The proof of equinumerosity by bijection between infinite sets, M and N,
is justified by mathematical induction: If every element of set M can be
related to one and only one corresponding element of set N and vice
versa, and if there is never an obstacle or halt in this process of
assignment, then both infinite sets are in bijection. "with respect to
this order we can talk about the nth algebraic number where not a single
one of this epitome (ω) has been forgotten." [E. Zermelo: "Georg Cantor
– Gesammelte Abhandlungen mathematischen und philosophischen Inhalts",
Springer, Berlin (1932) p. 116]

A supertask is a countably infinite sequence of operations that occur
sequentially within a finite interval of time.

Can you demonstrate a difference? (Cantor was born less than 200 years ago.)

Regards, WM

On Monday, 13 November 2023 at 10:49:51 UTC+1, WM wrote:

20+ years of the same nonsense and you still insist?

> The proof of equinumerosity by bijection between infinite sets, M and N,
> is justified by mathematical induction:

Wrong, equinumerosity by bijection is a *definition* not any theorem.

> A supertask is a countably infinite sequence of operations that occur
> sequentially within a finite interval of time.

Wrong, a "supertask" is the *limit* of any such sequence,
and "time" is utterly irrelevant if not as an expository device.

Julio

On 13.11.2023 12:42, Julio Di Egidio wrote:
> On Monday, 13 November 2023 at 10:49:51 UTC+1, WM wrote:
>
>> The proof of equinumerosity by bijection between infinite sets, M and N,
>> is justified by mathematical induction:
>
> Wrong, equinumerosity by bijection is a *definition* not any theorem.

The bijection between naturals and algebraics is defined only? It cannot
be proved?
>
>> A supertask is a countably infinite sequence of operations that occur
>> sequentially within a finite interval of time.
>
> Wrong, a "supertask" is the *limit* of any such sequence,
> and "time" is utterly irrelevant if not as an expository device.

A supertask is a task, not a limit. Look it up. Finite time belongs to
its definition. But that is irrelevant. Every step can be done in half
time of the previous step. Cantor did it within less than 200 years.

Regards, WM

On Monday, 13 November 2023 at 13:08:43 UTC+1, WM wrote:
> On 13.11.2023 12:42, Julio Di Egidio wrote:
> > On Monday, 13 November 2023 at 10:49:51 UTC+1, WM wrote:
> >
> >> The proof of equinumerosity by bijection between infinite sets, M and N,
> >> is justified by mathematical induction:
> >
> > Wrong, equinumerosity by bijection is a *definition* not any theorem.
>
> The bijection between naturals and algebraics is defined only? It cannot
> be proved?

That there is a bijection one has to prove, that equinumerous iff biject
is a *definition* (of "standard" mathematics).

> >> A supertask is a countably infinite sequence of operations that occur
> >> sequentially within a finite interval of time.
> >
> > Wrong, a "supertask" is the *limit* of any such sequence,
> > and "time" is utterly irrelevant if not as an expository device.
>
> A supertask is a task, not a limit.

Nope, you are wrong: but you are right that more nonsense has been
written about what a supertask even is than there are stars in our galaxy.

Julio

On 13/11/2023 13:23, Julio Di Egidio wrote:
> On Monday, 13 November 2023 at 13:08:43 UTC+1, WM wrote:
>> On 13.11.2023 12:42, Julio Di Egidio wrote:
>>> On Monday, 13 November 2023 at 10:49:51 UTC+1, WM wrote:
<snip>
>>>> A supertask is a countably infinite sequence of operations that occur
>>>> sequentially within a finite interval of time.
>>>
>>> Wrong, a "supertask" is the *limit* of any such sequence,
>>> and "time" is utterly irrelevant if not as an expository device.
>>
>> A supertask is a task, not a limit.
>
> Nope, you are wrong: but you are right that more nonsense has been
> written about what a supertask even is than there are stars in our galaxy.

