Jim Burns schrieb am Dienstag, 5. April 2022 um 22:58:24 UTC+2:
> On 4/5/2022 2:06 PM, WM wrote:
>
> > It is only possible to express numbers finitely.
> It is possible to make a claim which is true of
> each one of infinitely-many.

For instance this one: Every FISON that exists has been unioned together with all others in the sequence 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}
....

That holds for all the infinitely many FISONs of the sequence as the ... indicate. Another union cannot increase the result.

> > Infinite sequences can neither be written nor could they,
> > if existing, be read.
> If
> there is always some finite number k for which
> k replications of distance y passes distance x, k*y > x
> then

this is given by a finite formula: "there is always some finite number k for which
> k replications of distance y passes distance x, k*y > x" with k and y to be chosen.

Regards, WM

On 4/6/2022 7:56 AM, FromTheRafters wrote:
> Eram semper recta brought next idea :
>> On Wednesday, 6 April 2022 at 08:38:23 UTC+3, Eram semper recta wrote:
>>> On Wednesday, 6 April 2022 at 08:30:25 UTC+3, Eram semper recta wrote:
>>>> On Tuesday, 5 April 2022 at 21:06:32 UTC+3, WM wrote:
>>>>> Eram semper recta schrieb am Dienstag, 5. April 2022 um 08:52:52 UTC+2:
>>>>>> 1. The expression "finite number" is both illogical and anti-mathematical nonsense. The adjective "finite" cannot appear before "number". A number
>>>>>> is represented by a fixed (not "finite" because then this might imply something else (pretty absurd) is possible) number of symbols.
>>>>> "Finite number" need not imply infinite number but may emphasize that infinity is excluded.  This is not what happens to a delusional mind. They
>>>>> use "finite number" so as to leave the door open to their drivel "infinite number".
>>>>>>
>>>>>> 2. The above drivel is equivalent to claiming that 1/3 cannot be represented in base 10 "finitely" which is a result of the same misguided
>>>>>> thinking. 1/3 either can or can't be represented in decimal. A very important number theorem tells us clearly that 1/3 has no representation in
>>>>>> decimal. To say:
>>>>>> "You can't represent 1/3 finitely" is drivel. I just did ... look: "1 / 3" .
>>>>>
>>>>> Of course. It is only possible to express numbers finitely. Infinite sequences can neither be written nor could they, if existing, be read.
>>> Also I do not like the expression "express numbers finitely" because finitely is a redundant adjective.
>>> I prefer to use "express numbers with a fixed number of symbols".
>>> After all, the symbols we use to express numbers are NAMES and it is absurd to think that a NAME can be understood in any other way, for example as a
>>> never-ending string of digits (incomplete decimal expansion).
>>> PhD students have been using "infinity" as a sugar tit far too long in order to obtain a degree which means absolutely nothing in real terms.  Since
>>> they have zero aptitude for mathematics, "infinity" provides a conduit for them to bullshit their way into the circle jerk of PhD grads.
>>>>
>>>>
>>>>>
>>>>> Regards, WM
>>
>> Imagine the following scenario:
>>
>> I want to express the number that has exactly the same digits as pi but varies in the trillionth digit only.
>>
>> How do I express that number?
>
> You just did, well, almost. That would actually be more than one number because we have a selection of more than one possible base ten digit to occupy
> that 'changed' digit. You need to narrow it down to get it to be only one number.

use binary instead.

0110101.10010010111000101000001010... and just flip the trillionth digit.

zelos...@gmail.com schrieb am Mittwoch, 6. April 2022 um 06:20:40 UTC+2:
> tisdag 5 april 2022 kl. 19:59:44 UTC+2 skrev WM:
> > zelos...@gmail.com schrieb am Dienstag, 5. April 2022 um 06:33:56 UTC+2:
> > > måndag 4 april 2022 kl. 12:48:38 UTC+2 skrev WM:
> > > > zelos...@gmail.com schrieb am Montag, 4. April 2022 um 09:48:17 UTC+2:
> >
> > > > > You assume only finite number of FISONs here, which is a false assumption
> > > > No, less than ℵo FISONs is dictated by mathematics.
> > >
> > > And the union of all is N
> > Fact is that
> the union of ALL FISONs, which is INFINITELY MANY, is N

If ℕ is potentially infinite. If however |ℕ| = ℵo, then all FISONs are not enough because never a union of all FISONs is larger than all unions of FISONs contained in the sequence of FISONs:
> > {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}
> > ...

