William schrieb am Freitag, 31. Dezember 2021 um 15:53:54 UTC+1:
> On Friday, December 31, 2021 at 5:34:43 AM UTC-4, WM wrote:
>
> > {1}
> > {2, 1}
> > {3, 2, 1}
> > {4, 3, 2, 1}
> > {5, 4, 3, 2, 1}
> > ...
> >
> > If there are ℵo natnumbers, then they are in the first column. If they are in the first column, then they are in the whole figure.
> > There are not any two lines in the figure which contain more than one of them. No line contains ℵo natnumbers.
> Correct the whole figure, the set, contains aleph_0 elements. But no line, the elements of the figure, contain aleph_0 elements. The set has a property that no element does.

In this case the set cannot be larger than all lines. They cover all definable numbers.
> >
> > But the set ℕ is much, much larger that ℕ_def:
> >
> > {1, 2, 3, 4, 5, ..., ω/1000} is not a FISON although it contains only numbers less than ω.
> Nope, ω/1000= w.

Not if ω belongs to mathematics.

>The only ordinals less than ω are the natural numbers.

Yes, but all multiples of ω are dark.

Regards, WM

On 12/31/2021 11:44 AM, WM wrote:
> William schrieb am Freitag, 31. Dezember 2021 um 15:53:54 UTC+1:
>> On Friday, December 31, 2021 at 5:34:43 AM UTC-4, WM wrote:
>>
>>> {1}
>>> {2, 1}
>>> {3, 2, 1}
>>> {4, 3, 2, 1}
>>> {5, 4, 3, 2, 1}
>>> ...
>>>
>>> If there are ℵo natnumbers, then they are in the first column. If they are in the first column, then they are in the whole figure.
>>> There are not any two lines in the figure which contain more than one of them. No line contains ℵo natnumbers.
>> Correct the whole figure, the set, contains aleph_0 elements. But no line, the elements of the figure, contain aleph_0 elements. The set has a property that no element does.
>
> In this case the set cannot be larger than all lines. They cover all definable numbers.
>>>
>>> But the set ℕ is much, much larger that ℕ_def:
>>>
>>> {1, 2, 3, 4, 5, ..., ω/1000} is not a FISON although it contains only numbers less than ω.
>> Nope, ω/1000= w.
>
> Not if ω belongs to mathematics.

wrong on that one, which means you fail to understand infinity, as we all already know.

>
>> The only ordinals less than ω are the natural numbers.
>
> Yes, but all multiples of ω are dark.
>
> Regards, WM

the dark areas, are areas of math you fail to understand.

William schrieb am Freitag, 31. Dezember 2021 um 15:54:03 UTC+1:
> On Friday, December 31, 2021 at 5:26:21 AM UTC-4, WM wrote:
> > William schrieb am Donnerstag, 30. Dezember 2021 um 20:13:38 UTC+1:
> > > On Thursday, December 30, 2021 at 2:29:24 PM UTC-4, WM wrote:
> > > > William schrieb am Donnerstag, 30. Dezember 2021 um 16:47:57 UTC+1:
> > >
> > > > > Correct, Each element of N_P is finite. However, N_P is infinite
> > > > How does it get actually infinite when all definable natural numbers fail?
> > > It does not "get actually infinite" . N_P is not the result of a process. N_P is a Peano set and has cardinality aleph_0
> > >
> > > N_P does not have a last element. Every element of N_P is the largest element of a set with last element.
> > > Thus the set N_P has a property that no element of N_P has.
> > But the set ℕ has a property that the set ℕ_def = ℕ_P
> nope |N_def, the set of natural numbers that can be written down, is a finite set,

No, it is potentially infinite since it has no largest element - just like N_P.

> >does not have, namely a number of elements that is larger than *all* elements.
>
> yes, your |N.which is not the natural numbers, has paradoxical elements that are greater than themselves. N_P only has finite elements,
> |N def has a last element, N_P has no last element

No, both have only temporarily last elements.

> > > > Doubling a natural number produces a natural number.
> > > Correct
> > >
> > > > But if there was a fixed set, then doubling would double its [cardinality]
> > >
> > > Simply false.
> > With cardinality yes.
> Cardinality is the only way to talk about the "Number of elements" in a set with no largest element.