Thanks for bringing that up:

<https://en.wikipedia.org/wiki/Talk:Supertask#Wrong_definition>

Julio

On 11/13/2023 7:08 AM, WM wrote:
> On 13.11.2023 12:42, Julio Di Egidio wrote:

>> [...]

> A supertask is a task, not a limit.
> Look it up.

https://plato.stanford.edu/Archives/sum2005/entries/spacetime-supertasks/

| A supertask may be defined as
| an infinite sequence of actions or operations
| carried out in a finite interval of time.

| The terms ‘action’ and ‘operation’ must not
| be understood in their usual sense,
| which involves a human agent.

| We will assume that at each instant of time
| the state of the world relevant to
| a specific action can be described by
| a set S of sentences.
| Now an action or operation applied to
| a state of the world results in a change in
| that state, that is, in
| the set S corresponding to it.
| Consequently, an arbitrary action a
| will be defined (Allis and Koetsier [1995]) as
| a change in the state of the world by which
| the latter changes
| from state S before the change
| to state a(S) after it.

| [...] Before t=1 the state of the lamp [...]
| can be described by the sentence ‘lamp on’,
| and after t=1 by the sentence ‘lamp off’, [...]

| A hypertask is a non-numerable infinite sequence
| of actions or operations carried out in a
| finite interval of time. Therefore,
| a supertask which is not a hypertask will be
| a numerable infinite sequence of actions or
| operations carried out in a finite interval of time.

| Finally, a task can be defined as
| a finite sequence of actions or operations
| carried out in a finite interval of time.

> A supertask is a task, not a limit.
> Look it up.

Describing and reasoning about a supertask
is a task, not a supertask,
at least, for those of us not Chuck Norris.

Indexing the fractions,
not merely describing and reasoning about
indexing the fractions,
is a supertask, but not a hypertask,
and not a task.

Running after a tortoise, or just running,
is a hypertask, involving non-numerably-many
states of the world.

On Monday, 13 November 2023 at 18:15:05 UTC+1, Jim Burns wrote:
> On 11/13/2023 7:08 AM, WM wrote:
> > On 13.11.2023 12:42, Julio Di Egidio wrote:
> >> [...]

<https://en.wikipedia.org/wiki/Talk:Supertask#Wrong_definition>

> > A supertask is a task, not a limit.
> > Look it up.
> <https://plato.stanford.edu/Archives/sum2005/entries/spacetime-supertasks/>
>
> | A supertask may be defined as
> | an infinite sequence of actions or operations
> | carried out in a finite interval of time.
<snip>
> Indexing the fractions, not merely
> describing and reasoning about
> indexing the fractions, is a supertask

Indeed, *utter* bullshit.

Julio

On 13.11.2023 18:15, Jim Burns wrote:

> Describing and reasoning about a supertask
> is a task, not a supertask,
>
> Indexing the fractions,
> not merely describing and reasoning about
> indexing the fractions,
> is a supertask,

Indexing the algebraic numbers is a supertask.

Describing the method of indexing is a task.
This holds for all of Cantor's enumerations as well as for my matrices.

If completeness is accomplished, then it is accomplished in both cases.
Or can you determine a difference?

Regards, WM

On 13/11/2023 19:10, WM wrote:

> Indexing the algebraic numbers is a supertask.

It is *not*. "Where does exactly Achilles reach the tortoise",
when the question is *formulated mathematically*, becomes, in
modern terms, a question of *limits*.

And it is *that* fact and discovering it, i.e. that the answer to
that question is, again using a modern terminology, *not finitely
representable*, that upset the Pythagoreans: as the legend goes.

I won't insist: WM the Pentcho Valev of mathematics... and co.

HTH and EOD.

Julio

On 11/13/2023 1:10 PM, WM wrote:
> On 13.11.2023 18:15, Jim Burns wcrote:

>> Describing and reasoning about a supertask
>> is a task, not a supertask,
>>
>> Indexing the fractions,
>> not merely describing and reasoning about
>> indexing the fractions,
>> is a supertask,
>
> Indexing the algebraic numbers is a supertask.

