On 3/30/2022 9:15 AM, WM wrote:
> Jim Burns schrieb am Mittwoch, 30. März 2022 um 03:30:19 UTC+2:
>> On 3/29/2022 10:02 AM, WM wrote:
>
>>> You intend to study all fractions and their indices.
>>> But you cannot find the dark fractions.
>>> So you claim that they have left the matrix.
>> I claim that
>> only r/c
>> for which collections exist with counting-orders
>> which begin at 1 and end at r and at c
>> are r/c
>> for which collections exist with counting-orders
>> which begin at 1 and end at r and at c
>
> That is true. But most places of the matrix don't share this privilege.

again misguided, because you use sets and matrix's as variables

>>
>> If you change what's in the matrix as we proceed through
>> the discussion, you are equivocating.
>
> Every positive fraction is and remains in the matrix.

no it does not, as you have shown before, none remain

>>>
>>> You want to forget them.
>> Do your dark fractions have r/c
>> for which collections exist with counting-orders
>> which begin at 1 and end at r and at c ?
>
> No, dark fractions don't have numbers with FISON-ends.

red herring.

>>
>> If they do, r/c refers to them, too.
>>
>> If they don't, r/c has never referred to them.
>
> Only a tine part of the matrix has definable co-ordinates:
>
> 11, 12, 13, ..., Dark....................................................
> 21, 22, 23, ..., Dark....................................................
> 31, 32, 33, ..., Dark....................................................
> ...... ...... ....... Dark.....................................................
> Dark..........................................................................
> ..................................................................................

Wrong. Your Dark numbers do not have values, and cannot be placed into any matrix.

>
>>>> Your argument is that
>>>> Cantor is wrong because
>>>> his system implies undefinable points,
>>>> so, if his system is used,
>>>> there will be points undefinable.
>>>
>>> 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.

>
>>>> Perhaps I missed it, but
>>>> you've never answered the question
>>>> "and then what?"
>>>
>>> Then the notion of countable sets is nonsense,
>> A finite non-empty collection
>> cannot match element-for-element
>> any of its proper subsets.
>
> It can if the finity is not fixed.
>>
>> We can say that more compactly as
>> "The set of FISONs is countably infinite"

<snip repeated old + wrong argument>

>
> Regards, WM
>
>

WM <askasker48@gmail.com> wrote:
> Alan Mackenzie schrieb am Dienstag, 29. März 2022 um 21:04:50 UTC+2:
>> WM <askas...@gmail.com> wrote:

>> As something to think about, consider a random digit generator
>> generating a decimal representation. I don't think you would deny the
>> sequence it produces exists. However there is no formula for it - it
>> can't be "given".

> Here you touch a borderlin case.
> The sequence will never become actually infinite.

The sequence is infinite.

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

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

> Of course every digit will have a successor, but they are not yet
> fixed.

That's a philosophical assumption, not a mathematical one. If the
generator works in a finite (or zero) period of time the digits will be
"fixed" (whatever that means).

> In the case SUM 1/n! we cannot calculate more than any digit at a
> finite position either, but we can be sure that the digit discovered at
> position n will be independent of who calculates it.

>> >> Just because there may not be a formula known for some decimal
>> >> representation, how does that imply non-existence?

>> > What do you understand by existence?

>> In mathematics, something exists if it can be mathematically described
>> and doesn't give rise to contradictions.

> Therefore every digit sequence that can be described by a finite
> formula like 1/3 or SUM 1/n! does exist.

Every digit sequence exists. It can be mathematically described, and
doesn't give rise to a contradiction.

> But sequences without formula are necessarily finite because infinity
> cannot be finished or completed.

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

>> > Does something that cannot be known exist?

>> Some somethings, yes. The random decimal expansion exists.

> Yes, this notion exists.

No. Not merely the notion, but also the random decimal expansion exists.

> But the digit sequence is not yet fixed and will never be fixed
> completely.

"Fixed" is here meaningless. You might just as well say the number 6
isn't fixed.

