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Set Theory is DEAD!

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Adam Polak

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Nov 6, 2023, 6:01:03 AM11/6/23
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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

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Nov 6, 2023, 6:47:47 AM11/6/23
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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

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Nov 6, 2023, 7:15:55 AM11/6/23
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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

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Nov 6, 2023, 7:46:53 AM11/6/23
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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

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Nov 6, 2023, 9:25:20 AM11/6/23
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"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

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Nov 6, 2023, 9:41:55 AM11/6/23
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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.
Message has been deleted

Dan Christensen

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Nov 6, 2023, 11:37:41 AM11/6/23
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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

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Nov 6, 2023, 4:00:28 PM11/6/23
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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

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Nov 6, 2023, 4:09:22 PM11/6/23
to
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

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Nov 6, 2023, 4:34:12 PM11/6/23
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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

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Nov 6, 2023, 4:39:54 PM11/6/23
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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

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Nov 6, 2023, 4:53:01 PM11/6/23
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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

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Nov 6, 2023, 5:00:42 PM11/6/23
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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

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Nov 6, 2023, 5:09:11 PM11/6/23
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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

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Nov 6, 2023, 5:16:19 PM11/6/23
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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

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Nov 6, 2023, 8:14:26 PM11/6/23
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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

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Nov 6, 2023, 8:19:23 PM11/6/23
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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

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Nov 6, 2023, 8:20:42 PM11/6/23
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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

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Nov 7, 2023, 7:21:22 AM11/7/23
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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

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Nov 7, 2023, 9:14:57 AM11/7/23
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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

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Nov 7, 2023, 9:17:23 AM11/7/23
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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

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Nov 7, 2023, 12:18:37 PM11/7/23
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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

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Nov 7, 2023, 1:45:15 PM11/7/23
to
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

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Nov 7, 2023, 6:10:03 PM11/7/23
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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

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Nov 7, 2023, 6:30:35 PM11/7/23
to
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

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Nov 7, 2023, 6:36:05 PM11/7/23
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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

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Nov 7, 2023, 6:44:07 PM11/7/23
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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

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Nov 7, 2023, 8:49:56 PM11/7/23
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It was dead at its beginning. What is a set with nothing in it?
What can math make out of it?

Fritz Feldhase

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Nov 7, 2023, 8:52:22 PM11/7/23
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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

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Nov 7, 2023, 11:29:49 PM11/7/23
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The empty set.

Python

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Nov 7, 2023, 11:57:38 PM11/7/23
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We could do memes from this :-)


FromTheRafters

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Nov 8, 2023, 6:48:09 AM11/8/23
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Python presented the following explanation :
I asked the grocer for an empty box, he asked "An empty box of what?"

Adam Polak

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Nov 8, 2023, 7:03:25 AM11/8/23
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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

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Nov 8, 2023, 7:11:37 AM11/8/23
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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

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Nov 8, 2023, 7:44:40 AM11/8/23
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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

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Nov 8, 2023, 8:09:15 AM11/8/23
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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

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Nov 8, 2023, 8:15:48 AM11/8/23
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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

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Nov 8, 2023, 9:52:49 AM11/8/23
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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

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Nov 8, 2023, 9:55:13 AM11/8/23
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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

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Nov 8, 2023, 11:55:47 AM11/8/23
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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

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Nov 8, 2023, 6:09:58 PM11/8/23
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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

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Nov 8, 2023, 7:18:32 PM11/8/23
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"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

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Nov 8, 2023, 10:27:52 PM11/8/23
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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

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Nov 8, 2023, 10:27:53 PM11/8/23
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Adam Polak expressed precisely :
Status confirmed.

Phil Carmody

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Nov 9, 2023, 7:55:29 PM11/9/23
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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

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Nov 9, 2023, 8:18:00 PM11/9/23