Meaning:
No finite order exists for
the set Indexing of actions
"place k on ⟨i,j⟩"
for ⟨i,j⟩ ∈ ℕ×ℕ
k = i+(i+j-1)(i+j-2)/2

In a finite order,
for each split,
there are last-before and first-after.
and there are first and last for the whole.
There are no orders of Indexing which have
all of that.

Either ℕ is minimally-each-accessible ℕ⅏
or ℕ is some superset ℕᵂᴹ ⊇ ℕ⅏
No finite order exists for either
Indexing(ℕ⅏×ℕ⅏) or Indexing(ℕᵂᴹ×ℕᵂᴹ)

ℕ⅏ is inductive:
'⅏' looks like an inductor. Ha ha?

∀⟨1,…,n⟩ ⊆ ℕ⅏
ℕ⅏ = ⋂{S⊆ℕ⅏| ∀⟨1,…,n⟩ ⊆ S }

Indexing of ℕ⅏×ℕ⅏ by ℕ⅏ is _complete_
in the sense that
for each ⟨i,j⟩ ∈ ℕ⅏×ℕ⅏
there is an action
"place k on ⟨i,j⟩" ∈ Indexing
with k ∈ ℕ⅏

> Describing the method of indexing is a task.

Meaning:
The required set of actions
"say blah-blah about set Indexing"
has a finite order:
first, last, last-befores, first-afters.

> This holds for all of Cantor's enumerations
> as well as for my matrices.
>
> If completeness is accomplished,
> then it is accomplished in both cases.

There is no finite order of
actions "place k on ⟨i,j⟩"

There are only finite orders of
actions "say blah-blah about set Indexing"

We can accomplish the second.
We cannot accomplish the first.

However, we can _reason about_ the first,
starting with accomplishing the second,
then augmenting with not-first-false claims.

On 14.11.2023 21:44, Jim Burns wrote:

>
> However, we can _reason about_ the first,
> starting with accomplishing the second,
> then augmenting with not-first-false claims.
>
>
Can you reason about the question in the OP? What is the difference
between a supertask and enumerating the algebraic numbers?

Regards, WM

On 11/14/2023 5:15 PM, WM wrote:
> On 14.11.2023 21:44, Jim Burns wrote:

>> However, we can _reason about_ the first,
>> starting with accomplishing the second,
>> then augmenting with not-first-false claims.
>
> Can you reason about the question in the OP?
> What is the difference between a supertask and
> enumerating the algebraic numbers?

There is no finite order of
actions "place k on ⟨i,j⟩"

There are only finite orders of
actions "say blah-blah about set Indexing"

We can accomplish the second.
We cannot accomplish the first.

The first is a supertask.
The second is a task.

On 14.11.2023 23:57, Jim Burns wrote:
> On 11/14/2023 5:15 PM, WM wrote:

>> What is the difference between a supertask and
>> enumerating the algebraic numbers?
>
> The first is a supertask.
> The second is a task.

Why is enumerating the algebraic numbers only a task?
Why is the game of billiards more than a task?

Regards, WM

On 11/15/2023 5:10 AM, WM wrote:
> On 14.11.2023 23:57, Jim Burns wrote:
>> On 11/14/2023 5:15 PM, WM wrote:

>>> What is the difference between a supertask and
>>> enumerating the algebraic numbers?
>>
>> The first is a supertask.
>> The second is a task.
>
> Why is enumerating the algebraic numbers
> only a task?
> Why is the game of billiards
> more than a task?

If the states of a task are to be accomplished,
there needs to be
a first state and a last state,
and there needs to be,
for each split F ᣔ<ᣔ H of the task,
states φᵢ‖φᵢ₊₁ last‖first in F‖H with
an action φᵢ→φᵢ₊₁ across split F ᣔ<ᣔ H

The linguistic actions which describe
enumerating the algebraic numbers or
playing your game of billiards have all that,
first, last, last-befores, first-afters.

The indexing actions which are described as
enumerating the algebraic numbers or
playing your game of billiards
_cannot_ have all that.

