't even exist a _standard_ model of integers, only extensions and fragments,
of models of integers", gets into why you should explored open-mindedly, while still of course
keeping the absolutes apart: empty and infinity. (... and that they spring from the same and
are one.)

"Hope and a bottle of ketchup isn't a hamburger."


Anyways there is _ordinary_ set theory and many are familiar with it and it suffices for their
platforms if not foundations itself, and there is _extra-ordinary_ set theory which is by definition
more of the universe of sets, and as after and about class/set distinction, and the lack thereof.

In a set theory it's a set theory, in a part theory a part theory, ..., it's set theory.

[WM]

Adam Polak schrieb am Montag, 13. November 2023 um 07:03:52 UTC+1:

> The claim that any of infinite sets should be perceived
quantitatively as more numerous than the infinite set of natural numbers
is a huge mistake.

In every finite section from 0 to n there are far more rational numbers
than natural numbers. Therefore, to infere the infinite from the finite,
there are far more rationals. Equinumerosity can only be accepted if
there is no complete reality of numbers but the potential infinity of
numbers created by us.

Regards, WM

[WM]

Adam Polak schrieb am Montag, 13. November 2023 um 07:03:52 UTC+1:

> The claim that any of infinite sets should be perceived
quantitatively as more numerous than the infinite set of natural numbers
is a huge mistake.

[Adam Polak]

Any "finite section" / finite set is not a representative object for conducting research and adjudicating on the specifics of any infinite set.
Almost without exaggeration, I will say that the similarity between any, even the largest, finite set and an infinite set, on a scale from 1 to 10 is ZERO.

[Ross Finlayson]

Ah, you should not ignore models of 'effective infinity', that are basically large finite numbers,
where all the precision of measurements is for example less than the root of it, it suffices
that for any measurement precision, that its increments are arbitrarily small.

For example, modern-day Democritan atomic theory, and Planck length/time and 1 AMU,
has that Democritus or Demokrites, theoretical atoms are after the infinitely-divisible,
but the way it works as for substances the regime of the atomic elements, that Avogadro's
number is arbitrarily large, 10^23, about that "atoms are on the order of 25 orders of magnitude,
smaller, than us".

Then, the idea is that "superstrings" or "superstring theory" are that much arbitrarily smaller
again, or around 50 orders of magnitude, while still, not "infinitely-divisible" or "infinite".

If you wonder, every few years the fundamental small atomic constants their dimensions
are updated or as NIST CODATA, they don't just get more precise, but, smaller.

So, anyways you should be aware that there are lots of notions of the infinitely-divisible
that aren't, but are "effectively" infinitely-divisible, then getting into all the mathematics
of the Euclidean and so on in the Planck regimes, of course they aren't infinite but many
of the usual results associated with properties of the infinitely-divisible hold for it.

(Which is basically a notion that there's an arbitrarily small iota i and arbitrarily large
infinity I that iI = 1, or that 1/oo = iota and the sum or multiples of iota-values,
makes exactly one.)


You can exclude them, "potentially infinite" from the "actually infinite", but the
"effectively infinite" is very usual as "least measurable quantity's inverse".

[WM]

On 13.11.2023 09:37, Adam Polak wrote:
> Any "finite section" / finite set is not a representative object for conducting research and adjudicating on the specifics of any infinite set.

If you mean actual infinity, that is right. But potential infinity is
only a never ending sequence of finite sets.

> Almost without exaggeration, I will say that the similarity between any, even the largest, finite set and an infinite set, on a scale from 1 to 10 is ZERO.

Yes, this is true for actual infinity because the finite part is
vanishing compared to the infinite part:
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = always actually infinite.
Nevertheless even in actually infinite sets basic logic is valid.
Example: If every definable natural number has infinitely many natural
successors, then all definable natural numbers have infinitely many
natural successors.

Regards, WM

[Jim Burns]

On 11/13/2023 2:24 PM, WM wrote:
> On 13.11.2023 09:37, Adam Polak wrote:

>> [...]

> Nevertheless
> even in actually infinite sets
> basic logic is valid.

