union of a subsequence with last element is not in the sequence.

On Friday, April 1, 2022 at 6:34:00 AM UTC-3, WM wrote:
> William schrieb am Donnerstag, 31. März 2022 um 20:40:18 UTC+2:
> > On Thursday, March 31, 2022 at 2:24:11 PM UTC-3, WM wrote:
>
> No single element *that you can define* contains the union.

Every element of the sequence can be defined.

"potential infinity of the sequence."

Absolute nonsense.

>
> But fact is that every union is contained in the sequence.

Nope. Any union of a subsequence with no last element is not in the sequence.

--
William Hughes

>
> --
> William Hughes

WM presented the following explanation :
> sergio schrieb am Donnerstag, 31. März 2022 um 20:43:39 UTC+2:
>
>> Which element is not in the union of all FISONs ?
>
> Which element is not in the sequence of all FISONs?

The last one, because there is no last one.

William schrieb:

>> ... every union is contained in the sequence.
>
> Nope. Any union of a subsequence with no last element is not in the sequence.

The imbecile clown WM wants:
....
{1} U {1,2} U {1,2,3} U {1,2,3,4} U {1,2,3,4,5} = {1,2,3,4,5}
{1} U {1,2} U {1,2,3} U {1,2,3,4} U {1,2,3,4,5} U {IN} = {IN}

on 4/1/2022, WM supposed :
> FromTheRafters schrieb am Donnerstag, 31. März 2022 um 22:26:42 UTC+2:
>> Each newly revealed
>> digit gets you closer, not further, from the number's actual value as a
>> decimal representation.
>
> Each revealed digit is in infinite distance from the goal which is forever
> dark.

Sixty cycles per second is a good frequency for household electrical
outlets.

Your turn again.

On Friday, 1 April 2022 at 08:27:48 UTC-3, WM wrote:
> horand....@gmail.com schrieb am Freitag, 1. April 2022 um 12:37:06 UTC+2:
> > On Thursday, 31 March 2022 at 18:02:26 UTC-3, WM wrote:
> > > William schrieb am Donnerstag, 31. März 2022 um 20:50:12 UTC+2:
> > > > On Thursday, March 31, 2022 at 3:01:50 PM UTC-3, WM wrote:
> > >
> > > > > To uphold this wrong assumption you must believe that the union over all FISONs contains more than all FISONs.
> > > > Nope you must believe that the union over all FISONs contains more than *any single FISON*.
> > > But it does not. In fact you must fix a FISON and then you can prove that the union is larger, Small wonder. But I do not fix a FISON but consider all of them. Each one is the union of all its predecessors. Therefore the union over all FISONs which can be unioned at all has already been done and is already in the sequence.
> > > {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}
> > > ...
> > > > This is equivalent to believing that there is no largest FISON.
> > > There is no largest FISON, but there is no ℵo-FISON. Hence the union over FISONs is not an ℵo-set.
> > Doesn't follow. There is no largest integer, but their number is still ℵo. No contradiction, your quantifier shifting below notwithstanding.
> > > You trick is too simple: Choose a FISON and show that it is not the union of all FISONs. I do not choose a FISON but use all available FISONs. But they are all finite. And the are all unions of all predecessors. Nothing more is available for the union. A union of ℵo different FISONs would require FISONs with more than any finite number of symbols:
> > >
> > > o
> > > oo
> > > ooo
> > > ...
> > >
> > > But so many are not available.
> > Why not?
> Pigeonhole principle. Denied by matheologians but nevertheless true for all finite sets. And FISONs are finite sets, no matter how many of them are concerned.

But the *SET* of all FISONs is not finite, you stupid moron. Here you are committing a category error.

On Friday, 1 April 2022 at 13:27:45 UTC+3, WM wrote:
> Eram semper recta schrieb am Freitag, 1. April 2022 um 06:57:16 UTC+2:
> > On Thursday, 31 March 2022 at 20:15:03 UTC+3, WM wrote:
>
> > > > > The formula is p/q. The number has got the name p/q.
> > > > Rubbish. formula =/= number.
> > > But both can and do carry the same name.
> > If you mean they are synonyms, no - the dictionary does not state as much.
> >
> > Formula: a mathematical relationship or rule expressed in symbols.
> Here by p/q or 1/3.

No. Both p/q and 1/3 are measures.