We know that they can't have all that by
augmenting the descriptions not-first-false-ly.

It is a task to describe either of them.
It is not a task to perform either of them.

On Tuesday, November 14, 2023 at 11:15:05 PM UTC+1, WM wrote:

> What is the difference between a supertask and enumerating the algebraic numbers?

1. Spielt bei Supertasks die ZEIT eine Rolle, Mückenheim

"a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time" (Wikipedia)

2. Geht "enumerating the algebraic numbers" nicht so vonstatten, dass man Zahl für Zahl von Hand durchnummeriert, Mückenheim.

On 15.11.2023 20:09, Fritz Feldhase wrote:
> On Tuesday, November 14, 2023 at 11:15:05 PM UTC+1, WM wrote:
>
>> What is the difference between a supertask and enumerating the algebraic numbers?
>
> 1. Spielt bei Supertasks die ZEIT eine Rolle

That is so in fact. But Dedekind and Cantor claim that all algebraics
can be enumerated within less than a lifetime.
>
> "a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time" (Wikipedia)
>
> 2. Geht "enumerating the algebraic numbers" nicht so vonstatten, dass man Zahl für Zahl von Hand durchnummeriert,

Doch, genau das ist der Fall, wenn man die Supertask vollständig
durchführen will. Begnügt man sich allerdings mit der Beschreibung wie
hier: "But when ordering the polynomials by their height H, i.e., the sum of
their degree n and all absolute values of their coefficients a_nu
H = n + |a0| + |a1| + |a2| + ... + |an|
then for every height H there is a finite number of polynomials. Every
polynomial gets its place in the enumeration, and since every polynomial
of degree n has at most n different roots, every root can be inserted
into a sequence containing all of them. So the set of roots of all
polynomials is countable. It is the set A of all algebraic numbers." [R.
Dedekind, private note (29 Nov 1873). Cantor, p. 116)]

Dann genügt auch meine Beschreibung:

Push the natnumbers of the first column into the field of fractions and
store the hit fraction always there where the natnumber has come from.
Try to push the natnumbers such that all matrix positions are occupied
by them. That is best done by creating a pattern like

1, 2, 4, ...
3, 5, 8, ...
6, 9, 13, ...
.... ,

According to this simple rule it is impossible, in eternity, to remove a
fraction from the matrix or to attach a natnumber to a fraction.

Findest Du da noch etwas zu unterscheiden? Streng Dich an. Dein Weltbild
bricht sonst zusammen.

Regards, WM

On 15.11.2023 19:47, Jim Burns wrote:
> On 11/15/2023 5:10 AM, WM wrote:

>> Why is enumerating the algebraic numbers
>> only a task?
>> Why is the game of billiards
>> more than a task?
>
> The linguistic actions which describe
> enumerating the algebraic numbers  or
> playing your game of billiards have all that,
> first, last, last-befores, first-afters.
>
> The indexing actions which are described as
> enumerating the algebraic numbers  or
> playing your game of billiards
> _cannot_ have all that.
>
> It is a task to describe either of them.
> It is not a task to perform either of them.

So there is no difference. If all algebraics are enumerated, then my
game is completed too. "All algebraics are indexed" is corresponding to
my matrix A: all indices are applied.

Regards, WM

On 11/16/2023 3:35 AM, WM wrote:
> On 15.11.2023 19:47, Jim Burns wrote:
>> On 11/15/2023 5:10 AM, WM wrote:

>>> Why is enumerating the algebraic numbers
>>> only a task?
>>> Why is the game of billiards
>>> more than a task?
>>
>> The linguistic actions which describe
>> enumerating the algebraic numbers  or
>> playing your game of billiards have all that,
>> first, last, last-befores, first-afters.
>>
>> The indexing actions which are described as
>> enumerating the algebraic numbers  or
>> playing your game of billiards
>> _cannot_ have all that.
>>
>> It is a task to describe either of them.
>> It is not a task to perform either of them.
>
> So there is no difference.

There is a difference.
It's a task to describe not-a-task.