[ .... ]

> Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).

FromTheRafters schrieb am Mittwoch, 30. März 2022 um 17:34:22 UTC+2:
> It happens that WM formulated :
> > 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?
> Induction. If you have an element, you also have a next element.

And none of them produce a set of cardinality larger than every natural number, i.e., which has ℵo elements.

> Nothing more is added to this notion to terminate the step-by-step
> process you are sure to envision. It keeps going -- no element is left
> in darkness.

No element produces a set of cardinality larger than every natural number, i.e., which has ℵo elements. But Cantor claims that that ℕ is actually infinite, contains more than all finite initial segments contain, namely cardinality ℵo.

Regards, WM

Tom Bola schrieb am Mittwoch, 30. März 2022 um 18:01:35 UTC+2:
> Clown WM drivels:
> > {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 too. This sequence however does
> > not contain an actually infinite finite initial segment.
> But it happened infinite many times there *).

Nothing happens. The sequence is a static object with infinitely many finite terms including all unions of all its terms.

Regards, WM

Alan Mackenzie schrieb am Mittwoch, 30. März 2022 um 19:02:58 UTC+2:
> WM <askas...@gmail.com> wrote:
> > Alan Mackenzie schrieb am Dienstag, 29. März 2022 um 21:04:50 UTC+2:
> >> WM <askas...@gmail.com> wrote:
>
> >> As something to think about, consider a random digit generator
> >> generating a decimal representation. I don't think you would deny the
> >> sequence it produces exists. However there is no formula for it - it
> >> can't be "given".
>
> > Here you touch a borderline case.
> > The sequence will never become actually infinite.
> The sequence is infinite.

The sequence is in process of production and never infinite.

> > 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.

> > Of course every digit will have a successor, but they are not yet
> > fixed.
> That's a philosophical assumption, not a mathematical one. If the
> generator works in a finite (or zero) period of time the digits will be
> "fixed" (whatever that means).

If it works in zero time, and the sequence is completed, then a last digit has been produced before the production ceased. Impossible.

> >> In mathematics, something exists if it can be mathematically described
> >> and doesn't give rise to contradictions.
>
> > Therefore every digit sequence that can be described by a finite
> > formula like 1/3 or SUM 1/n! does exist.
> Every digit sequence exists. It can be mathematically described, and
> doesn't give rise to a contradiction.

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

> > But sequences without formula are necessarily finite because infinity
> > cannot be finished or completed.
> Wrong. The sequence produced by the random digit generator is infinite.

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

> > But the digit sequence is not yet fixed and will never be fixed
> > completely.
> "Fixed" is here meaningless. You might just as well say the number 6
> isn't fixed.

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

Regards, WM

On 3/30/2022 12:48 PM, WM wrote:
> Alan Mackenzie schrieb am Mittwoch, 30. März 2022 um 19:02:58 UTC+2:
>> WM <askas...@gmail.com> wrote:
>>> Alan Mackenzie schrieb am Dienstag, 29. März 2022 um 21:04:50 UTC+2:
>>>> WM <askas...@gmail.com> wrote:
>>
>>>> As something to think about, consider a random digit generator
>>>> generating a decimal representation. I don't think you would deny the
>>>> sequence it produces exists. However there is no formula for it - it
>>>> can't be "given".
>>
>>> Here you touch a borderline case.
>>> The sequence will never become actually infinite.
>> The sequence is infinite.
>
> The sequence is in process of production and never infinite.

says you. {1,2,3...} is an infinite sequence, not some motor running in the background

>
>>> 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.

there is no last term in an infinite sequence, so you are wrong again.

>
>>> Of course every digit will have a successor, but they are not yet
>>> fixed.
>> That's a philosophical assumption, not a mathematical one. If the
>> generator works in a finite (or zero) period of time the digits will be
>> "fixed" (whatever that means).
>
> If it works in zero time, and the sequence is completed, then a last digit has been produced before the production ceased. Impossible.

there is no last digit in an infinite sequence, when will you learn the basics of Math ?

