The imbecile Clown WM drivels:

> It is a description of an activity

LOL - in math there is no "activity".

Piss off, idiot.

On Thursday, March 31, 2022 at 8:01:18 AM UTC-3, WM wrote:
> William schrieb am Donnerstag, 31. März 2022 um 00:07:03 UTC+2:
> > On Wednesday, March 30, 2022 at 9:14:48 AM UTC-3, WM wrote:
> > > david...@gmail.com schrieb am Mittwoch, 30. März 2022 um 06:08:44 UTC+2:
> >
> > > {1} = {1}
> > > {1} U {1, 2} = {1, 2}
> > > {1} U {1, 2} U {1, 2, 3} = {1, 2, 3}
> > > {1} U {1, 2} U {1, 2, 3} U {1, 2, 3, 4} = {1, 2, 3, 4}
> > > {1} U {1, 2} U {1, 2, 3} U {1, 2, 3, 4} U {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5}
> > > ...
> > > That implies that the sequence of *all* finite initial segments
> > is a sequence without largest element,
> but with all elements.

Correct, and since the sequence contains all elements it does not contain a largest element, so the union is larger than any single element.
Every element of the sequence is a proper subset of the union. (Note that this does not imply that there is an *element in the union* that is not in any element of the sequence, it does imply that there is at least one *subset of the union* that is not contained in any single element of the sequence. There exist subsets of the union which do not have a largest element. No single element of the sequence can contains such a subset),

Recall that a "potentially infinite set" is nonsense.

--
William Hughes

On 3/31/2022 6:12 AM, WM wrote:
> Alan Mackenzie schrieb am Mittwoch, 30. März 2022 um 20:27:08 UTC+2:
>> WM <askas...@gmail.com> wrote:
>
>>> The sequence is in process of production and never infinite.
>> The sequence is always infinite. There is no final digit.
>
> Therefore the limit cannot be determined from the sequence.

wrong. Some you can, some you can't.

>
>>>>> So we cannot claim that a certain digit sequence exists.
>>
>>>> We can. The sequence produced by the digit generator exists.
>>
>>> It exists if all terms have been produced, i.e., if no further term can
>>> be produced. That is never the case.
>> Huh? It exists. Consider, it has been described mathematically, und
>> does not lead to a contradiction.
>
> A limit has been described if there is a formula.

One may be able to determine the limit to a infinite sequence.

> A random generator does not describe a limit.

Wrong, it can solve various classes of problems.

>
>> It's an infinite sequence. There is no last digit.
>
> And no limit can be detemined. Hence it is not a real number.

Wrong. (see above)

>>
>>> The contradiction is that between the production and the ceasing there
>>> must be an end.
>> That's incoherent.
>
> No, that is obvious: Between action and rest, there must have been a last action.

No. We are not making sausages here.

>
>>>> Wrong. The sequence produced by the random digit generator is infinite.
>>
>>> Show such a sequence. If it exists, you can write it.
>> Such a sequence cannot be shown. I cannot write it, but since it gives
>> rise to no contradiction, it exists.
>
> It gives rise to a contradiction to believe that it defines a real number.

you assume no numbers exist at all, However others assume numbers exist.

Who does Math better, with numbers that dont exist, or those that do exist ?

> Definable sequences can have limits like pi.

wrong! by your own daffynition: you haven't completed writing down all the digits of Pi, so it does not exist

> Your proposed sequence cannot have a limit.

and you are full of BS.

>
>>> It is, contrary to your sequence which cannot be shown and cannot be
>>> described.
>> The sequence has been described; it is an infinite sequence of decimal
>> digits produced by a random digit generator. What is that if not a
>> description?
>
> It is a description of an activity but it is neither a description nor a specification of a limit, i.e., of a real number.

so you cannot accept being wrong. Typical Troll

>
> Regards, WM

FromTheRafters schrieb am Mittwoch, 30. März 2022 um 20:16:09 UTC+2:
> sergio wrote :

> >>>> My main point is that he claims that
> >>>> all fractions are definable and can be enumerated.
> >>> k = (r+c-1)*(r+2-2)/2 + r
> >>
> >> please check the formula
> >
> > it is called an EQUATION.
> I agree, the equals sign was a big hint. :)
>
Yes some formulas are equations. They contain equals signs.

