Chris M. Thomasson submitted this idea :
> On 11/23/2023 10:06 AM, FromTheRafters wrote:
>> Fritz Feldhase formulated on Thursday :
>>> On Thursday, November 23, 2023 at 4:59:24 PM UTC+1, FromTheRafters wrote:
>>>
>>>>> { 1, 3, 4/4, 18 }
>>>>>
>>>> Four symbols separated by three commas. I would say four elements.
>>>> Cardinality four.
>>>
>>> C'mon. Don't be an idiot!
>>>
>>> Hint: If a = b = c = d, then card({a, b, c, d}) = 1.
>>>
>>> <faceplam>
>>
>> Sure, but then it would not have been a set, but a multiset.
>
> As a programmer, I see 4 elements. However, lets reduce and remove
> duplicates:

There are no duplicates in a ZFC set. In a multiset say:

[a,a,a,b,b,c,c,c,c]

You add the multiplicities, three a's two b's and four c's for a
cardinality of nine.

> { 1, 3, 4/4, 18 }
>
> Can be:
>
> { 1, 3, 1, 18 }
>
> Remove duplicates:
>
> { 1, 3, 18 }
>
> ?

A set with cardinality of three.

After serious thinking Adam Polak wrote :
> czwartek, 23 listopada 2023 o 16:59:24 UTC+1 FromTheRafters napisał(a):
>> Adam Polak presented the following explanation :
>>> czwartek, 23 listopada 2023 o 11:32:25 UTC+1 FromTheRafters napisał(a):
>>>> After serious thinking Adam Polak wrote :
>>>>> czwartek, 23 listopada 2023 o 02:58:10 UTC+1 olcott napisał(a):
>>>>>> On 11/22/2023 5:18 PM, Fritz Feldhase wrote:
>>>>>>> On Wednesday, November 22, 2023 at 11:31:32 PM UTC+1, olcott wrote:
>>>>>>>
>>>>>>>> Russell's Paradox, the Paradox of the set of all sets,
>>>>>>>> These two have been abolished by (axiomatic set theory)
>>>>>>>> https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>>>>>>>> a very long time ago.
>>>>>>>
>>>>>>> Though there are axiomatic set theories which do allow for a set of all
>>>>>>> sets.
>>>>>>>
>>>>>>> This set does not necessarily lead to a set theoretic antinomy (though
>>>>>>> "the Russell set" does and always will do so).
>>>>>>>
>>>>>>> Of course, if V is the set of all sets then V e V. (Still no problem in
>>>>>>> an appropriate axiomatic set theory).
>>>>>>>
>>>>>>> In the context of ZF(C), of course, there's no such set.
>>>>>> My key unique innovation to this is that a set that contains itself
>>>>>> is isomorphic to a can of soup that contains itself such that this can
>>>>>> of soup has no outside surface.
>>>>>>
>>>>>> So we simply must forbid a set from containing itself as incoherent.
>>>>>> --
>>>>>> Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
>>>>>> hits a target no one else can see." Arthur Schopenhauer
>>>>>
>>>>>
>>>>> "So we simply must forbid a set from containing itself as incoherent."
>>>>>
>>>>> So you want to forbid exactly this:
>>>>> { { } }
>>>> No, that is not a set containing itself.
>>>>
>>>> Under Frege, this {1,2,3} is.
>>>
>>>
>>> " { { } } <- No, that is not a set containing itself. "
>>> Is that what you think? :) I know this.
>>> let's start with one very simple question:
>>> how many elements does this set contain? : :
>>> { 1, 3, 4/4, 18 }
>> Four symbols separated by three commas. I would say four elements.
>> Cardinality four.
>>
>> How about this one?
>>
>> {{},elephant,gorrila,jackass}
>
> Set: { 1, 3, 4/4, 18 } how many elements contain ?
>> Four symbols separated by three commas. I would say four elements.
>> Cardinality four.
> OK, let me ask you another question befor I'll aswer your:
> Set: { 1, 3, 1.0000... , 18 } how many elements contain ?

A four element multiset.

On 11/23/2023 11:23 PM, FromTheRafters wrote:
> Chris M. Thomasson submitted this idea :
>> On 11/23/2023 10:06 AM, FromTheRafters wrote:
>>> Fritz Feldhase formulated on Thursday :
>>>> On Thursday, November 23, 2023 at 4:59:24 PM UTC+1, FromTheRafters
>>>> wrote:
>>>>
>>>>>> { 1, 3, 4/4, 18 }
>>>>>>
>>>>> Four symbols separated by three commas. I would say four elements.
>>>>> Cardinality four.
>>>>
>>>> C'mon. Don't be an idiot!
>>>>
>>>> Hint: If a = b = c = d, then card({a, b, c, d}) = 1.
>>>>
>>>> <faceplam>
>>>
>>> Sure, but then it would not have been a set, but a multiset.
>>
>> As a programmer, I see 4 elements. However, lets reduce and remove
>> duplicates:
>
> There are no duplicates in a ZFC set. In a multiset say:
>
> [a,a,a,b,b,c,c,c,c]
>
> You add the multiplicities, three a's two b's and four c's for a
> cardinality of nine.
>
>> { 1, 3, 4/4, 18 }
>>
>> Can be:
>>
>> { 1, 3, 1, 18 }
>>
>> Remove duplicates:
>>
>> { 1, 3, 18 }
>>
>> ?
>
> A set with cardinality of three.