>
>>>> In mathematics, something exists if it can be mathematically described
>>>> and doesn't give rise to contradictions.
>>
>>> Therefore every digit sequence that can be described by a finite
>>> formula like 1/3 or SUM 1/n! does exist.
>> Every digit sequence exists. It can be mathematically described, and
>> doesn't give rise to a contradiction.
>
> The contradiction is that between the production and the ceasing there must be an end.

you turn to "production" and "ceasing" to insert a time factor, a smoke screen for your defective math.

>
>>> But sequences without formula are necessarily finite because infinity
>>> cannot be finished or completed.
>> Wrong. The sequence produced by the random digit generator is infinite.
>
> Show such a sequence. If it exists, you can write it.

this is probably too advanced for you;
https://www.random.org/

>
>>> But the digit sequence is not yet fixed and will never be fixed
>>> completely.
>> "Fixed" is here meaningless. You might just as well say the number 6
>> isn't fixed.
>
> It is, contrary to your sequence which cannot be shown and cannot be described.

You Deny Variables. You are out of Math again.

>
> Regards, WM

The imbecile Clown WM drivels:

> No element produces a set of cardinality larger than every natural number,

LOL - there is no "production" in Math.

Piss off already, idiot.

Clown WM drivels:

> Tom Bola:

>> Clown WM drivels:
>>> {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 too. This sequence however does
>>> not contain an actually infinite finite initial segment.

>> But it happened infinite many times there *).
>
> Nothing happens.

Right - in Math there is no "production".

Piss of, asshole.

On 3/29/2022 11:57 PM, David Petry wrote:
> On Sunday, March 27, 2022 at 1:59:14 PM UTC-7,
> Jim Burns wrote:
>> On 3/27/2022 12:48 PM, David Petry wrote:

>>> Let me remind everybody about the argument that
>>> Poincare (and others) made:
>>
>> It seems to me that the correctness of
>> the mimicry of the intended _something_
>> is more important than being finite or infinite
>> or uncountable or inaccessible.
>
> I'm sure that's very deep, but I don't understand it.

Your objection
(the one you attribute to Poincaré and others)
sounds to me like
"Language cannot describe an uncountable set".

Imagine language making a claim we know is true of one of
uncountably-many, without needing to choose which of
those uncountably-many is referred to.

Imagine language saying that a thing is in a set
if and only if that "uncountably-many" claim
is true of it.

I would say that this refutes the argument that
language cannot describe an uncountable set.

You probably disagree. Why do you disagree?

> "He ignored things he shouldn't have ignored,
> and I (J.B.) have pointed out what those things are."

I don't know what Poincaré's argument is.
I see what you (DP) say it is.
I vaguely recall you quoting Poincaré to support
your claim he said such-and-such, when, seen in context,
he denied meaning such-and-such.
I don't see why I should trust your report.

Still, there is an argument, whoever made it.

The argument reminds me of Ed Nelson's attempt to
prove that arithmetic is inconsistent.

Terry Tao pointed out an error (a subtle one, to
my eyes), and Ed Nelson withdrew his claim.

https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html
| | Basically, I think the conceptual error here is to believe
| that the quantity 𝑙=𝑙(𝑇) provided by Chaitin’s theorem is
| monotone in the sense that if a given 𝑙 works for a theory 𝑇,
| then it would also work for all restricted subtheories of 𝑇.
| This is not the case, because a subtheory can in fact be
| much more complicated than the original theory in the sense
| that it requires a much longer proof verifier.