Regards, WM

FromTheRafters schrieb am Mittwoch, 30. März 2022 um 20:27:42 UTC+2:
> on 3/30/2022, WM supposed :

> > And none of them produce a set of cardinality larger than every natural
> > number, i.e., which has ℵo elements.
> FISONs have a natural number of elements, and aleph_zero is not a
> natural number.

Therefore it indicates more than any FISON.

> Aleph_zero, treated as a 'number of elements' rather
> than a notion of size cannot fit into a FISON but there is always a
> next FISON

that will fail to complete an ℵo-set.

Regrads, WM

Jim Burns schrieb am Donnerstag, 31. März 2022 um 00:34:00 UTC+2:

> Bob does not get to a place without numbers
> with FISON-ends.

Then he remains in the matrix. But Canto claims that all O's will disappear..

Regards, WM

zelos...@gmail.com schrieb am Donnerstag, 31. März 2022 um 06:41:09 UTC+2:
> onsdag 30 mars 2022 kl. 13:42:44 UTC+2 skrev WM:
> > zelos...@gmail.com schrieb am Mittwoch, 30. März 2022 um 07:05:15 UTC+2:
> > > tisdag 29 mars 2022 kl. 16:02:25 UTC+2 skrev WM:
> >
> > > > But it is still true that there are undefinable elements in actual infinity (or not existing elements in potential infinity).
> > > >
> > > You have yet to show any "dark" stuff exists
> > Here is a simple proof:
> >
> > The finite initial segments {1, 2, 3, ..., n} of natural numbers ℕ contain the union of all their predecessors:
> >
> > {1} = {1}
> > {1} U {1, 2} = {1, 2}
> > {1} U {1, 2} U {1, 2, 3} = {1, 2, 3}
> > {1} U {1, 2} U {1, 2, 3} U {1, 2, 3, 4} = {1, 2, 3, 4}
> > {1} U {1, 2} U {1, 2, 3} U {1, 2, 3, 4} U {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5}
> > ...
> >
> > That implies that the sequence of *all* finite initial segments contains the union of all finite initial segments. This sequence however does not contain an infinite finite initial segment. If ℕ is actually infinite, it must contain more than all finite initial segments in addition. What is this additional contents?
> >
> Just because all FISONs are finite does that not mean that the union of all of them isn't all of N,

ry to find a wrong step here:
Every FISON is finite.
Every FISON contains all its predecessors.
The sequence of all FISONs contains all FISONs including all successors of all FISONs.
The sequence of all FISONs contains the union of all FISONs.

Regards, WM

zelos...@gmail.com schrieb am Donnerstag, 31. März 2022 um 06:41:35 UTC+2:
> onsdag 30 mars 2022 kl. 13:47:46 UTC+2 skrev WM:
> > zelos...@gmail.com schrieb am Mittwoch, 30. März 2022 um 07:04:58 UTC+2:
> > > tisdag 29 mars 2022 kl. 15:51:08 UTC+2 skrev WM:
> > > > Jim Burns schrieb am Montag, 28. März 2022 um 22:07:25 UTC+2:
> > > >
> > > > > Note that a finite number of words can describe infinitely-
> > > > > -many things.
> > > > But only collectively, not individually. If you describe the set you don't describe its elements, unless there is only one or none.
> > > >
> > > your "collectively" and "individually" is meaningless.
> > n ∈ ℕ describes natural numbers collectively. 2 ∈ ℕ describes an individual.
>
> So it is meaningless because they are the same.

Only if n = 2. But that is not the case because 2 < 3 can be proved but n < 3 cannot be proved.

Regards, WM

Eram semper recta schrieb am Donnerstag, 31. März 2022 um 07:14:14 UTC+2:
> On Wednesday, 30 March 2022 at 17:03:04 UTC+3, WM wrote:
> > Eram semper recta schrieb am Mittwoch, 30. März 2022 um 15:18:28 UTC+2:
> >
> > > > Yes p/q is a symbolic fraction, but not an individual like 2.
> > > No. p/q is very much an individual fraction.
> > For 2 you can find out whether it is larger than 10, for p/q you cannot find out whether it is larger than 10.
>
> Your reply is strange because it does not address the topic.