3-ary finite set... Fair enough?

piątek, 24 listopada 2023 o 08:25:12 UTC+1 FromTheRafters napisał(a):
> After serious thinking Adam Polak wrote :
> > czwartek, 23 listopada 2023 o 16:59:24 UTC+1 FromTheRafters napisał(a):
> >> Adam Polak presented the following explanation :
> >>> czwartek, 23 listopada 2023 o 11:32:25 UTC+1 FromTheRafters napisał(a):
> >>>> After serious thinking Adam Polak wrote :
> >>>>> czwartek, 23 listopada 2023 o 02:58:10 UTC+1 olcott napisał(a):
> >>>>>> On 11/22/2023 5:18 PM, Fritz Feldhase wrote:
> >>>>>>> On Wednesday, November 22, 2023 at 11:31:32 PM UTC+1, olcott wrote:
> >>>>>>>
> >>>>>>>> Russell's Paradox, the Paradox of the set of all sets,
> >>>>>>>> These two have been abolished by (axiomatic set theory)
> >>>>>>>> https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
> >>>>>>>> a very long time ago.
> >>>>>>>
> >>>>>>> Though there are axiomatic set theories which do allow for a set of all
> >>>>>>> sets.
> >>>>>>>
> >>>>>>> This set does not necessarily lead to a set theoretic antinomy (though
> >>>>>>> "the Russell set" does and always will do so).
> >>>>>>>
> >>>>>>> Of course, if V is the set of all sets then V e V. (Still no problem in
> >>>>>>> an appropriate axiomatic set theory).
> >>>>>>>
> >>>>>>> In the context of ZF(C), of course, there's no such set.
> >>>>>> My key unique innovation to this is that a set that contains itself
> >>>>>> is isomorphic to a can of soup that contains itself such that this can
> >>>>>> of soup has no outside surface.
> >>>>>>
> >>>>>> So we simply must forbid a set from containing itself as incoherent.
> >>>>>> --
> >>>>>> Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
> >>>>>> hits a target no one else can see." Arthur Schopenhauer
> >>>>>
> >>>>>
> >>>>> "So we simply must forbid a set from containing itself as incoherent."
> >>>>>
> >>>>> So you want to forbid exactly this:
> >>>>> { { } }
> >>>> No, that is not a set containing itself.
> >>>>
> >>>> Under Frege, this {1,2,3} is.
> >>>
> >>>
> >>> " { { } } <- No, that is not a set containing itself. "
> >>> Is that what you think? :) I know this.
> >>> let's start with one very simple question:
> >>> how many elements does this set contain? : :
> >>> { 1, 3, 4/4, 18 }
> >> Four symbols separated by three commas. I would say four elements.
> >> Cardinality four.
> >>
> >> How about this one?
> >>
> >> {{},elephant,gorrila,jackass}
> >
> > Set: { 1, 3, 4/4, 18 } how many elements contain ?
> >> Four symbols separated by three commas. I would say four elements.
> >> Cardinality four.
> > OK, let me ask you another question befor I'll aswer your:
> > Set: { 1, 3, 1.0000... , 18 } how many elements contain ?
> A four element multiset.

to justify your answer: "... four elements. Cardinality four."
wrong answer in the context of the Set theory that is based on ZF(C) axiomatics,
you moved to the area of "multiset" which is incompatible with ZF(C) so with the Set theory.

We learn from this that one can disagree with ZF(C) Set theory and nothing terrible happen.
Good to notice.

"
Set: { 1, 3, 4/4, 18 } how many elements contain ?
> Four symbols separated by three commas. I would say four elements.
> Cardinality four.
"
what happened that you didn't recognize 4/4 as 1
as "the number ONE"
and that, in accordance with Set theory, you did not answer: three elements ?
just think it over

On 24.11.2023 06:22, Adam Polak wrote:

> OK, let me ask you another question befor I'll aswer your:
> Set: { 1, 3, 1.0000... , 18 } how many elements contain ?

That depends on what you look for. There are four representations of
three quantities.

If we try to index all fractions in

1, 1/2, 1/3, 1/4, ...
2, 2/2, 2/3, 2/4, ...
3, 3/2, 3/3, 3/4, ...
4, 4/2, 4/3, 4/4, ...
5, 5/2, 5/3, 5/4, ...
....

then we can apply the closed formula k = (m + n - 1)(m + n - 2)/2 + m
leading to the sequence
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2,
5/1, 1/6, 2/5, 3/4, 4/3, 5/2, 6/1, ... .
If we try to index the positive rational numbers, then we have to delete
the doublets.