> >
> > Number: an arithmetical value, expressed by a word, symbol, or figure,
> Here by p/q or 1/3.
>
> Regards, WM

On Friday, 1 April 2022 at 13:36:19 UTC+3, WM wrote:
> Eram semper recta schrieb am Freitag, 1. April 2022 um 07:10:06 UTC+2:
>
> > Your arguments about Cantor are 100% sound.
> It is relieving to know that mathematicians can understand this.
> > The number is NOT the formula.
> Correct. But both can have the same notation. Like Millers and millers.

Not in this case. Any formula you give for pi never gives the number pi, only a recognised constant.

>
> Regards, WM

FromTheRafters schrieb am Freitag, 1. April 2022 um 13:42:42 UTC+2:
> WM presented the following explanation :
> > sergio schrieb am Donnerstag, 31. März 2022 um 20:43:39 UTC+2:
> >
> >> Which element is not in the union of all FISONs ?
> >
> > Which element is not in the sequence of all FISONs?
> The last one, because there is no last one.

Alas, the same is missing in the union of all FISONs.

The claim that the union could contain more than the sequence, and that the sequence could contain more than is already unioned in a term of the sequence is wrong. It shows a fanatic of matheology with blocked brain.

Regards, WM

William schrieb am Freitag, 1. April 2022 um 13:32:35 UTC+2:
> union of a subsequence with last element is not in the sequence.
> On Friday, April 1, 2022 at 6:34:00 AM UTC-3, WM wrote:
> > William schrieb am Donnerstag, 31. März 2022 um 20:40:18 UTC+2:
> > > On Thursday, March 31, 2022 at 2:24:11 PM UTC-3, WM wrote:
> >
> > No single element *that you can define* contains the union.
> Every element of the sequence can be defined.
> "potential infinity of the sequence."
> Absolute nonsense.
> >
> > But fact is that every union is contained in the sequence.
> Nope. Any union of a subsequence with no last element is not in the sequence.

Please name natnumbers or FISONs which *are* in the union and not in the sequence.
In fact your claim is wrong. Every term of the sequence has a last element, but there is no last element of all terms of the sequence. Nevertheless all terms are finite.

The claim that the union could contain more than the sequence, and that the sequence could contain more than is already unioned in a term of the sequence is wrong.

Regards, WM

Eram semper recta schrieb am Freitag, 1. April 2022 um 15:29:17 UTC+2:
> On Friday, 1 April 2022 at 13:36:19 UTC+3, WM wrote:
> > Eram semper recta schrieb am Freitag, 1. April 2022 um 07:10:06 UTC+2:
> >
> > > Your arguments about Cantor are 100% sound.
> > It is relieving to know that mathematicians can understand this.
> > > The number is NOT the formula.
> > Correct. But both can have the same notation. Like Millers and millers.
> Not in this case. Any formula you give for pi never gives the number pi, only a recognised constant.

We had agreed long ago that it is a matter of taste whether irrationals are called numbers. At least I had agreed.

Regards, WM

horand....@gmail.com schrieb am Freitag, 1. April 2022 um 14:16:07 UTC+2:
> On Friday, 1 April 2022 at 08:27:48 UTC-3, WM wrote:

> > > > o
> > > > oo
> > > > ooo
> > > > ...
> > > >
> > > > But so many are not available.
> > > Why not?
> > Pigeonhole principle. Denied by matheologians but nevertheless true for all finite sets. And FISONs are finite sets, no matter how many of them are concerned.
> But the *SET* of all FISONs is not finite,

As I showed above, every contrary opinion had to accept infinite FISONs. Do you?
(There is no infinite set but only a potentially infinite collection of FISONs.)

Regards, WM

On Friday, 1 April 2022 at 10:51:12 UTC-3, WM wrote:
> horand....@gmail.com schrieb am Freitag, 1. April 2022 um 14:16:07 UTC+2:
> > On Friday, 1 April 2022 at 08:27:48 UTC-3, WM wrote:
>
> > > > > o
> > > > > oo
> > > > > ooo
> > > > > ...
> > > > >
> > > > > But so many are not available.
> > > > Why not?
> > > Pigeonhole principle. Denied by matheologians but nevertheless true for all finite sets. And FISONs are finite sets, no matter how many of them are concerned.
> > But the *SET* of all FISONs is not finite,
> As I showed above, every contrary opinion had to accept infinite FISONs.