Your memory is short. This is the topic: You said that p and q are NOT variables any more than say the x in x + 3 = 5 is a "variable".

That is wrong because for 2 you can find out whether it is larger than 10, for p/q you cannot find out whether it is larger than 10.

> p/q is NOT a formula because a number is not a formula and a formula is not a number.

A formula or prescription can produce a number. The formula is p/q. The number has got the name p/q.

Regards, WM

WM <askasker48@gmail.com> wrote:
> Alan Mackenzie schrieb am Mittwoch, 30. März 2022 um 20:27:08 UTC+2:
>> WM <askas...@gmail.com> wrote:

>> > The sequence is in process of production and never infinite.
>> The sequence is always infinite. There is no final digit.

> Therefore the limit cannot be determined from the sequence.

>> >> > So we cannot claim that a certain digit sequence exists.

>> >> We can. The sequence produced by the digit generator exists.

>> > It exists if all terms have been produced, i.e., if no further term can
>> > be produced. That is never the case.
>> Huh? It exists. Consider, it has been described mathematically, und
>> does not lead to a contradiction.

> A limit has been described if there is a formula.

A limit has been "described" (what?) whether or not there is a formula.
If you wish to assert such a limit does not exist, you must show what
contradiction its existence causes.

> A random generator does not describe a limit.

It "describes" (what) a random limit. :-)

>> It's an infinite sequence. There is no last digit.

> And no limit can be detemined. Hence it is not a real number.

What leads you to such crazy ideas? Just because a limit cannot be
determined doesn't imply its non-existence. I suppose it's the sort of
thing you might call a "dark" limit. It's existence gives rise to no
contradiction. Therefore it exists, and is a real number.

>> > The contradiction is that between the production and the ceasing there
>> > must be an end.

>> That's incoherent.

> No, that is obvious: Between action and rest, there must have been a
> last action.

For one thing, we're talking mathematics here, where "action" and "rest"
have no meaning. For another, there needn't be a last action. If you
"perform" an infinite sequence of events, event n happening at time 1 -
2^(-n), you will be finished at time 1, yet there is no last event.

>> >> Wrong. The sequence produced by the random digit generator is
>> >> infinite.

>> > Show such a sequence. If it exists, you can write it.

>> Such a sequence cannot be shown. I cannot write it, but since it gives
>> rise to no contradiction, it exists.

> It gives rise to a contradiction to believe that it defines a real
> number. Definable sequences can have limits like pi. Your proposed
> sequence cannot have a limit.

You haven't demonstrated the supposed contradiction. You haven't even
given any hint of its nature. To be blunt, there is no such
contradiction, and the infinite sequence of random digits, regarded as a
decimal expansion, most certainly has a limit, even though we cannot know
what it is.

>> > It is, contrary to your sequence which cannot be shown and cannot be
>> > described.

>> The sequence has been described; it is an infinite sequence of decimal
>> digits produced by a random digit generator. What is that if not a
>> description?

> It is a description of an activity but it is neither a description nor
> a specification of a limit, i.e., of a real number.

Not sure what you mean by "activity", but the infinite sequence of random
digits most certainly converges to a limit, and that limit is a real
number.

> Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).

On Thursday, 31 March 2022 at 19:40:39 UTC+3, WM wrote:
> Eram semper recta schrieb am Donnerstag, 31. März 2022 um 07:14:14 UTC+2:
> > On Wednesday, 30 March 2022 at 17:03:04 UTC+3, WM wrote:
> > > Eram semper recta schrieb am Mittwoch, 30. März 2022 um 15:18:28 UTC+2:
> > >
> > > > > Yes p/q is a symbolic fraction, but not an individual like 2.
> > > > No. p/q is very much an individual fraction.
> > > For 2 you can find out whether it is larger than 10, for p/q you cannot find out whether it is larger than 10.
> >
> > Your reply is strange because it does not address the topic.
> Your memory is short.

