måndag 31 oktober 2022 kl. 16:56:06 UTC+1 skrev WM:
> Gus Gassmann schrieb am Montag, 31. Oktober 2022 um 15:12:11 UTC+1:
> > On Monday, 31 October 2022 at 10:49:42 UTC-3, WM wrote:
> > > Gus Gassmann schrieb am Sonntag, 30. Oktober 2022 um 11:43:55 UTC+1:
> > > > On Sunday, 30 October 2022 at 06:21:39 UTC-3, WM wrote:
> > > > [...]
> > > > > A completed set of size ℵo cannot consist of natural numbers each of which has ℵo successors.
> > > > I am continually astonished by the ease with which you mess up quantifiers by switching them without justification.
> > > Take all natural numbers with ℵo successors and collect them. Note that there is no quantifier swap, but you simply have to take one by one every natnumber with ℵo successors. Is that possible? Even for a brain severely damaged by set theory?
> > Yes, of course it is possible. Every natural number has ℵo successors, so the collection you are after is, of course, |N.
> No, there are ℵo numbers missing by definiton up to the last number collected by you.
> > So it obviously does exist. What you seem to want is to construe the "collection" as a stepwise process, which is your bullshit approach to infinity.
> It is Cantor's approach. For the bijection of |N and Q and the more for |N alone he states:
> Die so definirte unendliche Reihe hat nun das merkwürdige an sich, sämmtliche positiven rationalen Zahlen und jede von ihnen nur einmal an einer bestimmten Stelle zu enthalten. [Cantor an Lipschitz (19 Nov 1883)]
>
> an einer bestimmten Stelle zu enthalten.
> an einer bestimmten Stelle
> einer bestimmten Stelle
> bestimmten Stelle
>
> What step could not be chosen for checking?
> > *INFINITY DOES NOT WORK THIS WAY*!
> Cantor's infinity does. Otherwise no bijection was possible. In particular no limits are involved.
> "Es lassen sich alsdann die Zahlen des Inbegriffes (ω), d. h. sämtliche algebraischen reellen Zahlen folgendermaßen anordnen: ... wobei keine einzige aus dem Inbegriffe (ω) vergessen ist." [Cantor, p. 116]
> Can you read? Not any singlenumber has been forgotten or confused with others in "the limit".
> > Your other mistake is to try to conclude from "For every n in |N card({m in |N : m > n}) = ℵo"
> Only such numbers are to be collected. That is impossible????
>
> Regards, WM
nothing is missing you retard!

On 10/31/2022 5:04 PM, WM wrote:
> Fritz Feldhase schrieb am Montag,
> 31. Oktober 2022 um 21:00:55 UTC+1:
>> On Monday, October 31, 2022
>> at 8:35:42 PM UTC+1, WM wrote:

>>> How many [natural] numbers must be outside
>>> of [the] collection in order to have
>>> not the set [IN]?
>>
>> One suffices,
>
> Not infinitely many?

One.

If |ℕ\𝐵| = 1 then 𝐵 ≠ ℕ

Extensionality means
𝐵 = ℕ if and only if
𝐵 and ℕ have the same elements,
that each in 𝐵 is in ℕ and
that each in ℕ is in 𝐵

|ℕ\𝐵| = 1 means
there is one element in ℕ not-in 𝐵
so
NOT each in ℕ is in 𝐵
so
NOT 𝐵 = ℕ

Consider
ℕ\{1}\{2}\{3}\{4}\{5}\ ...
ℕ is not
ℕ\{1} or
ℕ\{1}\{2} or
ℕ\{1}\{2}\{3} or
....

Variables do not vary, despite their name,
not within the scope of any particular use.
They are more like pronouns than arrows.