Symmetry is the only way. The number of rationals between 0 an 1 is the same as between 7 and 8, not even one more or less.

Cardinality is self-contradictory as we see when scrutinizing Cantors "bijection"
{1, 2, 3, 4, 5, ...} --> {1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...}.

Collect all rationals of the interval (0, 1] into a reservoir, map one of them by 1 and remove it from the reservoir. Then collect all rationals of the interval (1, 2] into the reservoir, map one of them by 2 and remove it from the reservoir. Then collect all rationals of the interval (2, 3] into the reservoir, map one of them by 3 and remove it from the reservoir. Continue until all intervals will have been worked through. Then the reservoir will be empty.

Of course this conclusion is contrary to mathematics. The reservoir will not be empty. But here the same mapping is applied as by Cantor. The only difference is that he puts all fractions into the reservoir before he begins. I use disjunct sets {(n, n+1] ∩ Q} in order to show how his operation fails.

Regards, WM

On 12/31/2021 11:53 AM, WM wrote:
> William schrieb am Freitag, 31. Dezember 2021 um 15:54:03 UTC+1:
>> On Friday, December 31, 2021 at 5:26:21 AM UTC-4, WM wrote:
>>> William schrieb am Donnerstag, 30. Dezember 2021 um 20:13:38 UTC+1:
>>>> On Thursday, December 30, 2021 at 2:29:24 PM UTC-4, WM wrote:
>>>>> William schrieb am Donnerstag, 30. Dezember 2021 um 16:47:57 UTC+1:
>>>>
>>>>>> Correct, Each element of N_P is finite. However, N_P is infinite
>>>>> How does it get actually infinite when all definable natural numbers fail?
>>>> It does not "get actually infinite" . N_P is not the result of a process. N_P is a Peano set and has cardinality aleph_0
>>>>
>>>> N_P does not have a last element. Every element of N_P is the largest element of a set with last element.
>>>> Thus the set N_P has a property that no element of N_P has.
>>> But the set ℕ has a property that the set ℕ_def = ℕ_P
>> nope |N_def, the set of natural numbers that can be written down, is a finite set,
>
> No, it is potentially infinite since it has no largest element - just like N_P.

no, ℕ has no largest element and it is infinite.

your logic is backwards.

potentially infinite = finite

ℕ is NOT finite

>
>>> does not have, namely a number of elements that is larger than *all* elements.
>>
>> yes, your |N.which is not the natural numbers, has paradoxical elements that are greater than themselves. N_P only has finite elements,
>> |N def has a last element, N_P has no last element
>
> No, both have only temporarily last elements.

there are no "temporarily last elements"

temporarily last Ants

>
>>>>> Doubling a natural number produces a natural number.
>>>> Correct
>>>>
>>>>> But if there was a fixed set, then doubling would double its [cardinality]
>>>>
>>>> Simply false.
>>> With cardinality yes.
>> Cardinality is the only way to talk about the "Number of elements" in a set with no largest element.
>
> Symmetry is the only way. The number of rationals between 0 an 1 is the same as between 7 and 8, not even one more or less.

then Prove it.

>

<snip misleading crap>

>
> Regards, WM

On 12/31/2021 9:44 AM, WM wrote:
> William schrieb am Freitag, 31. Dezember 2021 um 15:53:54 UTC+1:
>> On Friday, December 31, 2021 at 5:34:43 AM UTC-4, WM wrote:
>>
>>> {1}
>>> {2, 1}
>>> {3, 2, 1}
>>> {4, 3, 2, 1}
>>> {5, 4, 3, 2, 1}
>>> ...
>>>
>>> If there are ℵo natnumbers, then they are in the first column. If they are in the first column, then they are in the whole figure.
>>> There are not any two lines in the figure which contain more than one of them. No line contains ℵo natnumbers.
>> Correct the whole figure, the set, contains aleph_0 elements. But no line, the elements of the figure, contain aleph_0 elements. The set has a property that no element does.
>
> In this case the set cannot be larger than all lines. They cover all definable numbers.
>>>
>>> But the set ℕ is much, much larger that ℕ_def:
>>>
>>> {1, 2, 3, 4, 5, ..., ω/1000} is not a FISON although it contains only numbers less than ω.
>> Nope, ω/1000= w.
>
> Not if ω belongs to mathematics.
>
>> The only ordinals less than ω are the natural numbers.
>
> Yes, but all multiples of ω are dark.