Regards, WM

Adam Polak wrote on 11/24/2023 :
> piątek, 24 listopada 2023 o 08:25:12 UTC+1 FromTheRafters napisał(a):
>> After serious thinking Adam Polak wrote :
>>> czwartek, 23 listopada 2023 o 16:59:24 UTC+1 FromTheRafters napisał(a):
>>>> Adam Polak presented the following explanation :
>>>>> czwartek, 23 listopada 2023 o 11:32:25 UTC+1 FromTheRafters napisał(a):
>>>>>> After serious thinking Adam Polak wrote :
>>>>>>> czwartek, 23 listopada 2023 o 02:58:10 UTC+1 olcott napisał(a):
>>>>>>>> On 11/22/2023 5:18 PM, Fritz Feldhase wrote:
>>>>>>>>> On Wednesday, November 22, 2023 at 11:31:32 PM UTC+1, olcott wrote:
>>>>>>>>>
>>>>>>>>>> Russell's Paradox, the Paradox of the set of all sets,
>>>>>>>>>> These two have been abolished by (axiomatic set theory)
>>>>>>>>>> https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>>>>>>>>>> a very long time ago.
>>>>>>>>>
>>>>>>>>> Though there are axiomatic set theories which do allow for a set of
>>>>>>>>> all sets.
>>>>>>>>>
>>>>>>>>> This set does not necessarily lead to a set theoretic antinomy
>>>>>>>>> (though "the Russell set" does and always will do so).
>>>>>>>>>
>>>>>>>>> Of course, if V is the set of all sets then V e V. (Still no problem
>>>>>>>>> in an appropriate axiomatic set theory).
>>>>>>>>>
>>>>>>>>> In the context of ZF(C), of course, there's no such set.
>>>>>>>> My key unique innovation to this is that a set that contains itself
>>>>>>>> is isomorphic to a can of soup that contains itself such that this can
>>>>>>>> of soup has no outside surface.
>>>>>>>>
>>>>>>>> So we simply must forbid a set from containing itself as incoherent.
>>>>>>>> --
>>>>>>>> Copyright 2023 Olcott "Talent hits a target no one else can hit;
>>>>>>>> Genius hits a target no one else can see." Arthur Schopenhauer
>>>>>>>
>>>>>>>
>>>>>>> "So we simply must forbid a set from containing itself as incoherent."
>>>>>>>
>>>>>>> So you want to forbid exactly this:
>>>>>>> { { } }
>>>>>> No, that is not a set containing itself.
>>>>>>
>>>>>> Under Frege, this {1,2,3} is.
>>>>>
>>>>>
>>>>> " { { } } <- No, that is not a set containing itself. "
>>>>> Is that what you think? :) I know this.
>>>>> let's start with one very simple question:
>>>>> how many elements does this set contain? : :
>>>>> { 1, 3, 4/4, 18 }
>>>> Four symbols separated by three commas. I would say four elements.
>>>> Cardinality four.
>>>>
>>>> How about this one?
>>>>
>>>> {{},elephant,gorrila,jackass}
>>>
>>> Set: { 1, 3, 4/4, 18 } how many elements contain ?
>>>> Four symbols separated by three commas. I would say four elements.
>>>> Cardinality four.
>>> OK, let me ask you another question befor I'll aswer your:
>>> Set: { 1, 3, 1.0000... , 18 } how many elements contain ?
>> A four element multiset.
>
> to justify your answer: "... four elements. Cardinality four."
> wrong answer in the context of the Set theory that is based on ZF(C)
> axiomatics, you moved to the area of "multiset" which is incompatible with
> ZF(C) so with the Set theory.
>
> We learn from this that one can disagree with ZF(C) Set theory and nothing
> terrible happen. Good to notice.
>
> "
> Set: { 1, 3, 4/4, 18 } how many elements contain ?
>> Four symbols separated by three commas. I would say four elements.
>> Cardinality four.
> "
> what happened that you didn't recognize 4/4 as 1
> as "the number ONE"

I didn't recognize that collection as a set, but as a multiset, because
ZFC sets don't have duplicate elements.

> and that, in accordance with Set theory, you did not answer: three elements ?
> just think it over

Yes, you are correct. I should have because the symbol 4/4 and the
symbol 1 are the same number just like 1/2 and .5 and .4(9) are all the
same number.

poniedziałek, 6 listopada 2023 o 12:01:03 UTC+1 Adam Polak napisał(a):
> 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

Three weeks have passed since the thread was initiated. In a few days it will be a month.

So far, no one has been able to deny that:

The statement that in Hilbert's infinite hotel, a new guest can be accommodated, and even infinitely many new guests, is mutually contradictory with the statement that it is not possible to find a natural number as a pair for the "new" real number "created" by Cantor's diagonal method.

If we claim that we can accommodate a new guest in Hilbert's infinite hotel we cannot deny that exactly the same operation can be performed for a New real number supposedly „created” by Cantor's Diagonal Method.

In light of the presented analysis - in the absence of counter-arguments - on the 6th of Dec. 2023 we will have to formally state that:

!!!
Cantor's Diagonal Method cannot be indicated and treated as evidence confirming in any way a greater quantity of elements in an infinite set of real numbers compared to the quantity of elements in an infinite set of natural numbers.
!!!

Regards,
Adam

poniedziałek, 6 listopada 2023 o 12:01:03 UTC+1 Adam Polak napisał(a):
> 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

Three weeks have passed since the thread was initiated. In a few days it will be a month.

So far, no one has been able to deny that:

The statement that in Hilbert's infinite hotel, a new guest can be accommodated, and even infinitely many new guests, is mutually contradictory with the statement that it is not possible to find a natural number as a pair for the "new" real number "created" by Cantor's diagonal method.

If we claim that we can accommodate a new guest in Hilbert's infinite hotel we cannot deny that exactly the same operation can be performed for a New real number supposedly „created” by Cantor's Diagonal Method.