The problem here is not with the claim itself,
but with what you mean by the claim.

What you (WM) mean by basic logic
isn't basic logic.
It is a quantifier shift.

Even in infinite sets,
a quantifier shift is unreliable (invalid).

> Example:
> If every definable natural number has
> infinitely many natural successors,
> then all definable natural numbers have
> infinitely many natural successors.

No.

Each definable natural has
infinitely-many definable natural successors.
∀n ∈ ℕ, ∃S ᐜ⊆ ℕ: n <ᣔ S

Each definable natural has
infinitely-many definable naturals which
it does not succeed.
∀n ∈ ℕ, ∃S ᐜ⊆ ℕ: S ᣔ≮ n

Each definable natural does not succeed
all definable naturals.
∀n ∈ ℕ: ¬(ℕ ᣔ≤ n)

All definable naturals have
zero-many definable natural successors.
¬∃n ∈ ℕ: ℕ ᣔ≤ n

[Chris M. Thomasson]

On 11/13/2023 11:24 AM, WM wrote:
> On 13.11.2023 09:37, Adam Polak wrote:
>> Any "finite section" / finite set is not a representative object for
>> conducting research and adjudicating on the specifics of any infinite
>> set.
>
> If you mean actual infinity, that is right. But potential infinity is
> only a never ending sequence of finite sets.

[...]

MORON!!!!!

[FromTheRafters]

Chris M. Thomasson was thinking very hard :

He never gets it. I thought he was going to get it this time, but no.
Well, maybe in another decade or so it will sink in.

Here is a question to ponder:

Consider the set of points in the real interval [pi,pi+1] and how many
irrationals that you can remove from consideration in defining a new
set as follows.

Usually in a real interval like [0,1] you can 'address' one or more
rational points by divide by two (halfway point) and then two more by
dividing by three (one third and two thirds points) and on and on; but
this pi to pi plus one interval won't work that way as the numbers
'addressed' in this manner cannot be in Q.

However, in building your new set, only countably many irrationals will
be excised from consideration this way giving you only rationals for
your new set unless there are more irrationals than there are
rationals.

How can there be only one size of infinte in his world?

[Fritz Feldhase]

On Monday, November 13, 2023 at 11:30:28 PM UTC+1, Jim Burns wrote:
> On 11/13/2023 2:24 PM, WM wrote:
> >
> > If every definable natural number has
> > infinitely many natural successors,
> > then all definable natural numbers have
> > infinitely many natural successors.
> >
> No.

Actually, YES.

Hint: If every x in A has the property P, then all x in A have the property P.

In the context of mathematics we (except of you, it seems) would formalize this claim the following way:

Ax e A: P(x) -> Ax e A: P(x).

This is clearly a substitution instance of the logical tautology

P -> P.

Hence it's certrainly /true/.

[Fritz Feldhase]

You think?

[Ross Finlayson]

Ah, that a "quantifer shift" may be valid is called according to the "transfer principle",
that what's true for "each" is true for "all", or as with respect to "heap/sorites" sometimes.
There's even where "quantifier shift" is "anti-transfer", or for variously when it does
or doesn't validly apply, like so.

There are even functions that according to usual convergence tests and laws of large
numbers, for example ranging from zero to one, appear to have a limit that equals one,
but actually go to zero, "functions that belie their finite inputs", if of course, not
necessarily, "standard real functions".

The "Ramsey theory" has a lot going on about various asymptotics.

[Jim Burns]

On 11/13/2023 6:33 PM, Fritz Feldhase wrote:
> On Monday, November 13, 2023
> at 11:30:28 PM UTC+1, Jim Burns wrote:
>> On 11/13/2023 2:24 PM, WM wrote:

>>> If every definable natural number has
>>> infinitely many natural successors,

WM: if ∀n ∈ ℕ, ∃Sₙ ᐜ⊆ ℕ: n <ᣔ Sₙ

>>> then all definable natural numbers have
>>> infinitely many natural successors.