> Repeating wrong stuff doesn't make it more right.

Why then do you? How should ℵo result if all unios of existing FISONs are in the sequence of FISONs and are therefore finite?

> Union of all FISONs is N

Repeating wrong stuff doesn't make it right.

Regards, WM

On 4/6/2022 7:58 AM, WM wrote:
> William schrieb am Dienstag, 5. April 2022 um 21:33:11 UTC+2:
>> On Tuesday, April 5, 2022 at 3:32:59 PM UTC-3, WM wrote:
>>> potentially infinite set
>>
>> Piffle.
>>
> Every element that exists has been unioned together with all others in the sequence 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}
> ...
>
> That holds for all the infinitely many FISONs of the sequence as the ... indicate. Another union cannot increase the result. Learn: cannot cannot cannot!
>
> Regards, WM
>

What is your problem with math, dude ?

there is no last FISON.

zelos...@gmail.com schrieb am Mittwoch, 6. April 2022 um 06:22:13 UTC+2:
> tisdag 5 april 2022 kl. 20:32:59 UTC+2 skrev WM:

> >But whether or not there are many unions or only one, everything that can be unioned in the set has been unioned in the sequence already. That is so by construction.
> >
> Nope, the union of all FISONs is not in your sequence.

Then there are more FISONs than are in the sequence. But even then their union is not a set of card ℵo.

Regards, WM

On 4/6/2022 8:00 AM, WM wrote:
> William schrieb am Dienstag, 5. April 2022 um 21:41:29 UTC+2:
>> On Tuesday, April 5, 2022 at 3:35:46 PM UTC-3, WM wrote:
>>
>>> ...every element that exists
>>
>> Piffle. Talking about "exists" is silly. The set of FISONs that "exists" does not change.
>>
> Every FISON has been unioned together with all others in the sequence 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}
> ...
>
> That holds for all the infinitely many FISONs of the sequence as the ... indicate. Another union cannot increase the result.
>
> Regards, WM

and....

WM laid this down on his screen :
> William schrieb am Dienstag, 5. April 2022 um 21:41:29 UTC+2:
>> On Tuesday, April 5, 2022 at 3:35:46 PM UTC-3, WM wrote:
>>
>>> ...every element that exists
>>
>> Piffle. Talking about "exists" is silly. The set of FISONs that "exists"
>> does not change.
>>
> Every FISON has been unioned together with all others in the sequence
> 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}
> ...
>
> That holds for all the infinitely many FISONs of the sequence as the ...
> indicate.

That holds for *each* of the infinitely many FISONs of the sequence.
The "..." merely shows that there are infinitely many such FISONs --
which the rest of us knew already.

Eram semper recta schrieb am Mittwoch, 6. April 2022 um 07:38:23 UTC+2:
> On Wednesday, 6 April 2022 at 08:30:25 UTC+3, Eram semper recta wrote:

> > > It is only possible to express numbers finitely. Infinite sequences can neither be written nor could they, if existing, be read.
> Also I do not like the expression "express numbers finitely" because finitely is a redundant adjective.

It is for sober brains, but here around it is useful to emphasize it.
>
> I prefer to use "express numbers with a fixed number of symbols".

Fixed could be understood as omega, in this environment.