In light of the presented analysis - in the absence of counter-arguments - on the 6th of Dec. 2023 we will have to formally state that:

!!!
Cantor's Diagonal Method cannot be indicated and treated as evidence confirming in any way a greater quantity of elements in an infinite set of real numbers compared to the quantity of elements in an infinite set of natural numbers.
!!!

Regards,
Adam

poniedziałek, 27 listopada 2023 o 07:45:52 UTC+1 Fritz Feldhase napisał(a):
> On Monday, November 27, 2023 at 7:41:58 AM UTC+1, Adam Polak wrote: <nonsense>
>
> Please go away, you silly troll.
Have a nice day too :)

poniedziałek, 27 listopada 2023 o 07:58:56 UTC+1 Fritz Feldhase napisał(a):
> On Monday, November 27, 2023 at 7:41:58 AM UTC+1, Adam Polak wrote:
> > The statement that in Hilbert's infinite hotel, a new guest can be accommodated, and even infinitely many new guests, is mutually contradictory with the statement that it is not possible to find a natural number as a pair for the "new" real number "created" by Cantor's diagonal method.
> So what? No one claims the latter (except you, it seems).
>
> 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 concerning Cantor's diagonal method. 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.)
>
> Hope this helps.

Relax.

In the first step, we are demonstrating the error of the claim that Cantor's Diagonal Method, by "constructing a new real number," proves that the number of elements in the infinite set of real numbers is greater than the number of elements in the infinite set of natural numbers.
I'm glad you understand and confirm :)

In the second step, we will demonstrate the error of the claim that Cantor's Diagonal Method is a tool that can generate "new real number." Of course CAN NOT.

In the subsequent steps, we will debunk further flawed claims of Set Theory..

Fritz Feldhase schrieb am Montag, 27. November 2023 um 07:58:56 UTC+1:
> On Monday, November 27, 2023 at 7:41:58 AM UTC+1, Adam Polak wrote:
> > The statement that in Hilbert's infinite hotel, a new guest can be
accommodated, and even infinitely many new guests, is mutually
contradictory with the statement that it is not possible to find a
natural number as a pair for the "new" real number "created" by Cantor's
diagonal method.
> So what? No one claims the latter (except you, it seems).
>
> Of course, we may/can always add the new number "'created' by the
diagonal method" to the list of real numbers we are considering.

Then we have to add another natural number to those already enumerating
the list. Why didn't we add it (and the infinitely many that can be
added again and again) in advance?
>
> But that's not the point concerning Cantor's diagonal method. The
point is:

Two honest tricksters try to trick each other: First show me your list,
then I will show you mine. Like the game scissors stone paper. Unworthy
of and degrading mathematics.

Regards, WM

On 27.11.2023 18:36, Fritz Feldhase wrote:

> There is no sequence/list of real numbers which contains _all_ real numbers.

There is no sequence/list of natural numbers that contains _all_ natural
numbers.

Proof 1: All rooms in Hilbert's hotel are occupied. One guest leaves.
But no room gets empty. No number can be dispensed.

Proof 2: Let all natural numbers enumerate all rooms in Hilbert's hotel.
All are occupied. Let another guest require an enumerated room.

Proof 3: Let all natural numbers occupy all rooms in Hilbert's hotel.
Let another natural number require a room.

Regards, WM

On 11/24/2023 12:06 AM, Chris M. Thomasson wrote:
> On 11/23/2023 11:23 PM, FromTheRafters wrote:
>> Chris M. Thomasson submitted this idea :
>>> On 11/23/2023 10:06 AM, FromTheRafters wrote:
>>>> Fritz Feldhase formulated on Thursday :
>>>>> On Thursday, November 23, 2023 at 4:59:24 PM UTC+1, FromTheRafters
>>>>> wrote:
>>>>>
>>>>>>> { 1, 3, 4/4, 18 }
>>>>>>>
>>>>>> Four symbols separated by three commas. I would say four elements.
>>>>>> Cardinality four.
>>>>>
>>>>> C'mon. Don't be an idiot!
>>>>>
>>>>> Hint: If a = b = c = d, then card({a, b, c, d}) = 1.
>>>>>
>>>>> <faceplam>
>>>>
>>>> Sure, but then it would not have been a set, but a multiset.
>>>
>>> As a programmer, I see 4 elements. However, lets reduce and remove
>>> duplicates:
>>
>> There are no duplicates in a ZFC set. In a multiset say:
>>
>> [a,a,a,b,b,c,c,c,c]
>>
>> You add the multiplicities, three a's two b's and four c's for a
>> cardinality of nine.
>>
>>> { 1, 3, 4/4, 18 }
>>>
>>> Can be:
>>>
>>> { 1, 3, 1, 18 }
>>>
>>> Remove duplicates:
>>>
>>> { 1, 3, 18 }
>>>
>>> ?
>>
>> A set with cardinality of three.
>
> 3-ary finite set... Fair enough?

Take any set:

Fwiw, just for fun, we can use those elements as a root in a n-ary tree.
Notice the levels of the tree here, take note of the root node at 0:

l[0] = { 0 } // root! :^)
l[1] = { 1, 2 }
l[2] = { 3, 4, 5, 6 }
l[3] = { 7, 8, 9, 10, 11, 12, 13, 14 }
l[4] = { 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, ect}
....