Whoever hired you should be thrown in a dark room for an infinite amount
of time...

On Friday, December 31, 2021 at 1:44:41 PM UTC-4, WM wrote:
> William schrieb am Freitag, 31. Dezember 2021 um 15:53:54 UTC+1:
> > On Friday, December 31, 2021 at 5:34:43 AM UTC-4, WM wrote:
> >
> > > {1}
> > > {2, 1}
> > > {3, 2, 1}
> > > {4, 3, 2, 1}
> > > {5, 4, 3, 2, 1}
> > > ...
> > >
> > > If there are ℵo natnumbers, then they are in the first column.. If they are in the first column, then they are in the whole figure.
> > > There are not any two lines in the figure which contain more than one of them. No line contains ℵo natnumbers.
> > Correct the whole figure, the set, contains aleph_0 elements. But no line, the elements of the figure, contain aleph_0 elements. The set has a property that no element does.
> In this case the set cannot be larger than all lines.

Correct.. "All lines" is the set which is infinite. Note the set can be larger than each line, i.e. each element.

--
William Hughes

On Friday, December 31, 2021 at 1:53:41 PM UTC-4, WM wrote:
> William schrieb am Freitag, 31. Dezember 2021 um 15:54:03 UTC+1:
> > |N_def, the set of natural numbers that can be written down, is a finite set,
> No, it is potentially infinite since it has no largest element - just like N_P.

Potentailly infinite set is nonsense. N_P is a set.

> > |N def has a last element, N_P has no last element

> No, both have only temporarily last elements.

Nope, "temporary element" of a set is nonsense. Sets do not change.

--
William Hughes

On Friday, December 31, 2021 at 11:26:44 PM UTC+1, William wrote:
> On Friday, December 31, 2021 at 1:53:41 PM UTC-4, WM wrote:
> > William schrieb am Freitag, 31. Dezember 2021 um 15:54:03 UTC+1:
> > >
> > > IN def has a last element, N_P has no last element
> > >
> > No, both [sets] have only temporarily last elements.
> >
> Nope, "temporary element" of a set is nonsense. Sets do not change.

Note that in Mückenheim's world sets age.

On Friday, December 31, 2021 at 2:26:44 PM UTC-8, William wrote:
> On Friday, December 31, 2021 at 1:53:41 PM UTC-4, WM wrote:
> > William schrieb am Freitag, 31. Dezember 2021 um 15:54:03 UTC+1:
> > > |N_def, the set of natural numbers that can be written down, is a finite set,
> > No, it is potentially infinite since it has no largest element - just like N_P.
> Potentailly infinite set is nonsense. N_P is a set.
> > > |N def has a last element, N_P has no last element
>
> > No, both have only temporarily last elements.
> Nope, "temporary element" of a set is nonsense. Sets do not change.
>
> --
> William Hughes

What's a continued fractal?

Ross A. Finlayson explained :
> On Friday, December 31, 2021 at 2:26:44 PM UTC-8, William wrote:
>> On Friday, December 31, 2021 at 1:53:41 PM UTC-4, WM wrote:
>>> William schrieb am Freitag, 31. Dezember 2021 um 15:54:03 UTC+1:
>>>>> N_def, the set of natural numbers that can be written down, is a finite
>>>>> set,
>>> No, it is potentially infinite since it has no largest element - just like
>>> N_P. Potentailly infinite set is nonsense. N_P is a set.
>>>>> N def has a last element, N_P has no last element
>>> No, both have only temporarily last elements.
>> Nope, "temporary element" of a set is nonsense. Sets do not change.
>>
>> --
>> William Hughes
>
> What's a continued fractal?

I thought that they all were, by definition.