Regards, WM

WM wrote :
> Jim Burns schrieb am Dienstag, 5. April 2022 um 22:58:24 UTC+2:
>> On 4/5/2022 2:06 PM, WM wrote:
>>
>>> It is only possible to express numbers finitely.
>> It is possible to make a claim which is true of
>> each one of infinitely-many.
>
> For instance this one: Every FISON that exists has been unioned together with
> all others in the sequence 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}
> ...
>
> That holds for all the infinitely many FISONs of the sequence as the ...
> indicate.

So, FISONs are finite then?

FromTheRafters schrieb am Mittwoch, 6. April 2022 um 15:20:47 UTC+2:
> WM wrote :
> > Jim Burns schrieb am Dienstag, 5. April 2022 um 22:58:24 UTC+2:
> >> On 4/5/2022 2:06 PM, WM wrote:
> >>
> >>> It is only possible to express numbers finitely.
> >> It is possible to make a claim which is true of
> >> each one of infinitely-many.
> >
> > For instance this one: Every FISON that exists has been unioned together with
> > all others in the sequence 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}
> > ...
> >
> > That holds for all the infinitely many FISONs of the sequence as the ...
> > indicate.
> So, FISONs are finite then?

That is dictated by their name: Finite initial segment.

Regards, WM

sergio pretended :
> On 4/6/2022 7:56 AM, FromTheRafters wrote:
>> Eram semper recta brought next idea :
>>> On Wednesday, 6 April 2022 at 08:38:23 UTC+3, Eram semper recta wrote:
>>>> On Wednesday, 6 April 2022 at 08:30:25 UTC+3, Eram semper recta wrote:
>>>>> On Tuesday, 5 April 2022 at 21:06:32 UTC+3, WM wrote:
>>>>>> Eram semper recta schrieb am Dienstag, 5. April 2022 um 08:52:52 UTC+2:
>>>>>>> 1. The expression "finite number" is both illogical and
>>>>>>> anti-mathematical nonsense. The adjective "finite" cannot appear
>>>>>>> before "number". A number is represented by a fixed (not "finite"
>>>>>>> because then this might imply something else (pretty absurd) is
>>>>>>> possible) number of symbols.
>>>>>> "Finite number" need not imply infinite number but may emphasize that
>>>>>> infinity is excluded.  This is not what happens to a delusional mind.
>>>>>> They use "finite number" so as to leave the door open to their drivel
>>>>>> "infinite number".
>>>>>>>
>>>>>>> 2. The above drivel is equivalent to claiming that 1/3 cannot be
>>>>>>> represented in base 10 "finitely" which is a result of the same
>>>>>>> misguided thinking. 1/3 either can or can't be represented in decimal.
>>>>>>> A very important number theorem tells us clearly that 1/3 has no
>>>>>>> representation in decimal. To say:
>>>>>>> "You can't represent 1/3 finitely" is drivel. I just did ... look: "1
>>>>>>> / 3" .
>>>>>>
>>>>>> Of course. It is only possible to express numbers finitely. Infinite
>>>>>> sequences can neither be written nor could they, if existing, be read.
>>>> Also I do not like the expression "express numbers finitely" because
>>>> finitely is a redundant adjective.
>>>> I prefer to use "express numbers with a fixed number of symbols".
>>>> After all, the symbols we use to express numbers are NAMES and it is
>>>> absurd to think that a NAME can be understood in any other way, for
>>>> example as a never-ending string of digits (incomplete decimal
>>>> expansion).
>>>> PhD students have been using "infinity" as a sugar tit far too long in
>>>> order to obtain a degree which means absolutely nothing in real terms. 
>>>> Since they have zero aptitude for mathematics, "infinity" provides a
>>>> conduit for them to bullshit their way into the circle jerk of PhD grads.
>>>>>
>>>>>
>>>>>>
>>>>>> Regards, WM
>>>
>>> Imagine the following scenario:
>>>
>>> I want to express the number that has exactly the same digits as pi but
>>> varies in the trillionth digit only.
>>>
>>> How do I express that number?
>>
>> You just did, well, almost. That would actually be more than one number
>> because we have a selection of more than one possible base ten digit to
>> occupy that 'changed' digit. You need to narrow it down to get it to be
>> only one number.
>
>
> use binary instead.
>
> 0110101.10010010111000101000001010... and just flip the trillionth digit.