My memory is very good. It's yours that is failing you.

> This is the topic: You said that p and q are NOT variables any more than say the x in x + 3 = 5 is a "variable".
>
> That is wrong

Nonsense. It is correct.

> because for 2 you can find out whether it is larger than 10, for p/q you cannot find out whether it is larger than 10.

Has no relevance whatsoever. Once p and q are replaced, nothing prevents you from comparing them. However, p/q refers to a number.

> > p/q is NOT a formula because a number is not a formula and a formula is not a number.
> A formula or prescription can produce a number.

Not without using numbers and even so, that does not make the prescription a number.

An ice-cream machine produces ice-cream, does that make the ice-cream machine ice cream???

> The formula is p/q. The number has got the name p/q.

Rubbish. formula =/= number.

>
> Regards, WM

On Thursday, March 31, 2022 at 6:26:21 PM UTC+2, WM wrote:

> Every FISON is finite.

True.
> Every FISON contains all its predecessors.

"contains"? In what sense?

As "subsets"? True.

Please try to be more specific, Mückenheim.

> The sequence of all FISONs contains all FISONs

"contains"? In what sense?

As "terms"? True.

Please try to be more specific, Mückenheim.

> including all successors of all FISONs.

Errrr...?

All FISONs are not enough?

> The sequence of all FISONs contains the union of all FISONs.

"contains"? In what sense?

As "term"? False.
As an "elemnt". False.
As a "subset" False.

Please try to be more specific, Mückenheim.

Hint: The union of all FISONs is IN. The sequence of all FISONs does not "contain" IN, neither as term nor as element or subset.

Alan Mackenzie schrieb am Donnerstag, 31. März 2022 um 18:44:08 UTC+2:

> You haven't even
> given any hint of its nature. To be blunt, there is no such
> contradiction, and the infinite sequence of random digits, regarded as a
> decimal expansion, most certainly has a limit, even though we cannot know
> what it is.

We cannot find the limit. We cannot find the completing digits after the definable digits, i.e., digits which can be known but are not completing the sequence. That means there are numbers and digits which cannot be knwons. I abbreviate that by dark numbers and dark digits.

> > It is a description of an activity but it is neither a description nor
> > a specification of a limit, i.e., of a real number.
> Not sure what you mean by "activity",

The action of your random digit generator.

> but the infinite sequence of random
> digits most certainly converges to a limit, and that limit is a real
> number.

If so, this number cannot be known by anybody. It is a dark real number.

Regards, WM

On Thursday, March 31, 2022 at 1:26:21 PM UTC-3, WM wrote:

> try to find a wrong step here:
> Every FISON is finite.
> Every FISON contains all its predecessors.
> The sequence of all FISONs contains all FISONs including all successors of all FISONs.
> The sequence of all FISONs contains the union of all FISONs.

all correct. However we also have

Every FISON has a successor of which it is a proper subset.

thus

No single element of the sequence of all FISONs contains the union of all FISONs.

--
William Hughes

Eram semper recta schrieb am Donnerstag, 31. März 2022 um 18:52:51 UTC+2:
> On Thursday, 31 March 2022 at 19:40:39 UTC+3, WM wrote:
> > Eram semper recta schrieb am Donnerstag, 31. März 2022 um 07:14:14 UTC+2:
> > > On Wednesday, 30 March 2022 at 17:03:04 UTC+3, WM wrote:
> > > > Eram semper recta schrieb am Mittwoch, 30. März 2022 um 15:18:28 UTC+2:
> > > >
> > > > > > Yes p/q is a symbolic fraction, but not an individual like 2.
> > > > > No. p/q is very much an individual fraction.
> > > > For 2 you can find out whether it is larger than 10, for p/q you cannot find out whether it is larger than 10.
> > >
> > > Your reply is strange because it does not address the topic.
> > Your memory is short.
> My memory is very good. It's yours that is failing you.
> > This is the topic: You said that p and q are NOT variables any more than say the x in x + 3 = 5 is a "variable".
> >
> > That is wrong
> Nonsense. It is correct.
> > because for 2 you can find out whether it is larger than 10, for p/q you cannot find out whether it is larger than 10.
> Has no relevance whatsoever.