On Saturday, January 1, 2022 at 2:33:01 AM UTC-4, Fritz Feldhase wrote:
> On Friday, December 31, 2021 at 11:26:44 PM UTC+1, William wrote:
> > On Friday, December 31, 2021 at 1:53:41 PM UTC-4, WM wrote:
> > > William schrieb am Freitag, 31. Dezember 2021 um 15:54:03 UTC+1:
> > > >
> > > > IN def has a last element, N_P has no last element
> > > >
> > > No, both [sets] have only temporarily last elements.
> > >
> > Nope, "temporary element" of a set is nonsense. Sets do not change.
> Note that in Mückenheim's world sets age.

Indeed, but we are not in Mückenheim's world. The claim is that there is a contradiction in ZF.

--
William Hughes

William schrieb am Freitag, 31. Dezember 2021 um 23:11:00 UTC+1:
> On Friday, December 31, 2021 at 1:44:41 PM UTC-4, WM wrote:

> > > > {1}
> > > > {2, 1}
> > > > {3, 2, 1}
> > > > {4, 3, 2, 1}
> > > > {5, 4, 3, 2, 1}
> > > > ...

> Correct.. "All lines" is the set which is infinite. Note the set can be larger than each line, i.e. each element.
>
The set is nothing but its elements. In case of FISONs inclusion monotony shows that the set cannot surpass all its elements.

Regards, WM

William schrieb am Samstag, 1. Januar 2022 um 16:39:17 UTC+1:
> On Saturday, January 1, 2022 at 2:33:01 AM UTC-4, Fritz Feldhase wrote:
> > On Friday, December 31, 2021 at 11:26:44 PM UTC+1, William wrote:
> > > On Friday, December 31, 2021 at 1:53:41 PM UTC-4, WM wrote:
> > > > William schrieb am Freitag, 31. Dezember 2021 um 15:54:03 UTC+1:
> > > > >
> > > > > IN def has a last element, N_P has no last element
> > > > >
> > > > No, both [sets] have only temporarily last elements.
> > > >
> > > Nope, "temporary element" of a set is nonsense. Sets do not change.
> > Note that in Mückenheim's world sets age.
> Indeed, but we are not in Mückenheim's world. The claim is that there is a contradiction in ZF.
>
This has been shown: Collect all rationals of the interval (0, 1] into a reservoir, map one of them by 1 and remove it from the reservoir. Then collect all rationals of the interval(1, 2] into the reservoir, map one of them by 2 and remove it from the reservoir. Then collect all rationals of the interval (2, 3] into the reservoir, map one of mthem by 3 and remove it from the reservoir. Continue until all intervals will have been worked through. Then the reservoir will be empty according to set theory.

Regards, WM

On 1/1/2022 1:34 PM, WM wrote:
> William schrieb am Freitag, 31. Dezember 2021 um 23:11:00 UTC+1:
>> On Friday, December 31, 2021 at 1:44:41 PM UTC-4, WM wrote:
>
>>>>> {1}
>>>>> {2, 1}
>>>>> {3, 2, 1}
>>>>> {4, 3, 2, 1}
>>>>> {5, 4, 3, 2, 1}
>>>>> ...
>
>> Correct.. "All lines" is the set which is infinite. Note the set can be larger than each line, i.e. each element.
>>
> The set is nothing but its elements.

Wrong, and you stay confused.

> In case of FISONs inclusion monotony shows that the set cannot surpass all its elements.

your sentence is malformed.

>
> Regards, WM

On 1/1/2022 1:37 PM, WM wrote:
> William schrieb am Samstag, 1. Januar 2022 um 16:39:17 UTC+1:
>> On Saturday, January 1, 2022 at 2:33:01 AM UTC-4, Fritz Feldhase wrote:
>>> On Friday, December 31, 2021 at 11:26:44 PM UTC+1, William wrote:
>>>> On Friday, December 31, 2021 at 1:53:41 PM UTC-4, WM wrote:
>>>>> William schrieb am Freitag, 31. Dezember 2021 um 15:54:03 UTC+1:
>>>>>>
>>>>>> IN def has a last element, N_P has no last element
>>>>>>
>>>>> No, both [sets] have only temporarily last elements.
>>>>>
>>>> Nope, "temporary element" of a set is nonsense. Sets do not change.
>>> Note that in Mückenheim's world sets age.
>> Indeed, but we are not in Mückenheim's world. The claim is that there is a contradiction in ZF.
>>
> This has been shown: Collect all rationals of the interval (0, 1] into a reservoir, map one of them by 1 and remove it from the reservoir. Then collect all rationals of the interval(1, 2] into the reservoir, map one of them by 2 and remove it from the reservoir. Then collect all rationals of the interval (2, 3] into the reservoir, map one of mthem by 3 and remove it from the reservoir. Continue until all intervals will have been worked through. Then the reservoir will be empty according to set theory.
>
> Regards, WM
>