Notice the pattern? So, say we have {1, 3, 8}. Each element can be a
infinite tree:

1 has two children { 3, 4 }
3 has two children { 7, 8 }
8 has two children { 17, 18 }

on and on:

https://youtu.be/oRrlRbGT-LU

So, there are three infinite independent 2-ary fractal trees for the
finite set of { 1, 3, 8 }

1
/ \
3 4
[.....]
_________

3
/ \
7 8
[.....]
_________

8
/ \
17 18
[.....]
_________

This is fun to me. These trees can be multi n-ary on a per node and/or
level basis. This example is for 2-ary.

Chris M. Thomasson expressed precisely :
> On 11/23/2023 11:23 PM, FromTheRafters wrote:
>> Chris M. Thomasson submitted this idea :
>>> On 11/23/2023 10:06 AM, FromTheRafters wrote:
>>>> Fritz Feldhase formulated on Thursday :
>>>>> On Thursday, November 23, 2023 at 4:59:24 PM UTC+1, FromTheRafters
>>>>> wrote:
>>>>>
>>>>>>> { 1, 3, 4/4, 18 }
>>>>>>>
>>>>>> Four symbols separated by three commas. I would say four elements.
>>>>>> Cardinality four.
>>>>>
>>>>> C'mon. Don't be an idiot!
>>>>>
>>>>> Hint: If a = b = c = d, then card({a, b, c, d}) = 1.
>>>>>
>>>>> <faceplam>
>>>>
>>>> Sure, but then it would not have been a set, but a multiset.
>>>
>>> As a programmer, I see 4 elements. However, lets reduce and remove
>>> duplicates:
>>
>> There are no duplicates in a ZFC set. In a multiset say:
>>
>> [a,a,a,b,b,c,c,c,c]
>>
>> You add the multiplicities, three a's two b's and four c's for a
>> cardinality of nine.
>>
>>> { 1, 3, 4/4, 18 }
>>>
>>> Can be:
>>>
>>> { 1, 3, 1, 18 }

Neither are correct roster form set notation. Sets don't have repeated
elements. If you want to indicate a multiset then use square brackets.

>>> Remove duplicates:
>>>
>>> { 1, 3, 18 }
>>>
>>> ?
>>
>> A set with cardinality of three.
>
> 3-ary finite set... Fair enough?

Sure, if you remove the duplicates, it is correct roster form notation.

On 11/9/2023 5:48 PM, Fritz Feldhase wrote:
> On Friday, November 10, 2023 at 2:18:00 AM UTC+1, Chris M. Thomasson wrote:
>> On 11/9/2023 4:55 PM, Phil Carmody wrote:
>>>
>>> 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!
>
> No joke, in Russellian Type Theory each type has its own "empty set".

{ 0 } vs {}

Empty wrt 0 and/or empty because nothing is there, damn it!

On Tuesday, November 21, 2023 at 12:23:54 AM UTC-8, Ross Finlayson wrote:
> On Saturday, November 18, 2023 at 10:57:56 PM UTC-8, Ross Finlayson wrote:
> > On Saturday, November 18, 2023 at 8:58:27 AM UTC-8, Fritz Feldhase wrote:
> > > On Saturday, November 18, 2023 at 3:29:05 AM UTC+1, Ross Finlayson wrote:
> > >
> > > > If you get to reading enough Heidegger it starts looking like Greek..
> > > Agree. Though the later Heidegger is quite cool.
> > Yeah, Heidegger's "Being and Time", about Da (here) and Dasein (being, here-being), is something, then though
> > what follows is Heidegger's extended translations of the Greeks, mostly coat-tailing the Greeks some 2000 years
> > later, while though Heidegger and for example Gadamer offer various and varying interpretations, in their
> > uncovering or the a-Letheia, about the Logos (of course, written usually enough in Greek script).
> >
> > Eleatics - Pythagorans - Platonists
> > Augustine-Scotus-Arabs-Spinoza
> > DesCartes-Leibnitz-Kant-Hegel-Heidegger
> > Popper-Priest
> >
> > I've been studying these because it's kind of an outline of the Western canon, philosophically.
> > Of course, it's sort of a curated line out from logical foundations into principled science,
> > "technical philosophy", for a usual sort of narrative of the development of a "The One True Logic"
> > or for "an idealized Platonism", a strong mathematical platonism.
> >
> > ("DaDa" was a sort of zeitgeist of some Continental head-set,
> > it sort of reminds of "Bob Dobbs and the pinks".)
> >
> >
> > https://www.youtube.com/watch?v=BbO2q9WKUDQ&list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5&index=37
> >
> > So, Heidegger is famous for "Dasein", presence and consciously, similarly to how there's
> > Goethe and "beholding" vis-a-vis "knowing", and he coat-tails a lot of canon in his fully
> > voluble deliberations, then, his following works are mostly great translations and
> > interpretations of Parmenides and the Greeks and exploring the antonymous
> > character of a-letheia, uncovering or revealing as for showing or knowing,
> > and into the logos, as after Heraclitus or "the great obfuscator", what is so profound.
> >
> > Such manners of didactic dialectic of course reflect earlier into the tradition,
> > back to the Vedics and Daoism as similarly, even more "obscure", obtuse,
> > to be so profound. It's a broad dialectic for "ideal Platonism", strong mathematical
> > platonism.
> >
> > It's all English to me, but, it's all one language, a language of words, logoi.
> >
> > MfG (Mit fielen Gruesse),
>
>
> Pretty sure this thread is dead, ....