Consider this definition of ℕ
| ∀𝐵 ⊆ ℕ :
| 𝐵 = ℕ ⟺
| 0 ∈ 𝐵 ∧ ∀j ∈ 𝐵 : j⁺⁺ ∈ 𝐵

This part
| 𝐵 = ℕ ⟺
| 0 ∈ 𝐵 ∧ ∀j ∈ 𝐵 : j⁺⁺ ∈ 𝐵
is a claim about one of the subsets
of what being defined, ℕ
We're told that in
| ∀𝐵 ⊆ ℕ :

In order for that to work out to true,
one thing needs to be true of ℕ
and its negation needs to true of
any non-ℕ subset.
𝐵 doesn't change.
𝐵 is ℕ or 𝐵 isn't ℕ
0 ∈ 𝐵 ∧ ∀j ∈ 𝐵 : j⁺⁺ ∈ 𝐵
or
NOT ( 0 ∈ 𝐵 ∧ ∀j ∈ 𝐵 : j⁺⁺ ∈ 𝐵)

----
That's how induction works.
We learn things about a subset of N

0 ∈ {k ∈ ℕ : k ≠ k⁺⁺}

j ∈ {k ∈ ℕ : k ≠ k⁺⁺} ⟹
j⁺⁺ ∈ {k ∈ ℕ : k ≠ k⁺⁺}

....and we know there is only one subset which
{k ∈ ℕ : k ≠ k⁺⁺} can be, ℕ itself.

By definition,
∀j ∈ {k ∈ ℕ : k ≠ k⁺⁺} : j ≠ j⁺⁺
and now we know
{k ∈ ℕ : k ≠ k⁺⁺} = ℕ
so
∀j ∈ ℕ : j ≠ j⁺⁺

It is not an infinite number-check.
Is is a bread-crumb-trail from a definition
| ∀𝐵 ⊆ ℕ :
| 𝐵 = ℕ ⟺
| 0 ∈ 𝐵 ∧ ∀j ∈ 𝐵 : j⁺⁺ ∈ 𝐵

On 10/31/2022 3:17 PM, WM wrote:
> Jim Burns schrieb am Montag,
> 31. Oktober 2022 um 17:25:53 UTC+1:

>> Suppose I am proving the Pythagorean theorem.
>> I say
>> "All triangles have three corners"
>> And you say, what?
>> "No"?
>
> Not all triangles are defined and
> have defined corners.

Being defined and having defined corners
are not essential properties of triangles.

Having three corners is
an essential property of triangles.

>>> Subtracting all successors from ℕ leaves nothing:
>>>
>>> ℕ \ {1, 2, 3, ...} = { }
>>> ℕ \ {1} \ {2} \ {3} \ ... = { }
>>> Or, since in ZF everything is a set
>>> ℕ \ 1 \ 2 \ 3 \ ... = { }
>>> Nothing remains.
>
>> All successors have successors.
>
> ℕ \ 1 \ 2 \ 3 \ ... = { }.
> Can you read this line?

I'm guessing you think you've shown
a counter-example.

A counter-example to
| All successors have successors.
would be i,j,k where
k = j⁺⁺ = i⁺⁺⁺⁺
where i exists, j exists, but k does not exist.

What are your counter-example's i,j,k ?

>> No successors are 0
>> No successors are the same.
>>
>>>> Each natural number has |ℕ| successors.
>>>
>>> That is nonsense.
>>> If infinity is actual, then |ℕ| =/= |ℕ| - 1.
>>
>> WM-actual infinity is where there are
>> objects not equal to themselves, correct?
>
> Not *known* to be equal to anything.

_What we're talking about_
when we talk about '=', has axioms
x = y ∧ P(x,x) ⟹ P(x,y)
and
x = x

The first allows us to substitute equal things.
The second says each thing equals itself.

Anything you _call_ '=' which doesn't have
these axioms _isn't_ '='

>>>> | Assume otherwise.
>>>> | Assume some m ∈ ℕ such that
>>>> | m does not have |ℕ| successors.
>>>
>>> You cannot define it,
>>> but you can treat it collectively, see above.
>>
>> Being able to be defined is not an essential
>> property of m
>
> But every FISON-end is defined.

Being defined is not an essential property
of a FISON-end.
Ending a FISON is an essential property
of a FISON-end.

From ending a FISON, it arises that
the FISON-ends which follow
can match all FISON-ends.