re-word that to be clearer.

On 1/1/2022 5:02 AM, FromTheRafters wrote:
> Ross A. Finlayson explained :
>> On Friday, December 31, 2021 at 2:26:44 PM UTC-8, William wrote:
>>> On Friday, December 31, 2021 at 1:53:41 PM UTC-4, WM wrote:
>>>> William schrieb am Freitag, 31. Dezember 2021 um 15:54:03 UTC+1:
>>>>>> N_def, the set of natural numbers that can be written down, is a
>>>>>> finite set,
>>>> No, it is potentially infinite since it has no largest element -
>>>> just like N_P. Potentailly infinite set is nonsense. N_P is a set.
>>>>>> N def has a last element, N_P has no last element
>>>> No, both have only temporarily last elements.
>>> Nope, "temporary element" of a set is nonsense. Sets do not change.
>>> --
>>> William Hughes
>>
>> What's a continued fractal?
>
> I thought that they all were, by definition.

Indeed, they are.

https://math.uchicago.edu/~dannyc/research/dynamics/continued_fractals.pdf

On Saturday, January 1, 2022 at 3:37:22 PM UTC-4, WM wrote:
> William schrieb am Samstag, 1. Januar 2022 um 16:39:17 UTC+1:
> > On Saturday, January 1, 2022 at 2:33:01 AM UTC-4, Fritz Feldhase wrote:
> > > On Friday, December 31, 2021 at 11:26:44 PM UTC+1, William wrote:
> > > > On Friday, December 31, 2021 at 1:53:41 PM UTC-4, WM wrote:
> > > > > William schrieb am Freitag, 31. Dezember 2021 um 15:54:03 UTC+1:
> > > > > >
> > > > > > IN def has a last element, N_P has no last element
> > > > > >
> > > > > No, both [sets] have only temporarily last elements.
> > > > >
> > > > Nope, "temporary element" of a set is nonsense. Sets do not change.
> > > Note that in Mückenheim's world sets age.
> > Indeed, but we are not in Mückenheim's world. The claim is that there is a contradiction in ZF.
> >
> This has been shown: Collect all rationals of the interval (0, 1] into a reservoir, map one of them by 1 and remove it from the reservoir. Then collect all rationals of the interval(1, 2] into the reservoir, map one of them by 2 and remove it from the reservoir. Then collect all rationals of the interval (2, 3] into the reservoir, map one of mthem by 3 and remove it from the reservoir. Continue until all intervals will have been worked through. Then the reservoir will be empty according to set theory.
>
This is not a contradition but a result you do not like. The bijection exists, it is not the result of a process. (It can be made the result of a process, but only by using a "limit" that is not the Calc 101 limit)..

--
William Hughes

On Saturday, January 1, 2022 at 3:34:40 PM UTC-4, WM wrote:
> William schrieb am Freitag, 31. Dezember 2021 um 23:11:00 UTC+1:
> > On Friday, December 31, 2021 at 1:44:41 PM UTC-4, WM wrote:
>
> > > > > {1}
> > > > > {2, 1}
> > > > > {3, 2, 1}
> > > > > {4, 3, 2, 1}
> > > > > {5, 4, 3, 2, 1}
> > > > > ...
> > Correct.. "All lines" is the set which is infinite. Note the set can be larger than each line, i.e. each element.
> >
> The set is nothing but its elements. In case of FISONs inclusion monotony shows that the set cannot surpass all its elements.

