Fritz Feldhase schrieb am Donnerstag, 31. März 2022 um 19:33:23 UTC+2:
> 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.

> > 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).

That is not a valid argument. The sequence of all FISONs does not contain a largest element either.

The union of all FISONs cannot contain more than the sequence because of the fact that the equence consists of all unions already
{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}
....
All that is in the union is also in the sequence as a term because the sequence consists of unions. If there are all terms in the sequence, then the union of all terms is in the sequence.

Regards, WM

horand....@gmail.com schrieb am Donnerstag, 31. März 2022 um 19:34:39 UTC+2:
> of course the union over all FISONs *IS* IN,

That has been disproved. To uphold this wrong assumption you must believe that the union over all FISONs contains more than all FISONs. That is in contradiction with set theory.

Regards, WM

On Thursday, March 31, 2022 at 2:24:11 PM UTC-3, WM wrote:
> 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.

True but does not contradict my statement. Every FISON has a successor (the successor is contained in the sequence) of which it is a proper subset.

> >
> > 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

True but does not contradict my statement. Every "single element" (every FISON) is *contained* in the union. No "single element" (No FISON) *contains* the union.

--
William Hughes

On 3/31/2022 1:01 PM, WM wrote:
> horand....@gmail.com schrieb am Donnerstag, 31. März 2022 um 19:34:39 UTC+2:
>> of course the union over all FISONs *IS* IN,
>
> That has been disproved. To uphold this wrong assumption you must believe that the union over all FISONs contains more than all FISONs. That is in contradiction with set theory.
>
> Regards, WM

Wrong. Which element is not in the union of all FISONs ?

On 3/31/2022 11:28 AM, WM wrote:
> 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

diversion. n < m .

On Thursday, March 31, 2022 at 3:01:50 PM UTC-3, WM wrote:
> horand....@gmail.com schrieb am Donnerstag, 31. März 2022 um 19:34:39 UTC+2:
> > of course the union over all FISONs *IS* IN,
> That has been disproved.

Nope. It is trivial.

> 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*. This is equivalent to believing that there is no largest FISON.

--
William Hughes

On 3/31/2022 11:40 AM, 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. 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".

True, x 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.

diversion.

Your requirement of only working with numbers you have written down on paper, restricts you to sheep counter, or rock counter.

>
>> 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*.

that is horrible! use the correct terminology!

>
> Regards, WM

On Thursday, 31 March 2022 at 15:01:50 UTC-3, WM wrote:
> horand....@gmail.com schrieb am Donnerstag, 31. März 2022 um 19:34:39 UTC+2:
> > of course the union over all FISONs *IS* IN,
> That has been disproved. To uphold this wrong assumption you must believe that the union over all FISONs contains more than all FISONs.

Bullshit. Quantifier dyslexia.

After serious thinking WM wrote :
> 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.

Some formulae contain an equals sign too. Do you see no difference
between A=BC and D=RT?

WM formulated on Thursday :
> 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.

Of course, if the 'F' in FISON has meaning.

>> 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.

Induction!

On 3/31/2022 12:14 PM, WM wrote:
> 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?

10 is too restrictive, use x instead...

"...for p/q you cannot find out whether it is larger than x."

see? much better.

>
>>> 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.

some people are like a pair of brown shoes.

>
>>> 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?

"formula" is spelt differently from "number" for several reasons, or do you deny that ?

>
> Regards, WM

torsdag 31. mars 2022 kl. 18:44:08 UTC+2 skrev Alan Mackenzie:
> WM <askas...@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).

Are you sure about this? Why should a random prosess lead to a limit? Is it because no random prosess is not actual random? Then the promise is broken.. A true random prosess does not lead to a limit, but it goes on forever :) And the number is a real number.
KON

konyberg <konyberg@online.no> wrote:
> torsdag 31. mars 2022 kl. 18:44:08 UTC+2 skrev Alan Mackenzie:
>> WM <askas...@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).

> Are you sure about this?

Yes.

> Why should a random prosess lead to a limit?

Why shouldn't it? It's not the randomness which is important here,
rather the addition of ever smaller quantities (the decimal digits).
Wasn't that one of Cauchy's theorems (I forget)? You will surely agree
that 0.023023023023... converges to a limit. So, why shouldn't
0.570039362554378... also converge to a limit?