WM: then ∃S ᐜ⊆ ℕ, ∀nₛ ∈ ℕ: nₛ <ᣔ S

>>
>> No.
>
> Actually, YES.
>
> Hint:
> If every x in A has the property P,
> then all x in A have the property P.

Yes.
However,
I think WM means something else.

> In the context of mathematics
> we (except of you, it seems) would formalize
> this claim the following way:
>
> Ax e A: P(x) -> Ax e A: P(x).
>
> This is clearly a substitution instance of
> the logical tautology
>
> P -> P.
>
> Hence it's certrainly /true/.

P -> P
which you (FF) think WM means,
is true.

∀n∈ℕ, ∃Sₙᐜ⊆ℕ: n<ᣔSₙ -> ∃Sᐜ⊆ℕ, ∀nₛ∈ℕ: nₛ<ᣔS
which I (JB) think WM means,
is not true.

[Ross Finlayson]

It's pretty simple, you can conscientiously be an ultra-finitist, and indeed for each
there's a certain requirement of finitude and boundedness for matters of comprehension,
but there's also plenty of understanding of large numbers and that there's not just
one "law of large numbers", then besides there's that the "infinite" has its places according
to geometry and perspective, number theory, set theory, and so on, and "infinitesimals"
have their places in topology, analysis, and so on. So, you're welcome to reject the infinite,
and be ultra-finitist, but not refute it, which would be a "retro-finitist crankety troll".

[mitchr...@gmail.com]

What does set theory mean in math?
A matrix is a set at an angle.
Why would that change anything?

[Adam Polak]

poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):

> If you mean actual infinity, that is right. But potential infinity is
> only a never ending sequence of finite sets.

> Nevertheless even in actually infinite sets basic logic is valid.
> Example: If every definable natural number has infinitely many natural
> successors, then all definable natural numbers have infinitely many
> natural successors.

Regarding logic. Yes, it is crucial and is completely missing in set theory and elsewhere, unfortunately.

"everyone" and "all" have no quantitative context when used to describe an infinite set (infinity), ONLY qualitative! If you interpret it differently, you are making a mistake.
Look,
these:
1, 2, 3, 4, 5, 6, 7, 8...
I, II, III, IV, 5, 6, 7, 8, ...
are QUANTITATIVELY
EXACTLY THE SAME SET/ THE SAME CONSTRUCTION
they differ only in the qualitative parameters of a few elements

consequently, this:
0, 1, 2, 3, 4, 5, 6, 7, ...
is also the same set (from QUANTITATIVE perspective!)
same as:
-2, -1, 0, 1, 2, 3, 4, 5, ...
and so on

It can be said that in a quantitative context there is ONLY ONE infinite set. Or to put it another way, there is only one model of the infinite set, with which - in a quantitative context! - EVERY infinite set, WITHOUT EXCEPTION, is consistent.
The differences between infinite sets come down only to differences in the qualitative parameters of individual objects that are elements of different infinite sets.

It is impossible to understand these issues if you do not understand a number of subtle but key issues, e.g.: as I wrote above: "all" "every" only in the context of qualitative parameters, never in the quantitative context in relation to an infinite set.

Another extremely important issue: Number and numerical value

"numerical value" - is not a mathematical object, it is only a qualitative parameter that an object may have,

"a number" - is a mathematical object with a qualitative parameter that is numerical value, preferably if it is the only qualitative parameter of the number,
if additional ones appear, problems will also arise
e.g.:
4 - "four" (number, mathematical object)
IV - "four" (number, mathematical object)
but!:
IV - "four Roman" (it is an object with an additional quality parameter "Roman", as a result it is no longer a correctly defined number, it is not "a number" anymore)

This may seem like little or nothing important all, but it's an illusion.
This exact error leads to the conclusion that there may be objects other than irrational numbers in the right column of the matrix (Cantor's Diagonal Method). This is simply not true.
This:
0.001
and this:
0.0010000000000000...
are two completely different objects
only the first of which is a properly constructed decimal, "a number".
Yes, both objects have the same parameter that is a numerical value: 1/1000, but what does it matter if they differ in other parameters and in effect constitute two completely different mathematical objects and only one can be called the number: "one thousandth"