I was going to mention that, but I wasted enough time on him already.
:)

On 4/6/2022 8:13 AM, WM wrote:
> zelos...@gmail.com schrieb am Mittwoch, 6. April 2022 um 06:22:13 UTC+2:
>> tisdag 5 april 2022 kl. 20:32:59 UTC+2 skrev WM:
>
>>> But whether or not there are many unions or only one, everything that can be unioned in the set has been unioned in the sequence already. That is so by construction.
>>>
>> Nope, the union of all FISONs is not in your sequence.
>
> Then there are more FISONs than are in the sequence. But even then their union is not a set of card ℵo.
>
> Regards, WM

No, you make mistake, the union of all or every FISONs is not a FISON.

On 4/6/2022 8:10 AM, WM wrote:
> zelos...@gmail.com schrieb am Mittwoch, 6. April 2022 um 06:20:40 UTC+2:
>> tisdag 5 april 2022 kl. 19:59:44 UTC+2 skrev WM:
>>> zelos...@gmail.com schrieb am Dienstag, 5. April 2022 um 06:33:56 UTC+2:
>>>> måndag 4 april 2022 kl. 12:48:38 UTC+2 skrev WM:
>>>>> zelos...@gmail.com schrieb am Montag, 4. April 2022 um 09:48:17 UTC+2:
>>>
>>>>>> You assume only finite number of FISONs here, which is a false assumption
>>>>> No, less than ℵo FISONs is dictated by mathematics.
>>>>
>>>> And the union of all is N
>>> Fact is that
>> the union of ALL FISONs, which is INFINITELY MANY, is N
>
> If ℕ is potentially infinite. If however |ℕ| = ℵo, then all FISONs are not enough because never a union of all FISONs is larger than all unions of FISONs contained in the sequence of FISONs:

you are still making the SAME mistake, the union of all FISONs is not a FISON.

>
>> Repeating wrong stuff doesn't make it more right.
>

>
>> Union of all FISONs is N
>
> Repeating wrong stuff doesn't make it right.

read that again, let it sink in.

>
> Regards, WM

On 4/6/2022 8:24 AM, WM wrote:
> FromTheRafters schrieb am Mittwoch, 6. April 2022 um 15:20:47 UTC+2:
>> WM wrote :
>>> Jim Burns schrieb am Dienstag, 5. April 2022 um 22:58:24 UTC+2:
>>>> On 4/5/2022 2:06 PM, WM wrote:
>>>>
>>>>> It is only possible to express numbers finitely.
>>>> It is possible to make a claim which is true of
>>>> each one of infinitely-many.
>>>
>>> For instance this one: Every FISON that exists has been unioned together with
>>> all others in the sequence 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}
>>> ...
>>>
>>> That holds for all the infinitely many FISONs of the sequence as the ...
>>> indicate.
>> So, FISONs are finite then?
>
> That is dictated by their name: Finite initial segment.
>
> Regards, WM

and....

WM wrote on 4/6/2022 :
> FromTheRafters schrieb am Mittwoch, 6. April 2022 um 15:20:47 UTC+2:
>> WM wrote :
>>> Jim Burns schrieb am Dienstag, 5. April 2022 um 22:58:24 UTC+2:
>>>> On 4/5/2022 2:06 PM, WM wrote:
>>>>
>>>>> It is only possible to express numbers finitely.
>>>> It is possible to make a claim which is true of
>>>> each one of infinitely-many.
>>>
>>> For instance this one: Every FISON that exists has been unioned together
>>> with all others in the sequence 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}
>>> ...
>>>
>>> That holds for all the infinitely many FISONs of the sequence as the ...
>>> indicate.
>> So, FISONs are finite then?
>
> That is dictated by their name: Finite initial segment.