Each FISON-end k uniquely matches k⁺⁺
its immediate successor.
This is part of the definition of '⁺⁺'

In this way, with matches ⟨k k⁺⁺⟩
all the FISON-ends after j CAN match
all the FISON-ends after j⁺⁺

For each FISON-end m
the FISON-ends after m CAN match all FISON-ends.
Otherwise, there are contradictions.

| Assume otherwise.
| Assume the FISON-ends after m
| CANNOT match all FISON-ends.
| | m is a FISON-end.
| There is a _first_ m₁ between 0 and m
| such that
| the FISON-ends after m₁
| CANNOT match all FISON-ends,
| BUT
| the FISON-ends after m₁⁻⁻
| CAN match all FISON-ends.
| | However,
| with matches ⟨k k⁺⁺⟩
| all the FISON-ends after m₁⁻⁻ CAN match
| all the FISON-ends after m₁⁻⁻⁺⁺
| | Since, for FISON-ends, m₁⁻⁻⁺⁺ = m₁
| this contradicts that m₁ CANNOT
| Contradiction.

Therefore,
for each FISON-end m
the FISON-ends after m CAN match all FISON-ends.

On 11/1/2022 11:15 AM, Jim Burns wrote:
> On 10/31/2022 3:17 PM, WM wrote:
>> Jim Burns schrieb am Montag,
>> 31. Oktober 2022 um 17:25:53 UTC+1:
>
>>> Suppose I am proving the Pythagorean theorem.
>>> I say
>>> "All triangles have three corners"
>>> And you say, what?
>>> "No"?
>>
>> Not all triangles are defined and
>> have defined corners.
>
> Being defined and having defined corners
> are not essential properties of triangles.
>
> Having three corners is
> an essential property of triangles.
[...]

I can define a triangle folded into a line. Three points on the line,
and the angles sum up to pi.

On 11/1/2022 8:30 PM, Chris M. Thomasson wrote:
> On 11/1/2022 11:15 AM, Jim Burns wrote:

>> Having three corners is
>> an essential property of triangles.

> I can define a triangle folded into a line.
> Three points on the line,
> and the angles sum up to pi.

You can define it that way, but
is it what we mean by "triangle"?

If we added "non-collinear" to "three points"
that would define your pi-angle out of the
triangles.

Should we or shouldn't we do that?
These aren't exactly math or logic questions.
However, it seems to me that we need to
answer them in order to do math or logic.

On Monday, October 31, 2022 at 10:40:17 AM UTC-7, JVR wrote:
> On Monday, October 31, 2022 at 4:56:06 PM UTC+1, WM wrote:
> > Gus Gassmann schrieb am Montag, 31. Oktober 2022 um 15:12:11 UTC+1:
> > > On Monday, 31 October 2022 at 10:49:42 UTC-3, WM wrote:
> > > > Gus Gassmann schrieb am Sonntag, 30. Oktober 2022 um 11:43:55 UTC+1:
> > > > > On Sunday, 30 October 2022 at 06:21:39 UTC-3, WM wrote:
> > > > > [...]
> > > > > > A completed set of size ℵo cannot consist of natural numbers each of which has ℵo successors.
> > > > > I am continually astonished by the ease with which you mess up quantifiers by switching them without justification.
> > > > Take all natural numbers with ℵo successors and collect them. Note that there is no quantifier swap, but you simply have to take one by one every natnumber with ℵo successors. Is that possible? Even for a brain severely damaged by set theory?
> > > Yes, of course it is possible. Every natural number has ℵo successors, so the collection you are after is, of course, |N.
> > No, there are ℵo numbers missing by definiton up to the last number collected by you.
> > > So it obviously does exist. What you seem to want is to construe the "collection" as a stepwise process, which is your bullshit approach to infinity.
> > It is Cantor's approach. For the bijection of |N and Q and the more for |N alone he states:
> > Die so definirte unendliche Reihe hat nun das merkwürdige an sich, sämmtliche positiven rationalen Zahlen und jede von ihnen nur einmal an einer bestimmten Stelle zu enthalten. [Cantor an Lipschitz (19 Nov 1883)]
> >
> > an einer bestimmten Stelle zu enthalten.
> > an einer bestimmten Stelle
> > einer bestimmten Stelle
> > bestimmten Stelle
> >
> > What step could not be chosen for checking?
> > > *INFINITY DOES NOT WORK THIS WAY*!
> > Cantor's infinity does. Otherwise no bijection was possible. In particular no limits are involved.
> > "Es lassen sich alsdann die Zahlen des Inbegriffes (ω), d. h. sämtliche algebraischen reellen Zahlen folgendermaßen anordnen: ... wobei keine einzige aus dem Inbegriffe (ω) vergessen ist." [Cantor, p. 116]
> > Can you read? Not any singlenumber has been forgotten or confused with others in "the limit".
> > > Your other mistake is to try to conclude from "For every n in |N card({m in |N : m > n}) = ℵo"
> > Only such numbers are to be collected. That is impossible????
> >
> > Regards, WM
> You are off the mark by 2 or 3 hundred years.
>
> It isn't Cantor's fault it's Galileo's. Galileo claimed that there are as many
> squares of whole numbers as there are whole numbers. And the rest
> is history. A hundred years later fools like Euler were writing nonsense
> like log(log (lazy 8)) and he didn't even know where the confluence of the
> Havel and the Elbe is. Now Mücke is the only person in all of Bavaria who knows that.
>
> If they had burned Galileo at the stake when they should have, for the unlicensed
> printing of the "Sidereus Nuncius", it would never have come to this. Then Cantor
> and Hilbert and their sycophants would not have had anybody to copy.
>
> But then there wouldn't have been anything left to be discovered by Mücke Mückenheim, the greatest
> Mathematician in all of Mückenhausen. No dark numbers, no grandiose list of everything, no fallacy
> of the binary tree and no 75'000 Usenet contributions. We would all be much poorer for it.