sergio wrote :
> On 3/30/2022 9:15 AM, WM wrote:
>> Jim Burns schrieb am Mittwoch, 30. März 2022 um 03:30:19 UTC+2:
>>> On 3/29/2022 10:02 AM, WM wrote:
>>
>>>> You intend to study all fractions and their indices.
>>>> But you cannot find the dark fractions.
>>>> So you claim that they have left the matrix.
>>> I claim that
>>> only r/c
>>> for which collections exist with counting-orders
>>> which begin at 1 and end at r and at c
>>> are r/c
>>> for which collections exist with counting-orders
>>> which begin at 1 and end at r and at c
>>
>> That is true. But most places of the matrix don't share this privilege.
>
> again misguided, because you use sets and matrix's as variables
>
>>>
>>> If you change what's in the matrix as we proceed through
>>> the discussion, you are equivocating.
>>
>> Every positive fraction is and remains in the matrix.
>
> no it does not, as you have shown before, none remain
>
>>>>
>>>> You want to forget them.
>>> Do your dark fractions have r/c
>>> for which collections exist with counting-orders
>>> which begin at 1 and end at r and at c ?
>>
>> No, dark fractions don't have numbers with FISON-ends.
>
> red herring.
>
>>>
>>> If they do, r/c refers to them, too.
>>>
>>> If they don't, r/c has never referred to them.
>>
>> Only a tine part of the matrix has definable co-ordinates:
>>
>> 11, 12, 13, ..., Dark....................................................
>> 21, 22, 23, ..., Dark....................................................
>> 31, 32, 33, ..., Dark....................................................
>> ...... ...... .......
>> Dark.....................................................
>> Dark..........................................................................
>> ..................................................................................
>
>
> Wrong. Your Dark numbers do not have values, and cannot be placed into any
> matrix.
>
>>
>>>>> Your argument is that
>>>>> Cantor is wrong because
>>>>> his system implies undefinable points,
>>>>> so, if his system is used,
>>>>> there will be points undefinable.
>>>>
>>>> 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. :)

With r and c being free range elements of the positive integers.

WM <askasker48@gmail.com> wrote:
> Alan Mackenzie schrieb am Mittwoch, 30. März 2022 um 19:02:58 UTC+2:
>> WM <askas...@gmail.com> wrote:
>> > Alan Mackenzie schrieb am Dienstag, 29. März 2022 um 21:04:50 UTC+2:
>> >> WM <askas...@gmail.com> wrote:

>> >> As something to think about, consider a random digit generator
>> >> generating a decimal representation. I don't think you would deny
>> >> the sequence it produces exists. However there is no formula for it
>> >> - it can't be "given".

>> > Here you touch a borderline case.
>> > The sequence will never become actually infinite.

>> The sequence is infinite.

> The sequence is in process of production and never infinite.

The sequence is always infinite. There is no final digit.

>> > 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.

>> > Of course every digit will have a successor, but they are not yet
>> > fixed.

>> That's a philosophical assumption, not a mathematical one. If the
>> generator works in a finite (or zero) period of time the digits will
>> be "fixed" (whatever that means).

> If it works in zero time, and the sequence is completed, then a last
> digit has been produced before the production ceased. Impossible.

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

>> >> In mathematics, something exists if it can be mathematically
>> >> described and doesn't give rise to contradictions.

>> > Therefore every digit sequence that can be described by a finite
>> > formula like 1/3 or SUM 1/n! does exist.

>> Every digit sequence exists. It can be mathematically described, and
>> doesn't give rise to a contradiction.

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

That's incoherent.

>> > But sequences without formula are necessarily finite because infinity
>> > cannot be finished or completed.

>> 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.

>> > But the digit sequence is not yet fixed and will never be fixed
>> > completely.

>> "Fixed" is here meaningless. You might just as well say the number 6
>> isn't fixed.

> 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?

Or did you mean "specified"? I explained the difference between the
English words describe and specify to you last night.

> Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).

on 3/30/2022, WM supposed :
> FromTheRafters schrieb am Mittwoch, 30. März 2022 um 17:34:22 UTC+2:
>> It happens that WM formulated :
>>> 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?
>> Induction. If you have an element, you also have a next element.
>
> 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. 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 just as there is always a next natural number.