We're finally getting somewhere*. Why do you think things which apply
to finite objects must necessarily also apply to infinite ones like the
set of *all* such finite objects?

* Yes, I know it is not really happening. He will still speak of all
infinite FISONs or some such garbage again in the future.

On 4/6/2022 8:51 AM, FromTheRafters wrote:
> WM wrote on 4/6/2022 :
>> FromTheRafters schrieb am Mittwoch, 6. April 2022 um 15:20:47 UTC+2:
>>> WM wrote :
>>>> Jim Burns schrieb am Dienstag, 5. April 2022 um 22:58:24 UTC+2:
>>>>> On 4/5/2022 2:06 PM, WM wrote:
>>>>>> It is only possible to express numbers finitely.
>>>>> It is possible to make a claim which is true of each one of infinitely-many.
>>>>
>>>> For instance this one: Every FISON that exists has been unioned together with  all others in the sequence 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} ...
>>>> That holds for all the infinitely many FISONs of the sequence as the ... indicate.
>>> So, FISONs are finite then?
>>
>> That is dictated by their name: Finite initial segment.
>
> We're finally getting somewhere*. Why do you think things which apply to finite objects must necessarily also apply to infinite ones like the set of
> *all* such finite objects?
>
>
>
>
>
> * Yes, I know it is not really happening. He will still speak of all infinite FISONs or some such garbage again in the future.

"No Matter Where You Go, There You Are" - Buckaroo Banzai

at least he remembered part of FISON, Finite Intital Segment Of Narwhals

On 4/6/2022 9:05 AM, WM wrote:
> Jim Burns schrieb
> am Dienstag, 5. April 2022 um 22:58:24 UTC+2:
>> On 4/5/2022 2:06 PM, WM wrote:

>>> It is only possible to express numbers finitely.
>>
>> It is possible to make a claim which is true of
>> each one of infinitely-many.
>
> For instance this one:
> Every FISON that exists has been unioned together
> with all others in the sequence already:

We know that
a FISON equals the union of itself and all earlier
FISONs, but not any later FISON

....a claim true of one of infinitely-many FISONs
without needing to choose which one.

> That holds for all the infinitely many FISONs
> of the sequence as the ... indicate.
> Another union cannot increase the result.

We know that
a FISON does not equals the union of all FISONs

....a claim true of one of infinitely-many FISONs
without needing to choose which one.

>>> Infinite sequences can neither be written nor
>>> could they, if existing, be read.
>>
>> If
>> there is always some finite number k for which
>> k replications of distance y passes distance x, k*y > x
>> then
>
> this is given by a finite formula:
> "there is always some finite number k for which
> k replications of distance y passes distance x, k*y > x"
> with k and y to be chosen.

Choosing is not needed.

We know that,
for distances x and y,
a finite number k exists such that k*y > x

....a claim true of two distances
without needing to choose which two.

Also, we know that,
for an infinite sequence of digits S
and a finite number j,
there is a finite initial segment S[j] of length j
which represents an interval I(S[j]) of length 1/10^j

....a claim true of S and j
without needing to choose which sequence and number.

We know that there aren't two points in
all the intervals represented by
finite initial segments of S,
because
two points are a distance y apart
a finite number k exists such that k*y > 1
an interval I(S[j]) exists such that 1/k > 1/10^j
and both points can't be in I(S[j])

....a claim true of S
without needing to choose which sequence

On Wednesday, April 6, 2022 at 10:00:20 AM UTC-3, WM wrote:

> Every FISON has been unioned together with all others in the sequence already:

Nope the set "Every FISON" is infinite. Each element of the sequence is finite. "already" is nonsense.

--
William Hughes

On Wednesday, April 6, 2022 at 10:13:07 AM UTC-3, WM wrote:
> their union is not a set of card ℵo.

Nope, the Peano set of FISONs, and the union of this set, the Peano set of natural numbers, both have cardinality ℵo.