You mean o, Big O, and Theta, or asymptotics, and duBois-Reymond's long line?

Hey don't be writing such lucid, cogent posts and not be expecting compliments eventually.

Jim Burns schrieb am Dienstag, 1. November 2022 um 17:05:12 UTC+1:
> On 10/31/2022 5:04 PM, WM wrote:

> |ℕ\𝐵| = 1 means
> there is one element in ℕ not-in 𝐵
> so
> NOT each in ℕ is in 𝐵
> so
> NOT 𝐵 = ℕ

If only elements with infinitely many successors are collected, then the successors are missing.

> Variables do not vary, despite their name,
> not within the scope of any particular use.
> They are more like pronouns than arrows.

Variables vary. Like the number of people. The values they are taking are constants.

Regards, WM

Jim Burns schrieb am Dienstag, 1. November 2022 um 19:15:26 UTC+1:
> On 10/31/2022 3:17 PM, WM wrote:

> > ℕ \ 1 \ 2 \ 3 \ ... = { }.
> > Can you read this line?
> I'm guessing you think you've shown
> a counter-example.
>
> A counter-example to
> | All successors have successors.

is this. ℕ \ 1 \ 2 \ 3 \ ... = { }.

Regards, WM

On Wednesday, 2 November 2022 at 06:55:50 UTC-3, WM wrote:
> Jim Burns schrieb am Dienstag, 1. November 2022 um 19:15:26 UTC+1:
[...]
> > A counter-example to
> > | All successors have successors.
> is [*NOT*] this. ℕ \ 1 \ 2 \ 3 \ ... = { }.

You are constantly using unjustified quantifier swaps, you fucking moron.

On 11/2/2022 5:53 AM, WM wrote:
> Jim Burns schrieb am Dienstag,
> 1. November 2022 um 17:05:12 UTC+1:
>> On 10/31/2022 5:04 PM, WM wrote:

>> |ℕ\𝐵| = 1 means
>> there is one element in ℕ not-in 𝐵
>> so
>> NOT each in ℕ is in 𝐵
>> so
>> NOT 𝐵 = ℕ
>
> If
> only elements with infinitely many
> successors are collected,
> then
> the successors are missing.

No.
Any element j of ℕ with only finitely
many eventual-successors in ℕ
has eventual-successors missing from ℕ

Let FISON 𝐹⌞⋮⌝ contain all of those
eventual-successors of j which are in ℕ

𝐹⌝ = k the end of 𝐹 either
is an eventual-successor of j and in ℕ, or
k is NOT an eventual-successor of j and in ℕ

(case 1)
𝐹⌝ = k the end of 𝐹
is an eventual-successor of j and in ℕ

𝐹⌝⁺⁺ = k⁺⁺ is not in 𝐹 (lemma)
Thus,
k⁺⁺ is either not an eventual-successor of j
or k⁺⁺ is not in ℕ

| If k is an eventual-successor of j
| then k⁺⁺ is an eventual-successor of j

k⁺⁺ is an eventual-successor of j
k⁺⁺ is not in ℕ
An eventual-successor of j is missing from ℕ