> Is it because no random process is not actual random?

There are too many negatives in that sentence. ;-). I think you mean
"Is it because no random process is actually random?". The random digit
generator I posited is an ideal machine with a uniform distribution over
the digits 0, ..., 9.

> Then the promise is broken. A true random process does not lead to a
> limit, but it goes on forever :)

The process of generating random digits doesn't lead to a limit. But
arranging them as a decimal expansion (as I did above) does lead to a
limit. That limit is 0.570039362554378....

> And the number is a real number.

Indeed!

> KON

--
Alan Mackenzie (Nuremberg, Germany).

It happens that konyberg formulated :
> torsdag 31. mars 2022 kl. 18:44:08 UTC+2 skrev Alan Mackenzie:
>> WM <askas...@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).
>
> Are you sure about this? Why should a random prosess lead to a limit? Is it
> because no random prosess is not actual random? Then the promise is broken. A
> true random prosess does not lead to a limit, but it goes on forever :) And
> the number is a real number. KON

In this case the sequence is the fractional part of a decimal
representation. Think of it as a fully formed number and think of the
generator as merely revealing each next digit. Each newly revealed
digit gets you closer, not further, from the number's actual value as a
decimal representation.

On Thursday, 31 March 2022 at 16:57:54 UTC-3, konyberg wrote:
> torsdag 31. mars 2022 kl. 18:44:08 UTC+2 skrev Alan Mackenzie:
> > WM <askas...@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).
>
> Are you sure about this? Why should a random prosess lead to a limit? Is it because no random prosess is not actual random? Then the promise is broken. A true random prosess does not lead to a limit, but it goes on forever :) And the number is a real number.

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.

torsdag 31. mars 2022 kl. 22:26:24 UTC+2 skrev Alan Mackenzie:
> konyberg <kony...@online.no> wrote:
> > torsdag 31. mars 2022 kl. 18:44:08 UTC+2 skrev Alan Mackenzie:
> >> WM <askas...@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).
>
> > Are you sure about this?
> Yes.
> > Why should a random prosess lead to a limit?
> Why shouldn't it? It's not the randomness which is important here,
> rather the addition of ever smaller quantities (the decimal digits).
> Wasn't that one of Cauchy's theorems (I forget)? You will surely agree
> that 0.023023023023... converges to a limit. So, why shouldn't
> 0.570039362554378... also converge to a limit?
>
> > Is it because no random process is not actual random?
>
> There are too many negatives in that sentence. ;-). I think you mean
> "Is it because no random process is actually random?". The random digit
> generator I posited is an ideal machine with a uniform distribution over
> the digits 0, ..., 9.
>
> > Then the promise is broken. A true random process does not lead to a
> > limit, but it goes on forever :)
> The process of generating random digits doesn't lead to a limit. But
> arranging them as a decimal expansion (as I did above) does lead to a
> limit. That limit is 0.570039362554378....
> > And the number is a real number.
> Indeed!
>
> > KON
> --
> Alan Mackenzie (Nuremberg, Germany).

But you can increase the base (radix) to infinite. Then what happens?
KON

torsdag 31. mars 2022 kl. 22:28:13 UTC+2 skrev horand....@gmail.com:
> On Thursday, 31 March 2022 at 16:57:54 UTC-3, konyberg wrote:
> > torsdag 31. mars 2022 kl. 18:44:08 UTC+2 skrev Alan Mackenzie:
> > > WM <askas...@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).
> >
> > Are you sure about this? Why should a random prosess lead to a limit? Is it because no random prosess is not actual random? Then the promise is broken. A true random prosess does not lead to a limit, but it goes on forever :) And the number is a real number.
> 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.

I can use two numbers generated at random, 0 and 1. A random number is 0.101001000100001...
Adding a 0 between the 1. This doesn't look random, but it could be. What is the limit?
KON

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.

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.

Regards, WM

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:
> > 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.
> True but does not contradict my statement. Every FISON has a successor (the successor is contained in the sequence) of which it is a proper subset.
> > >
> > > 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
> True but does not contradict my statement. Every "single element" (every FISON) is *contained* in the union. No "single element" (No FISON) *contains* the union.

Wrong. Contradicted by the fact that only such terms of the sequence that are unioned in one term of the sequence can be unioned (again).

{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}
....