BR, Adam

[WM]

On 14.11.2023 07:09, Adam Polak wrote:
> poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):
>
>> If you mean actual infinity, that is right. But potential infinity is
>> only a never ending sequence of finite sets.
>
>> Nevertheless even in actually infinite sets basic logic is valid.
>> Example: If every definable natural number has infinitely many natural
>> successors, then all definable natural numbers have infinitely many
>> natural successors.
>
> Regarding logic. Yes, it is crucial and is completely missing in set theory and elsewhere, unfortunately.
>
> "everyone" and "all" have no quantitative context when used to describe an infinite set (infinity), ONLY qualitative! If you interpret it differently, you are making a mistake.
> Look,
> these:
> 1, 2, 3, 4, 5, 6, 7, 8...
> I, II, III, IV, 5, 6, 7, 8, ...
> are QUANTITATIVELY
> EXACTLY THE SAME SET/ THE SAME CONSTRUCTION
> they differ only in the qualitative parameters of a few elements

They differ only in the form of naming the first elements.

>
> consequently, this:
> 0, 1, 2, 3, 4, 5, 6, 7, ...
> is also the same set (from QUANTITATIVE perspective!)
> same as:
> -2, -1, 0, 1, 2, 3, 4, 5, ...
> and so on

If you wish to call 0 now -2, and so on, then both sets are the same. If
you use the names for the well-known numbers, then both sets differ,
namely by two elements of the second set which are missing in the first set.

Regards, WM

[WM]

On 14.11.2023 01:17, Jim Burns wrote:
> On 11/13/2023 6:33 PM, Fritz Feldhase wrote:
>> On Monday, November 13, 2023
>> at 11:30:28 PM UTC+1, Jim Burns wrote:
>>> On 11/13/2023 2:24 PM, WM wrote:
>
>>>> If every definable natural number has
>>>> infinitely many natural successors,

> Yes.
> However,
> I think WM means something else.

There is no other meaning. ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
That means there are ℵo dark numbers following upon all visible numbers.
Otherwise there would be a visible number without ℵo dark numbers
following upon it.

Regards, WM

[Adam Polak]

And don't wish anything. I'm just explaining.

Look, from a !quantitative perspective!
these sets:
(A) 1, 2, 3, 4, 5, 6, 7, 8, ...
(B) I, II, III, 4, 5, 6, 7, 8, ...
(C) &, P, !, 4, 5, 6, 7, 8, ...
are equivalent.
I hope this is understandable.

Note, changing the "names" (i.e. quality parameters) of three elements of the set did not bringed change the number of elements.

This exact rule is ALWAYS in force!
even when it comes to changing the "name" of each element in the set.

This set: { 1, 2, 3, 144 }
is equinumerous!, with this set: { 1, 2, 7, 144 }
and with this one: { 5, 6, 7, 144 }
and with this one: { 777, 1490, @, P, 12 }

When you understand that changing the "names", i.e. changing the quality parameters of objects, does not change the number of elements in the set in ANY case!
you will no longer be surprised that:
this set:
1, 2, 3, 4, 5, 6, 7, 8, ...
is equivalent to this one:
2, 4, 6, 8, 10, 12, 14, 16...
and with this one:
1, 2, 3, 4, 5, 6, 7, 8, ...
and with the one that "contains all" these elements:
1/1, 1/2, 1/3, ...
2/1, 2/2, 2/3, ...
3/1, 3/2, 3/3, ...
... , ... , ... , ...

Simply, all sets containing n elements (for n belonging to N) are quantitatively equal.
Exactly like all infinite sets - interpreted as "containing oo number" of elements, they are quantitativeli equal.

BR,
Adam

[FromTheRafters]

WM expressed precisely :

No it doesn't, it means that a difference set between the naturals and
a finite subset of the naturals has cardinality aleph_zero.

