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

Adam Polak <mt69ap211@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.

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

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).

"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.

poniedziałek, 6 listopada 2023 o 13:46:53 UTC+1 Fritz Feldhase napisał(a):
> 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).

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.

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 be a finite set. Then an infinite set is just one the 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

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

Adam Polak <mt69ap211@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.

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.

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

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.

poniedziałek, 6 listopada 2023 o 22:00:28 UTC+1 Ben Bacarisse napisał(a):
> 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.

On 11/6/2023 1:09 PM, 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.
>
> 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 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.,

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!

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."

Dan

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

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)?

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?

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?

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".

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.

On Tuesday, November 7, 2023 at 2:14:26 AM UTC+1, Fritz Feldhase wrote:
> 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.

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

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

Adam Polak <mt69ap211@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.

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

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).

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!

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:

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!

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.