FromTheRafters schrieb:

> on 3/30/2022, WM supposed :
>> FromTheRafters schrieb am Mittwoch, 30. März 2022 um 17:34:22 UTC+2:
>>> It happens that WM formulated :
>>>> 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?
>>> Induction. If you have an element, you also have a next element.
>>
>> 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. 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 just as there is always a next natural number.

I told him the same in German a few minutes in the German math newsgroup:

Geht nicht, weil du DAFÜR nirgendwo anfangen kannst runterzuzählen,
die omegas oder alephs haben keinen Vorgänger und jede andere Wahl
für deinen Anfang hätte unendlich viele Nachfolger.

Alan Mackenzie schrieb:

> Or did you mean "specified"? I explained the difference between the
> English words describe and specify to you last night.

He got things explained in the german math newsgroup for about 30 years.

Yes, for 30 years - looks like mathematicians are idiots - just as WM says...

Tom Bola wrote :
> FromTheRafters schrieb:
>
>> on 3/30/2022, WM supposed :
>>> FromTheRafters schrieb am Mittwoch, 30. März 2022 um 17:34:22 UTC+2:
>>>> It happens that WM formulated :
>>>>> 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?
>>>> Induction. If you have an element, you also have a next element.
>>>
>>> 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. 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 just as there is always a next natural number.
>
> I told him the same in German a few minutes in the German math newsgroup:
>
> Geht nicht, weil du DAFÜR nirgendwo anfangen kannst runterzuzählen,
> die omegas oder alephs haben keinen Vorgänger und jede andere Wahl
> für deinen Anfang hätte unendlich viele Nachfolger.

He's insane in any language.

On 3/30/2022 1:05 PM, Jim Burns wrote:
> On 3/29/2022 11:57 PM, David Petry wrote:
>> On Sunday, March 27, 2022 at 1:59:14 PM UTC-7,
>> Jim Burns wrote:
>>> On 3/27/2022 12:48 PM, David Petry wrote:
>
>>>> Let me remind everybody about the argument that
>>>> Poincare (and others) made:
>>>
>>> It seems to me that the correctness of
>>> the mimicry of the intended _something_
>>> is more important than being finite or infinite
>>> or uncountable or inaccessible.
>>
>> I'm sure that's very deep, but I don't understand it.
>
> Your objection
> (the one you attribute to Poincaré and others)
> sounds to me like
> "Language cannot describe an uncountable set".

that is not true anyway. Fractal generation.
the digits of Pi... etc etc

>
> Imagine language making a claim we know is true of one of
> uncountably-many, without needing to choose which of
> those uncountably-many is referred to.
>
> Imagine language saying that a thing is in a set
> if and only if that "uncountably-many" claim
> is true of it.
>
> I would say that this refutes the argument that
> language cannot describe an uncountable set.
>
> You probably disagree. Why do you disagree?
>
>> "He ignored things he shouldn't have ignored,
>> and I (J.B.) have pointed out what those things are."
>
> I don't know what Poincaré's argument is.
> I see what you (DP) say it is.
> I vaguely recall you quoting Poincaré to support
> your claim he said such-and-such, when, seen in context,
> he denied meaning such-and-such.
> I don't see why I should trust your report.
>
> Still, there is an argument, whoever made it.
>
> The argument reminds me of Ed Nelson's attempt to
> prove that arithmetic is inconsistent.
>
> Terry Tao pointed out an error (a subtle one, to
> my eyes), and Ed Nelson withdrew his claim.
>
> https://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html
> |
> | Basically, I think the conceptual error here is to believe
> | that the quantity 𝑙=𝑙(𝑇) provided by Chaitin’s theorem is
> | monotone in the sense that if a given 𝑙 works for a theory 𝑇,
> | then it would also work for all restricted subtheories of 𝑇.
> | This is not the case, because a subtheory can in fact be
> | much more complicated than the original theory in the sense
> | that it requires a much longer proof verifier.
>