Hello. Well, I started reading Gadamer and his hermeneutics on Plato. From Smith the translator's
introduction, it seems that getting into Gadamer's Being and Truth bit has it's sort of a Tarski truth,
so it sort of seems for a reading through a lens where Gadamer's elusive truth is let out under
Kant's sublime, so that there's still makings for a theory with a real truth, while at the same time
acknowledging human limits, about the mutual provability and lack thereof or the undecideability,
that there really is a true truth then of course that there must be, or there isn't. Then though
Gadamer as Heidegger's student, he lived later in the 20'th century so he makes some readings
and so on from his desk, while though he seems rather stern, that it's possible to be quite
serious and rather, sincere, about hermeneutics of truth, and philosophy and its canon,
while not necessarily having that the entire gravity of it is without its sublime brilliance
for what it is.

Then, I'll be reading it for what he denies himself, then figure it's as of a usual approach
of an object/subject distinction, just to have it so that for the extra-ordinary logic and
the usual reading of a Plato's "One" besides a usual multitudinous "Two", that they're
to be read as both, each other, for the dialog and dialectic, what results a dialectic as
reflective, and not just another lengthening shadow, in the cave.

On 11/27/2023 11:10 PM, Fritz Feldhase wrote:
> On Tuesday, November 28, 2023 at 7:46:21 AM UTC+1, Chris M. Thomasson wrote:
>> On 11/9/2023 5:48 PM, Fritz Feldhase wrote:
>>> On Friday, November 10, 2023 at 2:18:00 AM UTC+1, Chris M. Thomasson wrote:
>>>> On 11/9/2023 4:55 PM, Phil Carmody wrote:
>>>>>
>>>>> 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!
>>>>
>>> No joke, in Russellian Type Theory each type has its own "empty set".
>>
>> { 0 } vs {}
>
> Nope. {} of type 0 (say) differs from {} of type 1 (say).

Ahhh shit! Thanks for the correction.

wtorek, 28 listopada 2023 o 16:27:59 UTC+1 Fritz Feldhase napisał(a):
> On Tuesday, November 28, 2023 at 3:56:17 PM UTC+1, Fritz Feldhase wrote:
> > On Tuesday, November 28, 2023 at 7:43:16 AM UTC+1, FromTheRafters wrote:
> > >
> > > Neither are correct roster form set notation.
> > >
> > Sure they are.
> You may check this, for example: https://proofwiki.org/wiki/Existence_of_Singleton_Set
> > We (usually?) do NOT require a_i =/= a_j here for i,j e {1, ..., n}, i =/= j.
> For example:
>
> "_Curly bracket notation_
>
> We often define sets by listing their elements [...]:
>
> • The symbol {x, y, z} describes the set whose elements are precisely x, y, and z."
>
> You see, x = y = z = 1 (say) is not excluded.
>
> [Btw. A formal version of this definition would just be: x e {a, b, c} <-> x e a v x e b v x e c.]
>
> Source: https://math.mit.edu/~jhirsh/top_lecture.pdf

"
> > Sure they are.
> You may check this, for example: https://proofwiki.org/wiki/Existence_of_Singleton_Set
"
Proofs based on axioms :))
It's really amusing.
Axioms, unfortunately, being in it's important part internally contradictory, so flawed, in a way that cannot be questioned.

To say that something is true or a correct description of reality just because an axiom says so, or it follows from an axiom,
is exactly the same as saying that
God exists because it follows from the content of the Roman Catholic Church catechism.

Is it really so difficult to notice, to realize that using such methodologies of proving pushes you out from the realm of SCIENCE into the realm of an unscientific RELIGIOUS SECT?

An axiom is not an argument, much less a proof or a basis for proof!

Have a nice day.
Regards,
Adam

wtorek, 28 listopada 2023 o 16:27:59 UTC+1 Fritz Feldhase napisał(a):
> On Tuesday, November 28, 2023 at 3:56:17 PM UTC+1, Fritz Feldhase wrote:
> > On Tuesday, November 28, 2023 at 7:43:16 AM UTC+1, FromTheRafters wrote:
> > >
> > > Neither are correct roster form set notation.
> > >
> > Sure they are.
> You may check this, for example: https://proofwiki.org/wiki/Existence_of_Singleton_Set
> > We (usually?) do NOT require a_i =/= a_j here for i,j e {1, ..., n}, i =/= j.
> For example:
>
> "_Curly bracket notation_
>
> We often define sets by listing their elements [...]:
>
> • The symbol {x, y, z} describes the set whose elements are precisely x, y, and z."
>
> You see, x = y = z = 1 (say) is not excluded.
>
> [Btw. A formal version of this definition would just be: x e {a, b, c} <-> x e a v x e b v x e c.]
>
> Source: https://math.mit.edu/~jhirsh/top_lecture.pdf

"
> > Sure they are.
> You may check this, for example: https://proofwiki.org/wiki/Existence_of_Singleton_Set
"
Proofs based on axioms :))
It's really amusing.
Axioms, unfortunately, being in it's important part internally contradictory, so flawed, in a way that cannot be questioned.