[FromTheRafters]

on 11/14/2023, Adam Polak supposed :

> wtorek, 14 listopada 2023 o 13:04:21 UTC+1 WM napisał(a):
>> On 14.11.2023 07:09, Adam Polak wrote:
>>> poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):
>
>> If you wish to call 0 now -2, and so on, then both sets are the same. If
>> you use the names for the well-known numbers, then both sets differ,
>> namely by two elements of the second set which are missing in the first set.
>>
>> Regards, WM
>
> And don't wish anything. I'm just explaining.
>
> Look, from a !quantitative perspective!
> these sets:
> (A) 1, 2, 3, 4, 5, 6, 7, 8, ...
> (B) I, II, III, 4, 5, 6, 7, 8, ...
> (C) &, P, !, 4, 5, 6, 7, 8, ...
> are equivalent.
> I hope this is understandable.
>
> Note, changing the "names" (i.e. quality parameters) of three elements of the
> set did not bringed change the number of elements.
>
> This exact rule is ALWAYS in force!
> even when it comes to changing the "name" of each element in the set.
>
> This set: { 1, 2, 3, 144 }
> is equinumerous!, with this set: { 1, 2, 7, 144 }
> and with this one: { 5, 6, 7, 144 }
> and with this one: { 777, 1490, @, P, 12 }

A set with four elements is equinumerous with one with five elements?

Maybe you should include a non-infinite emptyset with your system so
that arithmetic works?

[WM]

On 14.11.2023 14:02, Adam Polak wrote:
> wtorek, 14 listopada 2023 o 13:04:21 UTC+1 WM napisał(a):
>> On 14.11.2023 07:09, Adam Polak wrote:
>>> poniedziałek, 13 listopada 2023 o 20:24:24 UTC+1 WM napisał(a):
>
>> If you wish to call 0 now -2, and so on, then both sets are the same. If
>> you use the names for the well-known numbers, then both sets differ,
>> namely by two elements of the second set which are missing in the first set.
>>
>

> And don't wish anything. I'm just explaining.

The you should see that {0, 1, 2, 3, 4, 5, 6, 7, ...} is in bijection
with itself and therefore {-2, -1, 0, 1, 2, 3, 4, 5, ..} has two more
elements. In actual infinity all elements are existing, independently of
who uses them in whatever constellation.

> Look, from a !quantitative perspective!
> these sets:
> (A) 1, 2, 3, 4, 5, 6, 7, 8, ...
> (B) I, II, III, 4, 5, 6, 7, 8, ...
> (C) &, P, !, 4, 5, 6, 7, 8, ...
> are equivalent

They are even equinumerous, because {4, 5, 6, 7, 8, ...} is in bijection
with itself and {1, 2, 3} and {I, II, III} and {&, P, !}, all have
cardinality 3.

>
> Note, changing the "names" (i.e. quality parameters) of three elements of the set did not bringed change the number of elements.

Of course. Cantor stated this already: "the cardinal number |M| remains
unchanged if in place of an element or of each of some elements, or even
of each of all elements m of M another thing is substituted." [E.
Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und
philosophischen Inhalts", Springer, Berlin (1932) p. 283]

>
> This exact rule is ALWAYS in force!
> even when it comes to changing the "name" of each element in the set.
>
> This set: { 1, 2, 3, 144 }
> is equinumerous!, with this set: { 1, 2, 7, 144 }
> and with this one: { 5, 6, 7, 144 }

Of course, we agree.

> and with this one: { 777, 1490, @, P, 12 }

Not quite.

>
> When you understand that changing the "names", i.e. changing the quality parameters of objects, does not change the number of elements in the set in ANY case!
> you will no longer be surprised that:
> this set:
> 1, 2, 3, 4, 5, 6, 7, 8, ...
> is equivalent to this one:
> 2, 4, 6, 8, 10, 12, 14, 16...

That is wrong because there are not enough even natural numbers that
could be substituted. There are only half as many even numbers as
natural numbers.

> and with the one that "contains all" these elements:
> 1/1, 1/2, 1/3, ...
> 2/1, 2/2, 2/3, ...
> 3/1, 3/2, 3/3, ...
> ... , ... , ... , ...