That is the difference. Are you really too dense to understand it?

> > A formula or prescription can produce a number.
> Not without using numbers and even so, that does not make the prescription a number.

A Miller can produce flour or can only carry the name.

> > The formula is p/q. The number has got the name p/q.
> Rubbish. formula =/= number.

But both can and does carry the same name. Have you the power to forbid it?

Regards, WM

On Thursday, March 31, 2022 at 7:11:51 PM UTC+2, William wrote:
> >
> > The sequence of all FISONs contains the union of all FISONs.
> >
> all correct.

Please elaborate.

The union of all FISONs is IN. In which sense does the sequence of all FISONs "contain" IN?

Hint: IN is not a term nor an element or a subset in this sequence.

Fritz Feldhase schrieb am Donnerstag, 31. März 2022 um 19:02:38 UTC+2:
> On Thursday, March 31, 2022 at 6:26:21 PM UTC+2, WM wrote:
>
> > Every FISON is finite.
>
> True.
> > Every FISON contains all its predecessors.
> "contains"? In what sense?
>
> As "subsets"? True.

> > The sequence of all FISONs contains all FISONs
> "contains"? In what sense?
>
> As "terms"? True.
>
> > including all successors of all FISONs.
> Errrr...?
>
> All FISONs are not enough?

For some matheologians obviously not.

> > The sequence of all FISONs contains the union of all FISONs.
> "contains"? In what sense?
>
> As "term"?

Of course.

> false.

No it is true. Why should it be wrong?

> Hint: The union of all FISONs is IN.

Wrong. The union of all FISONs is less than ℕ. Only the union of ℵo FISONs would be ℕ but there are not ℵo (nore than every finite number of) FISONs without an infinite FISON.

> The sequence of all FISONs does not "contain" IN

True, at least if |ℕ| = ℵo.

Regards, WM

William schrieb am Donnerstag, 31. März 2022 um 19:11:51 UTC+2:
> On Thursday, March 31, 2022 at 1:26:21 PM UTC-3, WM wrote:
>
>
>
> > try to find a wrong step here:
> > Every FISON is finite.
> > Every FISON contains all its predecessors.
> > The sequence of all FISONs contains all FISONs including all successors of all FISONs.
> > The sequence of all FISONs contains the union of all FISONs.
> all correct. However we also have
>
> Every FISON has a successor of which it is a proper subset.

All successors are also contained in the sequence.
>
> thus
>
> No single element of the sequence of all FISONs contains the union of all FISONs.

Note that only single elements are in the union. What is not in the sequence cannot be in the union. The contrary belief is mistaken matheology.

Regards, WM

On Thursday, March 31, 2022 at 7:18:26 PM UTC+2, Fritz Feldhase wrote:
> On Thursday, March 31, 2022 at 7:11:51 PM UTC+2, William wrote:
> > >
> > > The sequence of all FISONs contains the union of all FISONs.
> > >
> > all correct.
> >
> Please elaborate.
>
> The union of all FISONs is IN. In which sense does the sequence of all FISONs "contain" IN?

Does the set {{1}, {2}, {3}} "contain" the set {1, 2, 3}?

Man, we do have a proper terminology in set theory!

Fritz Feldhase schrieb am Donnerstag, 31. März 2022 um 19:18:26 UTC+2:
> On Thursday, March 31, 2022 at 7:11:51 PM UTC+2, William wrote:
> > >
> > > The sequence of all FISONs contains the union of all FISONs.
> > >
> > all correct.
> Please elaborate.
>
> The union of all FISONs is IN.

This assumption has turned out wrong.
The union of all terms cannot be more than all terms. This however is not an ℵo-set.

> Hint: IN is not a term nor an element or a subset in this sequence.

Hint: The union does not blow up the terms. You believe that that by unioning the terms of the sequence they are blown up to produce ℕ. That is nonsense. What you think to be the blow up in fact is the inclusion of dark numbers.