> If all algebraics are enumerated,
> then my game is completed too.

It's a task to describe your not-a-task game.

Your not-a-task game
cannot be ordered with
first and last and, for each split,
last-before and first-after.

Your game's description
can be ordered with
first and last and, for each split,
last-before and first-after.

It's a task to describe
not-a-task "k to ⟨iₖ,jₖ⟩" such that
sₖ = ⌈½+(2⋅k+¼)¹ᐟ²⌉
iₖ = k-(sₖ-1)(sₖ-2)/2
jₖ = sₖ-iₖ
sᵢⱼₖ = iₖ+jₖ
kᵢⱼₖ = iₖ+(sᵢⱼₖ-1)(sᵢⱼₖ-2)/2
k = kᵢⱼₖ

Done describing.
There are first and last symbols, and,
for each split,
last-before and first-after symbols.

For the k and for the ⟨iₖ,jₖ⟩
no order exists, dark or lit, such that
first and last and, for each split,
last-before and first-after all exist.

> "All algebraics are indexed"
> is corresponding to my matrix A:
> all indices are applied.

"All ⟨iₖ,jₖ⟩ are indexed"
is not a task.
Describing "All ⟨iₖ,jₖ⟩ are indexed"
is a task.

On 16.11.2023 13:05, Jim Burns wrote:
> On 11/16/2023 3:35 AM, WM wrote:

>>> It is a task to describe either of them.
>>> It is not a task to perform either of them.
>>
>> So there is no difference.
>
> There is a difference.

All indices are applied to matrix positions.
All indices are applied to roots of polynomials.

What is the difference?

Regards, WM

On 11/16/2023 9:23 AM, WM wrote:
> On 16.11.2023 13:05, Jim Burns wrote:
>> On 11/16/2023 3:35 AM, WM wrote:

>>>> It is a task to describe either of them.
>>>> It is not a task to perform either of them.
>>>
>>> So there is no difference.
>>
>> There is a difference.
>
> All indices are applied to matrix positions.
> All indices are applied to roots of polynomials.
>
> What is the difference?

"All indices are applied to matrix positions"
is not a task.

You (WM) reason from a description of it
as though it is a task.
Your dark elements are at the far end
you imagine but does not exist.

Describing
"All indices are applied to matrix positions"
is a task,
whether or not
"All indices are applied to matrix positions"
is a task.
Describing and augmenting not-first-falsely
is how we know what we know about
"All indices are applied to matrix positions".
How we know it isn't a task, for example.
Describing and augmenting is a task.

"All indices are applied to roots of polynomials"
is not a task.

Second verse, same as the first.

On 16.11.2023 18:03, Jim Burns wrote:
> On 11/16/2023 9:23 AM, WM wrote:

>>> There is a difference.
>>
>> All indices are applied to matrix positions.
>> All indices are applied to roots of polynomials.
>>
>> What is the difference?
>
> "All indices are applied to matrix positions"
> is not a task.
>
> "All indices are applied to roots of polynomials"
> is not a task.

Call it as you like. Above you claimed that there is a difference. What
is it?

Regards, WM

On 11/16/2023 1:11 PM, WM wrote:
> On 16.11.2023 18:03, Jim Burns wrote:
>> On 11/16/2023 9:23 AM, WM wrote:

>>>>>> It is a task to describe either of them.
>>>>>> It is not a task to perform either of them.

>>>> There is a difference.
>>>
>>> All indices are applied to matrix positions.
>>> All indices are applied to roots of polynomials.
>>>
>>> What is the difference?
>>
>> "All indices are applied to matrix positions"
>> is not a task.
>>
>> "All indices are applied to roots of polynomials"
>> is not a task.
>
> Call it as you like.

The difference between a task and a supertask
is what you (WM) have been objecting to.

A supertask can disappear Bob.
A task can't.

"The swaps _up to_ a swap" is a task.
"All the swaps" is a supertask.

> Above you claimed that there is a difference.
> What is it?

Using finitely-many finite-length claims
to describe a supertask is not itself a supertask.
It is a task.