No, that is wrong! In every line there are |ℕ| elements. In every column
there are |ℕ| elements. The complete matrix contains |ℕ|^2 elements. For
proof that it cannot be enumerated see "A game like billiards"
https://groups.google.com/g/sci.math/c/lH60V21yyi4.

>
> Simply, all sets containing n elements (for n belonging to N) are quantitatively equal.
> Exactly like all infinite sets - interpreted as "containing oo number" of elements, they are quantitativeli equal.
>

oo denotes potential infinity. All such sets can be put in bijection.
But actual infinity forbids that.
|ℕ| - 1 =/= |ℕ| =/= |ℕ| + 1 and {0, 1, 2, 3, ..., ω} \ ℕ = {0, ω}.

Regards, WM

[WM]

On 14.11.2023 15:30, FromTheRafters wrote:
> WM expressed precisely :

>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>> That means there are ℵo dark numbers following upon all visible
>> numbers. Otherwise there would be a visible number without ℵo dark
>> numbers following upon it.
>
> No it doesn't, it means that a difference set between the naturals and a
> finite subset of the naturals has cardinality aleph_zero.

Every definable initial segment of the naturals is finite. The union of
definable initial segments is potentially infinite, namely *always
finite*. Therefore the difference set is dark and actually infinite.

Try to find a natural number of the union of finite initial segments
which has not a set of ℵo dark numbers following it.

Regards, WM

[Fritz Feldhase]

On Tuesday, November 14, 2023 at 7:09:14 AM UTC+1, Adam Polak wrote:

> Regarding logic. Yes, it is crucial and is completely missing in set theory and <bla>

No, it's not "missing in set theory". Actually, the logical framework is CRUCIAL for modern/axiomatic set theory, idiot.

> 1, 2, 3, 4, 5, 6, 7, 8...
> I, II, III, IV, 5, 6, 7, 8, ...

> are [...] THE SAME SET[...]

It seems to me, that what you want to express is just: {1, 2, 3, 4, 5, 6, 7, 8...} = {I, II, III, IV, 5, 6, 7, 8, ...} .

This certainly is true, if the symbols "I", "II", "III", "IV" are used to refer to 1, 2, 3, 4, resp.

> 0, 1, 2, 3, 4, 5, 6, 7, ...

> is [...] the same set [...] as:

> -2, -1, 0, 1, 2, 3, 4, 5, ...

No, the set {0, 1, 2, 3, 4, 5, 6, 7, ...} is usually NOT considered identical with the set {-2, -1, 0, 1, 2, 3, 4, 5, ...}, since usually we assume that -2, -1 !e {0, 1, 2, 3, 4, 5, 6, 7, ... }.

> "all" "every"

(in the present context) refer to the /universal quantifier/. See: https://en.wikipedia.org/wiki/Universal_quantification

> 0.001
> and this:
> 0.0010000000000000...
> are two completely different objects

No, they aren't. Both expressions "0.001" and "0.0010000000000000..." refer to (denote) the very same /object/, namely the rational number 1/1000.

[Adam Polak]

Of course, you're right, I inserted one element too many in this example: { 777, 1490, @, P, 12 }
It should be e.g.: { 777, 1490, @, P }
Thank you for remark

BR, Adam

[Jim Burns]

On 11/14/2023 7:10 AM, WM wrote:
> On 14.11.2023 01:17, Jim Burns wrote:
>> On 11/13/2023 6:33 PM, Fritz Feldhase wrote:
>>> On Monday, November 13, 2023
>>> at 11:30:28 PM UTC+1, Jim Burns wrote:
>>>> On 11/13/2023 2:24 PM, WM wrote:

>>>>> If every definable natural number has
>>>>> infinitely many natural successors,

>>>>> then all definable natural numbers have
>>>>> infinitely many natural successors.

>>> If every x in A has the property P,
>>> then all x in A have the property P.

>> Yes.
>> However,
>> I think WM means something else.
>
> There is no other meaning.
> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> That means there are ℵo dark numbers
> following upon all visible numbers.

∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = ℵ₀
That means that,
for each visible number,
there are ℵ₀ visible numbers following.
Which is true.

∀⟨1,…,n⟩ ⊆ ℕ_def
ℕ_def = ⋂{S⊆ℕ_def| ∀⟨1,…,n⟩ ⊆ S }

> Otherwise
> there would be a visible number without
> ℵo dark numbers following upon it.

There are only visible numbers in ℕ_def
Nevertheless,
∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = ℵ₀

[FromTheRafters]

Adam Polak used his keyboard to write :

I believe it is only by convention that we drop the trailing zeros in
the infinite continued decimal expansion representation and call it
terminated.

0.0009999... or 0.000(9) is another unending CDE for that same
mathematical object.

[Fritz Feldhase]

On Tuesday, November 14, 2023 at 10:21:57 PM UTC+1, Jim Burns wrote:
> On 11/14/2023 7:10 AM, WM wrote:
> >

> > ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo .

> >
> > That means there are ℵo dark numbers
> > following upon all visible numbers.

Errr...

What WM (this fucking asshole full of shit) keeps to ignore:

∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo .

That means that for each and every natural number n the difference between IN and {1, ..., n} is countably infinte.

Equivalently: For each and every natural number n there are ℵ₀ natural numbers following (larger than) n.

_____________________________________________________________

This fucking asshole full of shit keep talking about "dark numbers" and "visible numbers", NO ONE ELSE is interested in.

[Adam Polak]

Yes, conventions, both good ones and those imperfect ones.

"Task for children" :) Find the difference between the pictures:

Pict No. 1: 1/2

Pict No. 2: 0.5


I'm sure everyone can spot it.
Each "picture" above shows a DIFFERENT object - these are two different mathematical objects! - the differences are noticeable at first glance. Both objects have the same qualitative feature, which is their numerical value.

They can be used interchangeably ONLY as long as only one of their properties, only one quality parameter of these objects (it's numerical value) is taken into account during applications, actions on/with these objects.
When another property is taken into account during activities, e.g., construction of object, e.g. meaning will be given to which of the objects has "/" as its element, these objects become completely different and independent.

This: 0.0009999... and this: 0.000(9)
it is the same only as long as you pretend not to notice the differences that are evident and completely ignore these differences in the actions on/with these objects.

This can not be puted in Cantor's Diagonal Method matrix:
0.001
isn't it?
and this one can:
0.0010000000000000...

Why? because it is determined by the structure of the object, not buy numerical value. Which means that in this case: 0.0010000000000000... is not correctly defined "the number" known as: "one thousandth".

Cantor's Diagonal Method matrix in it's well known form is flaweded.

[WM]

On 14.11.2023 22:21, Jim Burns wrote:
> On 11/14/2023 7:10 AM, WM wrote:

>> ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>> That means there are ℵo dark numbers
>> following upon all visible numbers.
>
> ∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = ℵ₀
> That means that,
> for each visible number,
> there are ℵ₀ visible numbers following.
> Which is true.

No that is wrong. ℕ_def is potentially infinite. There are oo visible
numbers following upon every visible number. But ℕ is complete. Best
this can be seen by the unit fractions.
The function NUF(x) cannot increase from NUF(0) = 0 to NUF(x>0) = ℵo.

You claim this antimathematical mantra of matheology. But it is refuted
by ∀n ∈ ℕ: 1/n - 1/(n+1) = 1/(n(n+1)) > 0, a finite distance.

Correct would be to say: For all x ∈ (0, 1] which are larger than at
least ℵo unit fractions and the gaps between them, NUF(x) = ℵo.
This cannot be all x > 0, because the unit fractions and the gaps
between them occupy points on the positive real axis.

>
> There are only visible numbers in ℕ_def
> Nevertheless,
> ∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = ℵ₀

Wrong. You will never have defined ℵ₀ natural numbers.

∀n ∈ ℕ_def: |ℕ_def\{1,2,3,...,n}| = oo.