Since you never in your life showed anything, let alone "above", this is non-responsive. You still do not know what a category error is. Hence you cannot avoid making them. And, yes, you committed one, and your shameful attempt at snipping and falsifying my contribution to cover up your shortcomings is duly noted.

Alan Mackenzie schrieb am Freitag, 1. April 2022 um 12:05:51 UTC+2:
> WM <askas...@gmail.com> wrote:
> > Alan Mackenzie schrieb am Donnerstag, 31. März 2022 um 22:26:24 UTC+2:
> >> You will surely agree
> >> that 0.023023023023... converges to a limit.
>
> > Not if it is a sequence without defining formula like repeat 023
> > without changing the digits. Otherwise the next digit could be 7.
> > 0.023023023023... is usually taken as such a formula.
>
> >> So, why shouldn't 0.570039362554378... also converge to a limit?
> > If it does, ....
>
> It does.
>
> > .... the limit is unknown (dark) and unknowable: remaining dark
> > forever.
>
> That is indeed the case.
>
> Hey, we agree on something! :-)

Would be nice if we agreed on this too:

The union of FISONs cannot contain more than the sequence of FISONs, and the sequence does not contain more than is already unioned in a term of the sequence. Therefore, if |ℕ| = ℵo then there must be dark natural numbers.

Regards, WM

On 4/1/2022 4:44 AM, WM wrote:
> Alan Mackenzie schrieb am Donnerstag, 31. März 2022 um 22:26:24 UTC+2:
>> You will surely agree
>> that 0.023023023023... converges to a limit.
>
> Not if it is a sequence without defining formula like repeat 023 without changing the digits. Otherwise the next digit could be 7. 0.023023023023... is usually taken as such a formula.

If written as 0.023023023(023)... converges to 23/999

>
>> So, why shouldn't
>> 0.570039362554378... also converge to a limit?

what do you mean by "converge" ?

>
> If it does, the limit is unknown (dark) and unknowable: remaining dark forever.

WM, the Miss Director, again is totally wrong. if it does converge, it is known. Fail.

>
> Regards, WM

On 4/1/2022 4:50 AM, WM wrote:
> horand....@gmail.com schrieb am Donnerstag, 31. März 2022 um 22:28:13 UTC+2:
>>
>> The standard way to use a random process is to generate digits, for instance from tossing a coin. With such a process you know that each iterate is contained in a small interval that gets cut in half at each iteration. By completeness of the real numbers, a limit exists.
>
> Forever unknown: dark.
>
> Regards, WM

You don't know it is dark, because you cannot check it.

On 4/1/2022 5:27 AM, WM wrote:
> Eram semper recta schrieb am Freitag, 1. April 2022 um 06:57:16 UTC+2:
>> On Thursday, 31 March 2022 at 20:15:03 UTC+3, WM wrote:
>
>>>>> The formula is p/q. The number has got the name p/q.
>>>> Rubbish. formula =/= number.
>>> But both can and do carry the same name.
>> If you mean they are synonyms, no - the dictionary does not state as much.
>>
>> Formula: a mathematical relationship or rule expressed in symbols.
>
> Here by p/q or 1/3.

No. those are called "fractions", they are not "formulas" nor "equations".

>
> Regards, WM

On Friday, April 1, 2022 at 10:51:12 AM UTC-3, WM wrote:

> (There is no infinite set but only a potentially infinite collection of FISONs.)

Piffle

--
William Hughes

On Friday, April 1, 2022 at 10:44:45 AM UTC-3, WM wrote:
> William schrieb am Freitag, 1. April 2022 um 13:32:35 UTC+2:

> > > But fact is that every union is contained in the sequence.
> > Nope. Any union of a subsequence with no last element is not in the sequence.
> Please name natnumbers or FISONs which *are* in the union and not in the sequence.

Given a subset of natural numbers, A, A is contained in a single FISON iff A has a largest element. *Each* element of A is contained within a single FISON, so A does not contain "more" than the sequence but there is no single FISON which contains *all* the elements of A. (It follows that A is contained within the union of a set of FISONs without largest FISON. Indeed, the union of a set of FISONs without largest FISON is N).
--
William Hughes

William schrieb am Freitag, 1. April 2022 um 19:46:29 UTC+2:
> On Friday, April 1, 2022 at 10:51:12 AM UTC-3, WM wrote:
>
> > (There is no infinite set but only a potentially infinite collection of FISONs.)
> Piffle

(1) The union of FISONs cannot contain more than the sequence of FISONs.
(2) The sequence does not contain more than is already unioned in at least one term of the sequence.
(3) Every term is finite.