Regards, WM

Fritz Feldhase schrieb am Donnerstag, 31. März 2022 um 19:25:17 UTC+2:
> On Thursday, March 31, 2022 at 7:18:26 PM UTC+2, Fritz Feldhase wrote:
> > On Thursday, March 31, 2022 at 7:11:51 PM UTC+2, William wrote:
> > > >
> > > > The sequence of all FISONs contains the union of all FISONs.
> > > >
> > > all correct.
> > >
> > Please elaborate.
> >
> > The union of all FISONs is IN. In which sense does the sequence of all FISONs "contain" IN?
> Does the set {{1}, {2}, {3}} "contain" the set {1, 2, 3}?

The set {1} U {1, 2} U {1, 2, 3} contains the set {1, 2, 3}. That is the point here.

Regards, WM

On Thursday, March 31, 2022 at 7:21:42 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Donnerstag, 31. März 2022 um 19:02:38 UTC+2:
> > >
> > > The sequence of all FISONs contains the union of all FISONs.
> > >
> > "contains"? In what sense?
> >
> > As "term"?
> >
> Of course.

So why don't you just SAY so, idiot?

> > false.
> >
> Why should it be wrong?

Because the union of all FISONs does not have a largest element, while on the other hand each and every FISON has a largest element (which just is the number of elements in it).

Hence

> > the sequence of all FISONs does not "contain" the union of all FISONs.
> >
> True, at least if |ℕ| = ℵo.

Well, actually |ℕ| = ℵo does hold in set theory.

On 3/31/2022 12:25 PM, Fritz Feldhase wrote:
> On Thursday, March 31, 2022 at 7:18:26 PM UTC+2, Fritz Feldhase wrote:
>> On Thursday, March 31, 2022 at 7:11:51 PM UTC+2, William wrote:
>>>>
>>>> The sequence of all FISONs contains the union of all FISONs.
>>>>
>>> all correct.
>>>
>> Please elaborate.
>>
>> The union of all FISONs is IN. In which sense does the sequence of all FISONs "contain" IN?
>
> Does the set {{1}, {2}, {3}} "contain" the set {1, 2, 3}?
>
> Man, we do have a proper terminology in set theory!

Extendable (Ants);

{{contents Ant},{contain Ants},{containerized Ants},{containable Ants}}

{none of those Ants can talk to each other}

On Thursday, 31 March 2022 at 14:11:51 UTC-3, William wrote:
> On Thursday, March 31, 2022 at 1:26:21 PM UTC-3, WM wrote:
>
>
>
> > try to find a wrong step here:
> > Every FISON is finite.
> > Every FISON contains all its predecessors.
> > The sequence of all FISONs contains all FISONs including all successors of all FISONs.
> > The sequence of all FISONs contains the union of all FISONs.
> all correct.

How can you even parse his gobbledygook? In what sense does "[t]he sequence of all FISONs [contain] the union of all FISONs"? The union of all FISONs is IN, which distinctly is not a term of the sequence. On the other hand, of course the union over all FISONs *IS* IN, so of course the union (or the set limit of the sequence) will contain IN as a(n improper) subset. Mucki thrives on such ambiguities, which is the main reason why he wallows in his convoluted natural language bullshit.

On 3/31/2022 12:29 PM, WM wrote:
> Fritz Feldhase schrieb am Donnerstag, 31. März 2022 um 19:18:26 UTC+2:
>> On Thursday, March 31, 2022 at 7:11:51 PM UTC+2, William wrote:
>>>>
>>>> The sequence of all FISONs contains the union of all FISONs.
>>>>
>>> all correct.
>> Please elaborate.
>>
>> The union of all FISONs is IN.
>
> This assumption has turned out wrong.
> The union of all terms cannot be more than all terms. This however is not an ℵo-set.
>
>> Hint: IN is not a term nor an element or a subset in this sequence.
>
> Hint: The union does not blow up the terms. You believe that that by unioning the terms of the sequence they are blown up to produce ℕ. That is nonsense. What you think to be the blow up in fact is the inclusion of dark numbers.
>
> Regards, WM

na, your imaginary little black smuggies are not required, it is nonsense.

no smuggies here;

{1,2,3...} = ℕ