Of course the set is nothing more than "all its elements" and cannot surpass itself. . It does not follow that because each of the elements of the set is finite. the set, " all its elements", must be finite.

--
William Hughes

William schrieb am Sonntag, 2. Januar 2022 um 15:34:04 UTC+1:
> On Saturday, January 1, 2022 at 3:34:40 PM UTC-4, WM wrote:
> > William schrieb am Freitag, 31. Dezember 2021 um 23:11:00 UTC+1:
> > > On Friday, December 31, 2021 at 1:44:41 PM UTC-4, WM wrote:
> >
> > > > > > {1}
> > > > > > {2, 1}
> > > > > > {3, 2, 1}
> > > > > > {4, 3, 2, 1}
> > > > > > {5, 4, 3, 2, 1}
> > > > > > ...
> > > Correct.. "All lines" is the set which is infinite. Note the set can be larger than each line, i.e. each element.
> > >
> > The set is nothing but its elements. In case of FISONs inclusion monotony shows that the set cannot surpass all its elements.
> Of course the set is nothing more than "all its elements" and cannot surpass itself. . It does not follow that because each of the elements of the set is finite. the set, " all its elements", must be finite.

The collection ℕ_def of natnumbers gathered by the FISONs cannot differ from all FISONs. One of the FISONs is the collection. Otherwise there would be at least one FISON that omitted at least one of the natnumbers collected by its predecessors.

But of course there is no last FISON. There is no completed set of definable natnumbers because every FISON is followed by an actually infinite endsegment.

Regards, WM

William schrieb am Sonntag, 2. Januar 2022 um 15:22:50 UTC+1:
> On Saturday, January 1, 2022 at 3:37:22 PM UTC-4, WM wrote:
> > William schrieb am Samstag, 1. Januar 2022 um 16:39:17 UTC+1:
> > > On Saturday, January 1, 2022 at 2:33:01 AM UTC-4, Fritz Feldhase wrote:
> > > > Note that in Mückenheim's world sets age.
> > > Indeed, but we are not in Mückenheim's world. The claim is that there is a contradiction in ZF.
> > >
> > This has been shown: Collect all rationals of the interval (0, 1] into a reservoir, map one of them by 1 and remove it from the reservoir. Then collect all rationals of the interval(1, 2] into the reservoir, map one of them by 2 and remove it from the reservoir. Then collect all rationals of the interval (2, 3] into the reservoir, map one of mthem by 3 and remove it from the reservoir. Continue until all intervals will have been worked through. Then the reservoir will be empty according to set theory.
> >
> This is not a contradition but a result you do not like.

It is a result that I do not like because it is provably wrong. I never met a student who accepted this mess.

> The bijection exists,

If a bijection existed, then every re-ordering of the image set would be a bijection. For instance taking one rational out of every unit interval, for instance simply chosing always n/1.

Cantor has endorsed this method: "The question by what transformations of a well-ordered set its ordinal number (Anzahl) is changed, by what it is not, is simply answered: Those and only those transformations leave the ordinal number unchanged which can be put down to a finite or infinite set of transpositions, i.e., exchanges of two elements." [Cantor, p. 214] Hence the well-order is not damaged by infinitely many transpositions.

> it is not the result of a process.

That is the usual lie. Of course every sequence is a process. Compare Cantor: "Zuordnungsgesetz, Zuordnungsmodus, Zuordnungsprozeß, Zuordnungsverfahren, Zuordnungsverhältnis". [Cantor, Collected Works, pp. 239, 283f, 286f, 291, 293, 305f, 413] With the usual "all must happen simultaneously" his theory would never have been accepted. He devised the sequence of counted rationals
{1, 2, 3, 4, 5, ...} --> {1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...}.
Had he only said abrcadabra, noone would have listened.

> (It can be made the result of a process, but only by using a "limit" that is not the Calc 101 limit)..

No, there is no limit possible. "two sets are ascribed the same cardinality if they mutually uniquely, element by element, can be mapped onto each other. ... If every element of set M can be related to one and only one corresponding element of set N and vice versa, and if there is never an obstacle or halt". Limits would destroy this. *All* elements must appear, element by element. But as my example shows they don't.