To say that something is true or a correct description of reality just because an axiom says so, or it follows from an axiom,
is exactly the same as saying that
God exists because it follows from the content of the Roman Catholic Church catechism.

Is it really so difficult to notice, to realize that using such methodologies of proving pushes you out from the realm of SCIENCE into the realm of an unscientific RELIGIOUS SECT?

An axiom is not an argument, much less a proof or a basis for proof!

Have a nice day.
Regards,
Adam

środa, 29 listopada 2023 o 09:26:50 UTC+1 Adam Polak napisał(a):
> wtorek, 28 listopada 2023 o 16:27:59 UTC+1 Fritz Feldhase napisał(a):
> > On Tuesday, November 28, 2023 at 3:56:17 PM UTC+1, Fritz Feldhase wrote:
> > > On Tuesday, November 28, 2023 at 7:43:16 AM UTC+1, FromTheRafters wrote:
> > > >
> > > > Neither are correct roster form set notation.
> > > >
> > > Sure they are.
> > You may check this, for example: https://proofwiki.org/wiki/Existence_of_Singleton_Set
> > > We (usually?) do NOT require a_i =/= a_j here for i,j e {1, ..., n}, i =/= j.
> > For example:
> >
> > "_Curly bracket notation_
> >
> > We often define sets by listing their elements [...]:
> >
> > • The symbol {x, y, z} describes the set whose elements are precisely x, y, and z."
> >
> > You see, x = y = z = 1 (say) is not excluded.
> >
> > [Btw. A formal version of this definition would just be: x e {a, b, c} <-> x e a v x e b v x e c.]
> >
> > Source: https://math.mit.edu/~jhirsh/top_lecture.pdf
>
> "
> > > Sure they are.
> > You may check this, for example: https://proofwiki.org/wiki/Existence_of_Singleton_Set
> "
> Proofs based on axioms :))
> It's really amusing.
> Axioms, unfortunately, being in it's important part internally contradictory, so flawed, in a way that cannot be questioned.
>
> To say that something is true or a correct description of reality just because an axiom says so, or it follows from an axiom,
> is exactly the same as saying that
> God exists because it follows from the content of the Roman Catholic Church catechism.
>
> Is it really so difficult to notice, to realize that using such methodologies of proving pushes you out from the realm of SCIENCE into the realm of an unscientific RELIGIOUS SECT?
>
> An axiom is not an argument, much less a proof or a basis for proof!
>
> Have a nice day.
> Regards,
> Adam

PS:
To be clear, I am not questioning here the existence of God.
As you can see, after all, thousands of years ago He predicted and described by inspired authors the emergence of the Set Theory Religion sect in the field of science:

Romans 1:22
Although they claimed to be wise, they became fools. (New International Version)
Claiming to be wise, they instead became utter fools. (New Living Translation)
:)

Regards,
Adam

środa, 29 listopada 2023 o 11:21:19 UTC+1 Fritz Feldhase napisał(a):
> On Wednesday, November 29, 2023 at 9:26:50 AM UTC+1, Adam Polak wrote:
> > Proofs based on axioms
>
> Since you seem to like this topic:
>
> http://de.metamath.org/
>
> Great site.
thank U

Adam Polak schrieb am Mittwoch, 29. November 2023 um 11:42:37 UTC+1:
> środa, 29 listopada 2023 o 11:21:19 UTC+1 Fritz Feldhase napisał(a):

Hi Adam, unfortunately your post to "A game like billiards" of today
does not appear in my news server. Therefore I answer it here:

Adam Polak schrieb am Mittwoch, 29. November 2023 um 10:34:28 UTC+1:

If the method involves STEP-by-STEP action, then the possibility of
performing an action on "ALL" elements of an infinite series or set is
excluded

Counting is a step-by-step procedure. _If_ Cantor is accepted, then this
procedure can be completed.

> the step-by-step method inherently requires minimal (delta t) one
after one, in which subsequent steps are taken.

By using the geometric series 1 + 1/2 + 1/4 + 1/8 + ... infinitely many
steps can be processed within 2 seconds.
>
> The only type of action on the elements of an infinite series or set
that can encompass "ALL" elements of such a series or set is an action
involving "ALL" elements simultaneously.

Mathematics is not time-dependent. If all steps are defined by the
algorithm, and if infinitely many natural numbers can be applied at all,
then this can be done like described in the OP. The only condition is
that all steps are determined and no further decisions are necessary.

Regards, WM

środa, 29 listopada 2023 o 21:12:35 UTC+1 WM napisał(a):
> Adam Polak schrieb am Mittwoch, 29. November 2023 um 11:42:37 UTC+1:
> > środa, 29 listopada 2023 o 11:21:19 UTC+1 Fritz Feldhase napisał(a):
> Hi Adam, unfortunately your post to "A game like billiards" of today
> does not appear in my news server. Therefore I answer it here:
>
> Adam Polak schrieb am Mittwoch, 29. November 2023 um 10:34:28 UTC+1:
>
> If the method involves STEP-by-STEP action, then the possibility of
> performing an action on "ALL" elements of an infinite series or set is
> excluded
>
Counting (one by one) the elements of an infinite set can never be brought to completion because it would inevitably involve necesity of finding an element in the infinite set that does not have a successor - and such an element, by definition, does not exist in an infinite set.
So, it's simply ruled out.