Regards, WM

On Thursday, March 31, 2022 at 6:08:29 PM UTC-3, WM wrote:

>... the fact that only such terms of the sequence that are unioned in one term of the sequence can be unioned

Absolute nonsense. All terms of the sequence can by "unioned" and the resulting union is not one term of the sequence.

--
William Hughes

konyberg wrote :
> torsdag 31. mars 2022 kl. 22:28:13 UTC+2 skrev horand....@gmail.com:
>> On Thursday, 31 March 2022 at 16:57:54 UTC-3, konyberg wrote:
>>> torsdag 31. mars 2022 kl. 18:44:08 UTC+2 skrev Alan Mackenzie:
>>>> WM <askas...@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).
>>>
>>> Are you sure about this? Why should a random prosess lead to a limit? Is it
>>> because no random prosess is not actual random? Then the promise is broken.
>>> A true random prosess does not lead to a limit, but it goes on forever :)
>>> And the number is a real number.
>> 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.
>
> I can use two numbers generated at random, 0 and 1. A random number is
> 0.101001000100001... Adding a 0 between the 1. This doesn't look random, but
> it could be. What is the limit? KON

That's not random, and as such can be otherwise described by a formula.
Chaotic is deterministic, random is not. Makes no difference to this
topic though. I suggested a similar one earlier where the number of
zeroes between ones followed the fractional digits of pi --
deterministic too though, it only looks random.

torsdag 31 mars 2022 kl. 13:12:44 UTC+2 skrev WM:
> 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 random generator does not describe a limit.
> > It's an infinite sequence. There is no last digit.
> And no limit can be detemined. Hence it is not 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.
> > >> 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.
> > > 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.
>
> Regards, WM
Your "definable" is meaningless

There are no "activity", no "action" in mathematics

torsdag 31 mars 2022 kl. 18:26:21 UTC+2 skrev 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.

Correct

> Every FISON contains all its predecessors.

Correct

> The sequence of all FISONs contains all FISONs including all successors of all FISONs.
Correct

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

False, the union of all FISONs is N but the sequence does not contain N

>
> Regards, WM

On Thursday, 31 March 2022 at 20:15:03 UTC+3, WM wrote:
> 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.

If you mean they are synonyms, no - the dictionary does not state as much.

Formula: a mathematical relationship or rule expressed in symbols.

Number: an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification.

> Have you the power to forbid it?

Irrelevant. If two are in a discussion and they use words meaning different things, then there is no point in the discussion.

There are many synonyms for formula but number is not one of them:

https://www.thesaurus.com/browse/formula

>
> Regards, WM

On Friday, 1 April 2022 at 07:57:16 UTC+3, Eram semper recta wrote:
> On Thursday, 31 March 2022 at 20:15:03 UTC+3, WM wrote:
> > 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.
> If you mean they are synonyms, no - the dictionary does not state as much..
>
> Formula: a mathematical relationship or rule expressed in symbols.
>
> Number: an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification.
> > Have you the power to forbid it?
> Irrelevant. If two are in a discussion and they use words meaning different things, then there is no point in the discussion.
>
> There are many synonyms for formula but number is not one of them:
>
> https://www.thesaurus.com/browse/formula
>
> >
> > Regards, WM

Back to the topic:

1/2 in base 10: ***1/2 x 10/10 = 10/2 x 1/10 = 5 x 1/10 = 0.5 ***

1/4 in base 10: ***1/4 x 10/10 = 10/4 x 1/10 = (2 + 1/2) x 1/10 = (2 x 1/10)+ (10/2 x 1/100)= (2 x 1/10)+(5 x 1/100) = 0.25 ***

It is glaringly obvious that the numbers 1/2 and 1/4 are not at all like the formula which written in words states:

Formula/Theorem/Method/Process/Recipe/Rubric ... add whatever synonym you choose...

i. Multiply by base/base
ii. Switch numerators by association
iii. Filter out first significant digit of measure
iv. Go back to (i).

Can you see how different are the starred processes from the formula?

Your arguments about Cantor are 100% sound. Your knowledge about number is very iffy.
For all the fuss that you make about "infinite decimal expansions" being garbage (which they are!), your perspective of number is still very much Cantorish in that it is as primitive as counting by bijective cardinality, that is, one pebble for each sheep.

The number is NOT the formula. Lerne das gut!