sergio submitted this idea :
> On 3/30/2022 1:05 PM, Jim Burns wrote:
>> On 3/29/2022 11:57 PM, David Petry wrote:
>>> On Sunday, March 27, 2022 at 1:59:14 PM UTC-7,
>>> Jim Burns wrote:
>>>> On 3/27/2022 12:48 PM, David Petry wrote:
>>
>>>>> Let me remind everybody about the argument that
>>>>> Poincare (and others) made:
>>>>
>>>> It seems to me that the correctness of
>>>> the mimicry of the intended _something_
>>>> is more important than being finite or infinite
>>>> or uncountable or inaccessible.
>>>
>>> I'm sure that's very deep, but I don't understand it.
>>
>> Your objection
>> (the one you attribute to Poincaré and others)
>> sounds to me like
>> "Language cannot describe an uncountable set".
>
> that is not true anyway. Fractal generation.
> the digits of Pi... etc etc

Those are countable aren't they?

> Let me remind everybody about the argument that Poincare (and others) made:
>
> 1) Above all else, mathematics is a language. That is, it's a system of symbols and rules for manipulating strings of symbols.
>
> 2) The purpose of the language is to give us a way to encode and communicate "ideas".
>
> 3) The things that can be said to "exist" in mathematics are ideas!

Correct

> Or in other words, the things that "exist" in the language of mathematics are things we can actually talk about, and identify, and describe.
>
> 4) Emphatically, ideas can be completely defined in a finite number of words of the language.
>

True, but a finite number of words can be used to talk about an infinite object.

> 5) If we take Cantor's ideas about infinite sets seriously, we would conclude that there cannot be more than "countably" many ideas, since there are not more than "countably" many finite strings of symbols.

Indeed , there are only a countable number of "ideas". But that does not stop us from using a finite number of words, an idea, to describe a set with an arbitrary number of elements.

>
> 6) But then Cantor goes on to "prove" that there "exist" sets (collections of objects) that are bigger than any "countable" collection.

Nope. Cantor's proof that there is no bijection from the integers to the binary sequences holds whether or not the binary sequences are considered subcountable.

a binary sequence is a function f:N->{0,1}
a list of binary sequences is a function g:NxN -> {0,1}

If you restrict f to objects that are finitely definable, the you have to restrict g as well.

given any list of binary sequences g, AD:N->{0,1} defined by AD(n) = g(n,n) + 1 (mod 2) is not in the list.

--
William Hughes

On 3/30/2022 4:35 PM, FromTheRafters wrote:
> sergio submitted this idea :
>> On 3/30/2022 1:05 PM, Jim Burns wrote:
>>> On 3/29/2022 11:57 PM, David Petry wrote:
>>>> On Sunday, March 27, 2022 at 1:59:14 PM UTC-7,
>>>> Jim Burns wrote:
>>>>> On 3/27/2022 12:48 PM, David Petry wrote:
>>>
>>>>>> Let me remind everybody about the argument that
>>>>>> Poincare (and others) made:
>>>>>
>>>>> It seems to me that the correctness of
>>>>> the mimicry of the intended _something_
>>>>> is more important than being finite or infinite
>>>>> or uncountable or inaccessible.
>>>>
>>>> I'm sure that's very deep, but I don't understand it.
>>>
>>> Your objection
>>> (the one you attribute to Poincaré and others)
>>> sounds to me like
>>> "Language cannot describe an uncountable set".
>>
>> that is not true anyway.  Fractal generation.
>>   the digits of Pi...  etc etc
>
> Those are countable aren't they?

yep, Pi using spigot algorithm, BBP digit extraction algorithm...