Regards, WM

On 1/3/2022 2:56 AM, WM wrote:
> William schrieb am Sonntag, 2. Januar 2022 um 15:34:04 UTC+1:
>> On Saturday, January 1, 2022 at 3:34:40 PM UTC-4, WM wrote:
>>> William schrieb am Freitag, 31. Dezember 2021 um 23:11:00 UTC+1:
>>>> On Friday, December 31, 2021 at 1:44:41 PM UTC-4, WM wrote:
>>>
>>>>>>> {1}
>>>>>>> {2, 1}
>>>>>>> {3, 2, 1}
>>>>>>> {4, 3, 2, 1}
>>>>>>> {5, 4, 3, 2, 1}
>>>>>>> ...
>>>> Correct.. "All lines" is the set which is infinite. Note the set can be larger than each line, i.e. each element.
>>>>
>>> The set is nothing but its elements. In case of FISONs inclusion monotony shows that the set cannot surpass all its elements.
>> Of course the set is nothing more than "all its elements" and cannot surpass itself. . It does not follow that because each of the elements of the set is finite. the set, " all its elements", must be finite.
>
> The collection ℕ_def of natnumbers gathered by the FISONs cannot differ from all FISONs. One of the FISONs is the collection. Otherwise there would be at least one FISON that omitted at least one of the natnumbers collected by its predecessors.

Learn to use Union.

"collected, gathered, collection, ℕ_def, completed" are meaningless in Math, however they are useful weasel words to trolls.

>
> But of course there is no last FISON. There is no completed set of definable natnumbers because every FISON is followed by an actually infinite endsegment.
>

wrong. The set of natural numbers is ℕ, and E(1).

> Regards, WM
>

On 1/3/2022 3:23 AM, WM wrote:
> William schrieb am Sonntag, 2. Januar 2022 um 15:22:50 UTC+1:
>> On Saturday, January 1, 2022 at 3:37:22 PM UTC-4, WM wrote:
>>> William schrieb am Samstag, 1. Januar 2022 um 16:39:17 UTC+1:
>>>> On Saturday, January 1, 2022 at 2:33:01 AM UTC-4, Fritz Feldhase wrote:
>
>>>>> Note that in Mückenheim's world sets age.
>>>> Indeed, but we are not in Mückenheim's world. The claim is that there is a contradiction in ZF.
>>>>
>>> This has been shown: Collect all rationals of the interval (0, 1] into a reservoir, map one of them by 1 and remove it from the reservoir. Then collect all rationals of the interval(1, 2] into the reservoir, map one of them by 2 and remove it from the reservoir. Then collect all rationals of the interval (2, 3] into the reservoir, map one of mthem by 3 and remove it from the reservoir. Continue until all intervals will have been worked through. Then the reservoir will be empty according to set theory.
>>>
>> This is not a contradition but a result you do not like.
>
> It is a result that I do not like because it is provably wrong. I never met a student who accepted this mess.

your leaky rubber sets...

>
>> The bijection exists,
>
> If a bijection existed, then every re-ordering of the image set would be a bijection. For instance taking one rational out of every unit interval, for instance simply chosing always n/1.
>
> Cantor has endorsed this method: "The question by what transformations of a well-ordered set its ordinal number (Anzahl) is changed, by what it is not, is simply answered: Those and only those transformations leave the ordinal number unchanged which can be put down to a finite or infinite set of transpositions, i.e., exchanges of two elements." [Cantor, p. 214] Hence the well-order is not damaged by infinitely many transpositions.
>
>> it is not the result of a process.
>
> That is the usual lie. Of course every sequence is a process.

wrong. this is where you make huge mistake, your rubber sets.

Compare Cantor: "Zuordnungsgesetz, Zuordnungsmodus, Zuordnungsprozeß, Zuordnungsverfahren, Zuordnungsverhältnis". [Cantor, Collected Works, pp. 239,
283f, 286f, 291, 293, 305f, 413] With the usual "all must happen simultaneously" his theory would never have been accepted. He devised the sequence of
counted rationals
> {1, 2, 3, 4, 5, ...} --> {1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...}.
> Had he only said abrcadabra, noone would have listened.

yes we all agree Cantor counted the rationals, but that is not the point here.