On 16.11.2023 19:55, Jim Burns wrote:
> On 11/16/2023 1:11 PM, WM wrote:
>> On 16.11.2023 18:03, Jim Burns wrote:
>>> On 11/16/2023 9:23 AM, WM wrote:
>
>>>>>>> It is a task to describe either of them.
>>>>>>> It is not a task to perform either of them.
>
>>>>> There is a difference.
>>>>
>>>> All indices are applied to matrix positions.
>>>> All indices are applied to roots of polynomials.
>>>>
>>>> What is the difference?
>>>
>>> "All indices are applied to matrix positions"
>>> is not a task.
>>>
>>> "All indices are applied to roots of polynomials"
>>> is not a task.
>>
>> Call it as you like.
>
> The difference between a task and a supertask
> is what you (WM) have been objecting to.
>
> A supertask can disappear Bob.

Spare your nonsense.

>> Above you claimed that there is a difference.
>> What is it?
>
> Using finitely-many finite-length claims
> to describe a supertask is not itself a supertask.
> It is a task.

Maybe. I asked for the difference betweenAll indices are applied to
matrix positions.
andAll indices are applied to roots of polynomials.

Regards, WM

On 11/16/2023 2:25 PM, WM wrote:
> On 16.11.2023 19:55, Jim Burns wrote:
>> On 11/16/2023 1:11 PM, WM wrote:
>>> On 16.11.2023 18:03, Jim Burns wrote:
>>>> On 11/16/2023 9:23 AM, WM wrote:

>>>>>>>> It is a task to describe either of them.
>>>>>>>> It is not a task to perform either of them.
>>
>>>>>> There is a difference.
>>>>>
>>>>> All indices are applied to matrix positions.
>>>>> All indices are applied to roots of polynomials.
>>>>>
>>>>> What is the difference?
>>>>
>>>> "All indices are applied to matrix positions"
>>>> is not a task.
>>>>
>>>> "All indices are applied to roots of polynomials"
>>>> is not a task.
>>>
>>> Call it as you like.
>>
>> The difference between a task and a supertask
>> is what you (WM) have been objecting to.
>>
>> A supertask can disappear Bob.
>
> Spare your nonsense.

All swaps: supertask.
Up to a swap: task.

Your objection is to
supertasks not acting like tasks.

To which I wish I could say:
"Better late than never".
YES.
That is the point of Hilbert's Hotel et al.
task ≠ supertask.

However,
you get that result, task ≠ supertask,
and apply the axiom "Mueckenheim is never wrong"
in order to prove dark numbers, somehow.

>>> Above you claimed that there is a difference.
>>> What is it?
>>
>> Using finitely-many finite-length claims
>> to describe a supertask is not itself a supertask.
>> It is a task.
>
> Maybe.

Why "maybe"?
Are some finite sequences of finite-length claims
not tasks?

> I asked for the difference between
> All indices are applied to matrix positions.
> and
> All indices are applied to roots of polynomials.

You asked about the difference I claimed above.
I have told you what the difference
I claimed above is. Repeatedly.
It didn't take, apparently.

Indexing the matrix is a supertask.
Describing "indexing the matrix" is a task.

Notably, we can use a task
to learn about a supertask.

On 16.11.2023 21:01, Jim Burns wrote:
> On 11/16/2023 2:25 PM, WM wrote:

>> I asked for the difference between
>> All indices are applied to matrix positions.
>> and
>> All indices are applied to roots of polynomials.
>
> You asked about the difference I claimed above.
> I have told you what the difference
> I claimed above is. Repeatedly.
> It didn't take, apparently.
>
> Indexing the matrix is a supertask.
> Describing "indexing the matrix" is a task.

Yes I have understood that. Here I apply that knowledge:

Indexing all algebraics is a supertask. Describing "indexing all
algebraics" is a task.

Correct? My question however is: Why can all algebraics be indexed or be
regarded as indexed, but all matrix positions cannot be indexed or be
regarded as indexed?

Regards, WM