On Wednesday, March 30, 2022 at 9:14:48 AM UTC-3, WM wrote:
> david...@gmail.com schrieb am Mittwoch, 30. März 2022 um 06:08:44 UTC+2:

> {1} = {1}
> {1} U {1, 2} = {1, 2}
> {1} U {1, 2} U {1, 2, 3} = {1, 2, 3}
> {1} U {1, 2} U {1, 2, 3} U {1, 2, 3, 4} = {1, 2, 3, 4}
> {1} U {1, 2} U {1, 2, 3} U {1, 2, 3, 4} U {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5}
> ...
> That implies that the sequence of *all* finite initial segments
is a sequence without largest element, so the union is larger than any element of the sequence (i.e. while every element of the union, n is in some element d(n) (if you change n you may have to change d(n)), every element of the sequence is a proper subset of the union). Note that no element of the sequence contains a dark element, so the union does not contain a dark element.

--
William Hughes

On 3/30/2022 10:15 AM, WM wrote:
> Jim Burns schrieb
> am Mittwoch, 30. März 2022 um 03:30:19 UTC+2:
>> On 3/29/2022 10:02 AM, WM wrote:

>>> You want to forget them.
>>
>> Do your dark fractions have r/c
>> for which collections exist with counting-orders
>> which begin at 1 and end at r and at c ?
>
> No, dark fractions don't have numbers with FISON-ends.

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

| Assume that Bob gets to a place without
| numbers-with-FISON-ends.
| | The places Bob was before that place can be cut LIT,DARK
| Bob steps from LIT to DARK.
| | A step from LIT to DARK has
| | r/c in LIT for which
| collections exist with counting-orders
| which begin at 1 and end at r and at c
| | and k/1 in DARK
| a collection does not exist with a counting-order
| which begins at 1 and end at k
| | such that k = (r+c-1)*(r+c-2)/2 + r
| | There are no r,c, where
| ⟨1,...,r⟩ and ⟨1,...,c⟩ exist,
| where ⟨1,...,k⟩ does not exist
| Contradiction.
|

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

sergio wrote :
> On 3/30/2022 4:35 PM, FromTheRafters wrote:
>> sergio submitted this idea :
>>> On 3/30/2022 1:05 PM, Jim Burns wrote:
>>>> On 3/29/2022 11:57 PM, David Petry wrote:
>>>>> On Sunday, March 27, 2022 at 1:59:14 PM UTC-7,
>>>>> Jim Burns wrote:
>>>>>> On 3/27/2022 12:48 PM, David Petry wrote:
>>>>
>>>>>>> Let me remind everybody about the argument that
>>>>>>> Poincare (and others) made:
>>>>>>
>>>>>> It seems to me that the correctness of
>>>>>> the mimicry of the intended _something_
>>>>>> is more important than being finite or infinite
>>>>>> or uncountable or inaccessible.
>>>>>
>>>>> I'm sure that's very deep, but I don't understand it.
>>>>
>>>> Your objection
>>>> (the one you attribute to Poincaré and others)
>>>> sounds to me like
>>>> "Language cannot describe an uncountable set".
>>>
>>> that is not true anyway.  Fractal generation.
>>>   the digits of Pi...  etc etc
>>
>> Those are countable aren't they?
>
> yep, Pi using spigot algorithm, BBP digit extraction algorithm...

Ah, but the claim was about the uncountable. Anyway, they can still be
described by finite language as being 'larger' by virtue of the
countable sets being embedded 'smaller' proper subsets of the 'larger'
reals.

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.

Non-sequitur. ??????

Your reply is strange because it does not address the topic. It's as if you were asked what colour is the sky and your response is 23.

While your reply is correct, it's not the issue. I am telling you that your opinion about p/q being a formula is bullshit.

All I can conclude is that you are responding to me as if I am part of the retarded collective in this thread - the morons who are arguing with you about Cantor's delusions.

p/q is NOT a formula because a number is not a formula and a formula is not a number. What part of this do you not understand?

I am in agreement with you on the basic issues. Stop slapping me when you are slapped by one of the other crap spewing idiots.

>
> Regards, WM

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

but with all elements.

> so the union is larger than any element of the sequence

That is a mistaken claim because the sequence contains all elements. There is nothing that could blow up the union to more than all FISONs present in the sequence.

Regards, 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