> Counting is a step-by-step procedure. _If_ Cantor is accepted, then this
> procedure can be completed.
>
> > the step-by-step method inherently requires minimal (delta t) one
> after one, in which subsequent steps are taken.
>
> By using the geometric series 1 + 1/2 + 1/4 + 1/8 + ... infinitely many
> steps can be processed within 2 seconds.
> >
Yes, but only as an "AT ONCE" operation, never as a "STEP by STEP" operation.

> > The only type of action on the elements of an infinite series or set
> that can encompass "ALL" elements of such a series or set is an action
> involving "ALL" elements simultaneously.
>
> Mathematics is not time-dependent. If all steps are defined by the
> algorithm, and if infinitely many natural numbers can be applied at all,
> then this can be done like described in the OP. The only condition is
> that all steps are determined and no further decisions are necessary.
>

If you define the operation as "STEP by STEP," it takes place in time - thats reality,
regardless of what you may think about it or say.

"STEP by STEP" means one AFTER another, forcing the existence of minimal time units one after another, in which successive operations, successive steps, take place.

Regards,
Adam

czwartek, 30 listopada 2023 o 12:52:33 UTC+1 Fritz Feldhase napisał(a):
> On Thursday, November 30, 2023 at 6:30:31 AM UTC+1, Adam Polak wrote:
> > środa, 29 listopada 2023 o 21:12:35 UTC+1 WM napisał(a):
> > > Adam Polak schrieb am Mittwoch, 29. November 2023 um 11:42:37 UTC+1:
> > > >
> > > > the step-by-step method inherently requires minimal (delta t) one
> > > > after one, in which subsequent steps are taken.
> Actually not. The time required for a step may shrink from step to step (from a logical point of view).
> > > By using the geometric series 1 + 1/2 + 1/4 + 1/8 + ... infinitely many
> > > steps can be processed within 2 seconds.
> What WM means here is:
>
> Time required for 1. step: 1 sec.
> Time required for 2. step: 1/2 sec.
> Time required for 3. step: 1/4 sec.
> etc.
>
> Hence, time required for all steps: 2 sec.

OK, I see.
I don't want to start a new big thread because I don't have time :) - nomen omen - for developing it now,
but I will only point out that such reasoning is flawed because:
1/oo is not a unit of time, it is not a unit of anything, it is not even .... (I will refrain here so as not to provoke a discussion that concerns a large and important topic and cannot be short.)

> > Yes, but only as an "AT ONCE" operation, never as a "STEP by STEP" operation.
> Nope (at least this is debetable). What WM has in mind here is a so called /supertask/.
>
> See: https://en.wikipedia.org/wiki/Supertask

:)
I can create a bijection between the elements of an infinite set of natural numbers:
1, 2, 3, 4, 5, 6, 7, ...
and the elements of an infinite set of integers arranged in this order:
1, -1, 2, -2, 3, -3, ...
same efficiently as if you arranged the integers in this order:
1, 2, 3, 4, 5, 6, 7, ... -1, -2, -3, -4, -5, -6, -7, ...

I can count integers arranged alternately (positive/negative) as effectively as I can count integers arranged first "all" positive, then "all" negative.

Can you?
Fritz?
Wolfgang?
anyone?

Adam Polak schrieb am Donnerstag, 30. November 2023 um 06:30:31 UTC+1:
> środa, 29 listopada 2023 o 21:12:35 UTC+1 WM napisał(a):
> > Adam Polak schrieb am Mittwoch, 29. November 2023 um 11:42:37 UTC+1:
> > > środa, 29 listopada 2023 o 11:21:19 UTC+1 Fritz Feldhase napisał(a):
> > Hi Adam, unfortunately your post to "A game like billiards" of today
> > does not appear in my news server. Therefore I answer it here:
> >
> > Adam Polak schrieb am Mittwoch, 29. November 2023 um 10:34:28 UTC+1:
> >
> > If the method involves STEP-by-STEP action, then the possibility of
> > performing an action on "ALL" elements of an infinite series or set is
> > excluded
> >
> Counting (one by one) the elements of an infinite set can never be brought to completion because it would inevitably involve necesity of finding an element in the infinite set that does not have a successor - and such an element, by definition, does not exist in an infinite set.
> So, it's simply ruled out.

Of course you are right. By the same argument it is impossible to enumerate something with all natural numbers. Inevitably a last one would appear.

> > Counting is a step-by-step procedure. _If_ Cantor is accepted, then this
> > procedure can be completed.

Note the IF.
> >
> > > the step-by-step method inherently requires minimal (delta t) one
> > after one, in which subsequent steps are taken.
> >
> > By using the geometric series 1 + 1/2 + 1/4 + 1/8 + ... infinitely many
> > steps can be processed within 2 seconds.
> > >
> Yes, but only as an "AT ONCE" operation, never as a "STEP by STEP" operation.

If all steps are predetermined and if Cantor is accepted, then completeness is accomplished in whatever way.

> If you define the operation as "STEP by STEP," it takes place in time - thats reality,
> regardless of what you may think about it or say.

Every mathematical activity requires matter and time although most mathematicians do not like that.
>
Regards, WM