What can't you read or understand?

If you can understand, then you know: if |ℕ| = ℵo then there must be dark natural numbers.

Regards, WM

William schrieb am Freitag, 1. April 2022 um 20:05:42 UTC+2:
> On Friday, April 1, 2022 at 10:44:45 AM UTC-3, WM wrote:
> > William schrieb am Freitag, 1. April 2022 um 13:32:35 UTC+2:
>
> > > > But fact is that every union is contained in the sequence.
> > > Nope. Any union of a subsequence with no last element is not in the sequence.
> > Please name natnumbers or FISONs which *are* in the union and not in the sequence.
> Given a subset of natural numbers, A, A is contained in a single FISON iff A has a largest element. *Each* element of A is contained within a single FISON, so A does not contain "more" than the sequence but there is no single FISON which contains *all* the elements of A.

That claim is mistaken because it is impossible to satisfy. FISONs obey inclusion monotony. Everything contained in some FISONs is contained in one FISON by the construction of FISONs.

That's why I have been using them for decades. F(1) ⊂ F(2) ⊂ F(3) ⊂ ... There is no last FISON, but there is no infinite FISON. It is impossible that two or more elements of A are contained in FISONs but are not contained in one FISON.

Regards, WM

On Friday, April 1, 2022 at 3:07:27 PM UTC-3, WM wrote:
> William schrieb am Freitag, 1. April 2022 um 19:46:29 UTC+2:
> > On Friday, April 1, 2022 at 10:51:12 AM UTC-3, WM wrote:
> >
> > > (There is no infinite set but only a potentially infinite collection of FISONs.)
> > Piffle
> (1) The union of FISONs cannot contain more than the sequence of FISONs.
> (2) The sequence does not contain more than is already unioned in at least one term of the sequence.

Piffle. There is no reason to think that the union of all elements of the sequence are contained in one element of the sequence.
Indeed a the union of a sequence of FISONs is contained in one element of the sequence, iff the sequence has a last element.

--
William Hughes

William schrieb am Freitag, 1. April 2022 um 20:33:55 UTC+2:
> On Friday, April 1, 2022 at 3:07:27 PM UTC-3, WM wrote:
> > William schrieb am Freitag, 1. April 2022 um 19:46:29 UTC+2:
> > > On Friday, April 1, 2022 at 10:51:12 AM UTC-3, WM wrote:
> > >
> > > > (There is no infinite set but only a potentially infinite collection of FISONs.)
> > > Piffle
> > (1) The union of FISONs cannot contain more than the sequence of FISONs.
> > (2) The sequence does not contain more than is already unioned in at least one term of the sequence.
> Piffle. There is no reason to think that the union of all elements of the sequence are contained in one element of the sequence.

It is so by construction.

> Indeed a the union of a sequence of FISONs is contained in one element of the sequence, iff the sequence has a last element.

It simply holds for every existing FISON: All predecessors are in this one FISON. That cannot change. Only for not existing FISONs it may be wrong. But I am not interested in them.

Regards, WM

On Friday, April 1, 2022 at 3:16:35 PM UTC-3, WM wrote:

> That claim is mistaken because it is impossible to satisfy. FISONs obey inclusion monotony. Everything contained in some FISONs is contained in one FISON by the construction of FISONs.

Nope This only holds for sequences with a last element. We have: The union of a sequence of FISONs is contained in a single element of the sequence iff the sequence has a last element.

--
William Hughes

On Friday, April 1, 2022 at 3:42:27 PM UTC-3, WM wrote:
> existing FISON:

More nonsense. The collection of FISONs is a set. The set of FISONs does not change. The set of FISONs does not contain a largest FISON. The union of the set of FISONs is not contained in a single FISON.

--
William Hughes

WM explained :
> FromTheRafters schrieb am Freitag, 1. April 2022 um 13:42:42 UTC+2:
>> WM presented the following explanation :
>>> sergio schrieb am Donnerstag, 31. März 2022 um 20:43:39 UTC+2:
>>>
>>>> Which element is not in the union of all FISONs ?
>>>
>>> Which element is not in the sequence of all FISONs?
>> The last one, because there is no last one.
>
> Alas, the same is missing in the union of all FISONs.

Have you never worked with inductive sets?

> The claim that the

[...]

Blah blah blah, and no, it isn't.