∀n ∈ ℕ_def: Number of definable unit fractions between 0 and the
definable unit fraction 1/n = oo.

Regards, WM

[FromTheRafters]

Adam Polak used his keyboard to write :

> środa, 15 listopada 2023 o 00:20:50 UTC+1 FromTheRafters napisał(a):

> Yes, conventions, both good ones and those imperfect ones.
>
> "Task for children" :) Find the difference between the pictures:
>

> Pict No. 1 Pict No. 2
>
> 1/2 0.5

>
> I'm sure everyone can spot it.
> Each "picture" above shows a DIFFERENT object - these are two different
> mathematical objects! - the differences are noticeable at first glance. Both
> objects have the same qualitative feature, which is their numerical value.

I disagree. Mathematically speaking those are two different symbolic
representations of the same mathematical object.

[...]

[Adam Polak]

You can, of course, disagree with reality, but you can't change it.
this: 1
this: 23/23
this: 1.00
this: 1.(0)
are completely different objects, which can be seen at first glance, having one common parameter: a numerical value.
Mathematically speaking, mathematics as a science honestly admits that it does not know what the object known as "the number ONE" is, and so on.
The common mistakes, I describe, are therefore not surprising. What is surprising that is the firmness of statements made in the absence of arguments,
because this: "Mathematically speaking" is not an argument at all.
Especially if you are aware that the truth and the official position of the scientific community is: mathematics as a science honestly admits that it does not know what the object known as "the number ONE" is.

BR,
Adam

[FromTheRafters]

After serious thinking Adam Polak wrote :

No, they are different symbols for the number one.

[Fritz Feldhase]

On Wednesday, November 15, 2023 at 3:21:19 PM UTC+1, FromTheRafters wrote:
> After serious thinking Adam Polak wrote :
> >

> > this: 1
> > this: 23/23
> > this: 1.00
> > this: 1.(0)
> > are completely different objects,

No, they aren't.

> No, they are different symbols for the number one.

Right.

1 = 23/23 = 1.00 = 1.(0)

Funny guy, our Adam.

[Fritz Feldhase]

On Wednesday, November 15, 2023 at 1:57:05 PM UTC+1, Adam Polak wrote:

> this: 1
> this: 23/23

> ...
> are completely different objects, which [...]

You are mixing up names/symbols/terms with the object(s) these names/symbols/terms refer to (denote).

The symbols "1", "23/23" both denote the number 1 (in a mathematical context like, say, analysis).

That's why we (may) write: 1 = 23/23.

On the other hand, clearly "1" =/= "23/23". (Hint: "1" consists of one character, while "23/23" consists of 5 characters.)

[Fritz Feldhase]

On Wednesday, November 15, 2023 at 10:11:00 AM UTC+1, Adam Polak wrote:
>
> Find the difference between the pictures:
>
> Pict No. 1: 1/2
>
> Pict No. 2: 0.5
>
>
> I'm sure everyone can spot it.

Sure.

What we SEE, are two different "symbolic expressions" (consisting of characters).

> the differences are noticeable at first glance.

Indeed! In "1/2" we spot the characters "1", "/", and "2". While in "0.5" we spot the characters "0", "." and "5".

> these are two different mathematical objects!

Nope. "1/2" and "0.5" are two different "symbols" (names, terms) which denote (in certain mathematical contexts like, say, analysis) one and the same mathematical object, namely the rational number 1/5.

Hence these symbols

> can be used interchangeably

(in certain mathematical contexts like, say, analysis).

Hint: 1/2 = 0.5 (there).

Same with

> This: 0.0009999... and this: 0.000(9)

Hint: 0.0009999... = 0.000(9) (there).

> <nonsense deleted>

[FromTheRafters]

on 11/15/2023, Fritz Feldhase supposed :

I look forward to reading the arguments between WM and Adam Polak.

[Fritz Feldhase]

On Wednesday, November 15, 2023 at 7:11:51 PM UTC+1, FromTheRafters wrote:

> I look forward to reading the arguments between WM and Adam Polak.

Same, same. Crank fight! :-)