--
William Hughes

On Wednesday, April 6, 2022 at 10:15:42 AM UTC-3, FromTheRafters wrote:
> WM laid this down on his screen :
> > William schrieb am Dienstag, 5. April 2022 um 21:41:29 UTC+2:
> >> On Tuesday, April 5, 2022 at 3:35:46 PM UTC-3, WM wrote:
> >>
> >>> ...every element that exists
> >>
> >> Piffle. Talking about "exists" is silly. The set of FISONs that "exists"
> >> does not change.
> >>
> > Every FISON has been unioned together with all others

Nope the union of any FISON with "all others" is infinite.

--
William Hughes

On Wednesday, 6 April 2022 at 16:20:25 UTC+3, WM wrote:
> Eram semper recta schrieb am Mittwoch, 6. April 2022 um 07:38:23 UTC+2:
> > On Wednesday, 6 April 2022 at 08:30:25 UTC+3, Eram semper recta wrote:
>
> > > > It is only possible to express numbers finitely. Infinite sequences can neither be written nor could they, if existing, be read.
> > Also I do not like the expression "express numbers finitely" because finitely is a redundant adjective.
> It is for sober brains, but here around it is useful to emphasize it.
> >
> > I prefer to use "express numbers with a fixed number of symbols".
> Fixed could be understood as omega, in this environment.

There is no hope for those in Cantor's Fools Paradise.

>
> Regards, WM

William schrieb am Mittwoch, 6. April 2022 um 17:54:29 UTC+2:
> On Wednesday, April 6, 2022 at 10:15:42 AM UTC-3, FromTheRafters wrote:

> > > Every FISON has been unioned together with all others
> Nope the union of any FISON with "all others" is infinite.

Show at least one FISON that is in the union of all FISONs but not in any finite union of the FISONs in the sequence.

Regards, WM

William schrieb am Mittwoch, 6. April 2022 um 17:44:16 UTC+2:
> On Wednesday, April 6, 2022 at 10:00:20 AM UTC-3, WM wrote:
>
> > Every FISON has been unioned together with all others in the sequence already:
> Nope the set "Every FISON" is infinite. Each element of the sequence is finite. "already" is nonsense.

"Already" means unioned in the sequence. Find a FISON that is not unioned already in the sequence. Fail.

Regards, WM

FromTheRafters schrieb am Mittwoch, 6. April 2022 um 15:52:34 UTC+2:
> WM wrote on 4/6/2022 :
> > FromTheRafters schrieb am Mittwoch, 6. April 2022 um 15:20:47 UTC+2:
> >> WM wrote :
> >>> Jim Burns schrieb am Dienstag, 5. April 2022 um 22:58:24 UTC+2:
> >>>> On 4/5/2022 2:06 PM, WM wrote:
> >>>>
> >>>>> It is only possible to express numbers finitely.
> >>>> It is possible to make a claim which is true of
> >>>> each one of infinitely-many.
> >>>
> >>> For instance this one: Every FISON that exists has been unioned together
> >>> with all others in the sequence 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}
> >>> ...
> >>>
> >>> That holds for all the infinitely many FISONs of the sequence as the ...
> >>> indicate.
> >> So, FISONs are finite then?
> >
> > That is dictated by their name: Finite initial segment.
> We're finally getting somewhere*. Why do you think things which apply
> to finite objects must necessarily also apply to infinite ones like the
> set of *all* such finite objects?

All such objects satisfy this specification: They are finite. They are unions of all predecessors. Find a FISON that is in the union of all FISONs but not in the sequence. Fail.

Regards, WM

Jim Burns schrieb am Mittwoch, 6. April 2022 um 16:36:03 UTC+2:
> On 4/6/2022 9:05 AM, WM wrote:

> We know that
> a FISON equals the union of itself and all earlier
> FISONs, but not any later FISON

But we know that this holds also for all later FISONs. None is left out.
>
> > Another union cannot increase the result.
> We know that
> a FISON does not equals the union of all FISONs

Find a FISON that violates my claim. Find a FISON in the union of all which is not unioned in the sequence.

Regards WM