>
>> (It can be made the result of a process, but only by using a "limit" that is not the Calc 101 limit)..
>
> No, there is no limit possible. "two sets are ascribed the same cardinality if they mutually uniquely, element by element, can be mapped onto each other. ... If every element of set M can be related to one and only one corresponding element of set N and vice versa, and if there is never an obstacle or halt". Limits would destroy this. *All* elements must appear, element by element. But as my example shows they don't.

your example fails.

>
> Regards, WM

On Monday, January 3, 2022 at 4:56:33 AM UTC-4, WM wrote:
> William schrieb am Sonntag, 2. Januar 2022 um 15:34:04 UTC+1:
> > On Saturday, January 1, 2022 at 3:34:40 PM UTC-4, WM wrote:
> > > William schrieb am Freitag, 31. Dezember 2021 um 23:11:00 UTC+1:
> > > > On Friday, December 31, 2021 at 1:44:41 PM UTC-4, WM wrote:
> > >
> > > > > > > {1}
> > > > > > > {2, 1}
> > > > > > > {3, 2, 1}
> > > > > > > {4, 3, 2, 1}
> > > > > > > {5, 4, 3, 2, 1}
> > > > > > > ...
> > > > Correct.. "All lines" is the set which is infinite. Note the set can be larger than each line, i.e. each element.
> > > >
> > > The set is nothing but its elements. In case of FISONs inclusion monotony shows that the set cannot surpass all its elements.
> > Of course the set is nothing more than "all its elements" and cannot surpass itself. . It does not follow that because each of the elements of the set is finite. the set, " all its elements", must be finite.
> The collection ℕ_def

of natural numbers that have the property they can be written down, is a subset of N_P the set of natural numbers that are the largest element of a FISON. N_P is infinite. even though its elements are finite.

of natnumbers gathered by the FISONs cannot differ from all FISONs. One of the FISONs is the collection.

Nope, the colliction is the set N_P. Each FISON has a last element . The set N_P has no last element. N_P is not a FISON. However, N_P is complete.

--
William Hughes

On 1/3/2022 4:23 AM, WM wrote:
> William schrieb
> am Sonntag, 2. Januar 2022 um 15:22:50 UTC+1:
>> On Saturday, January 1, 2022 at 3:37:22 PM UTC-4,
>> WM wrote:

>>> This has been shown:
>>> Collect all rationals of the interval (0, 1]
[...]

>> The bijection exists,
>
> If a bijection existed, then every re-ordering of the
> image set would be a bijection. For instance taking
> one rational out of every unit interval, for instance
> simply chosing always n/1.

You are taking a property which is true for some sets
-- sets which can be totally-ordered with two ends
in which each _cut_ is a _step_ --
and you're looking for it in sets without that property.
And you find it's not there.

You haven't found a contradiction.
You've found things which cannot be totally-ordered
with two ends in which each cut is a step.

We have reasons to examine both kinds of things,
different reasons, in some cases.

>> it is not the result of a process.
>
> That is the usual lie.
> Of course every sequence is a process.

....and every sheep-number is a sheep.

No, wait. Maybe not.

Maybe there is an important difference between
representing a process and being a process.

On 1/3/2022 4:23 AM, WM wrote:
> William schrieb
> am Sonntag, 2. Januar 2022 um 15:22:50 UTC+1:

>> This is not a contradition but a result you do not like.
>
> It is a result that I do not like because it is provably
> wrong. I never met a student who accepted this mess.

You never met a student who...

You never met a student enrolled in your class who...

You never met a student enrolled in your class
and required to repeat back whatever you feel like
telling them, who...

....who accepted this mess.

....who was willing to publicly disagree with the old man
who would give them an easy A if they managed to keep
their mouth shut.

This version of peer review, your version, certainly
has advantages over the more common version,
advantages to you, anyway.

One advantage, to you, at least:
You could announce to your students that 2 + 2 = 5
and they would dutifully nod and write that down.
Your classroom is a safe space for you, in which
you face zero chance of have any errors pointed out.
I'm sure you love that.