(case 2)
𝐹⌝ = k the end of 𝐹
is NOT an eventual-successor of j and in ℕ

𝐹 is a FISON.
There is a _first_ k₁ in F such that
k₁ is NOT an eventual-successor of j and in ℕ
and
k₁⁻⁻ IS an eventual-successor of j and in ℕ

k₁ is either not an eventual-successor of j
or k₁ is not in ℕ

| If k₁⁻⁻ is an eventual-successor of j
| then k₁⁻⁻⁺⁺ is an eventual-successor of j

k₁⁻⁻⁺⁺ is an eventual-successor of j
k₁⁻⁻⁺⁺ is not in ℕ
An eventual-successor of j is missing from ℕ

Therefore,
any element j of ℕ with only finitely
many eventual-successors in ℕ
has eventual-successors missing from ℕ

>> Variables do not vary, despite their name,
>> not within the scope of any particular use.
>> They are more like pronouns than arrows.
>
> Variables vary.

If we vary a variable --
within the scope of a particular use --
we are committing the fallacy of equivocation.

> Like the number of people.
> The values they are taking are constants.

Either
the number of people is 7,985,359,035,
or
the number of people is not 7,985,359,035.

One or the other, not both.

Consider a particle with trajectory 𝒓(𝑡)

At each point in time 𝑡, the particle has
a single position 𝒓(𝑡).
Its position _at that time_ does not vary.

How we express things which vary, such as
positions of particles, is by pairing
a unique position with each point in time.

On Wednesday, November 2, 2022 at 6:28:00 PM UTC-7, Jim Burns wrote:
> On 11/2/2022 5:53 AM, WM wrote:
> > Jim Burns schrieb am Dienstag,
> > 1. November 2022 um 17:05:12 UTC+1:
> >> On 10/31/2022 5:04 PM, WM wrote:
>
> >> |ℕ\𝐵| = 1 means
> >> there is one element in ℕ not-in 𝐵
> >> so
> >> NOT each in ℕ is in 𝐵
> >> so
> >> NOT 𝐵 = ℕ
> >
> > If
> > only elements with infinitely many
> > successors are collected,
> > then
> > the successors are missing.
> No.
> Any element j of ℕ with only finitely
> many eventual-successors in ℕ
> has eventual-successors missing from ℕ
>
> Let FISON 𝐹⌞⋮⌝ contain all of those
> eventual-successors of j which are in ℕ
>
> 𝐹⌝ = k the end of 𝐹 either
> is an eventual-successor of j and in ℕ, or
> k is NOT an eventual-successor of j and in ℕ
>
> (case 1)
> 𝐹⌝ = k the end of 𝐹
> is an eventual-successor of j and in ℕ
>
> 𝐹⌝⁺⁺ = k⁺⁺ is not in 𝐹 (lemma)
> Thus,
> k⁺⁺ is either not an eventual-successor of j
> or k⁺⁺ is not in ℕ
>
> | If k is an eventual-successor of j
> | then k⁺⁺ is an eventual-successor of j
>
> k⁺⁺ is an eventual-successor of j
> k⁺⁺ is not in ℕ
> An eventual-successor of j is missing from ℕ
>
> (case 2)
> 𝐹⌝ = k the end of 𝐹
> is NOT an eventual-successor of j and in ℕ
>
> 𝐹 is a FISON.
> There is a _first_ k₁ in F such that
> k₁ is NOT an eventual-successor of j and in ℕ
> and
> k₁⁻⁻ IS an eventual-successor of j and in ℕ
>
> k₁ is either not an eventual-successor of j
> or k₁ is not in ℕ
>
> | If k₁⁻⁻ is an eventual-successor of j
> | then k₁⁻⁻⁺⁺ is an eventual-successor of j
>
> k₁⁻⁻⁺⁺ is an eventual-successor of j
> k₁⁻⁻⁺⁺ is not in ℕ
> An eventual-successor of j is missing from ℕ
>
>
> Therefore,
> any element j of ℕ with only finitely
> many eventual-successors in ℕ
> has eventual-successors missing from ℕ
> >> Variables do not vary, despite their name,
> >> not within the scope of any particular use.
> >> They are more like pronouns than arrows.
> >
> > Variables vary.
> If we vary a variable --
> within the scope of a particular use --
> we are committing the fallacy of equivocation.
> > Like the number of people.
> > The values they are taking are constants.
> Either
> the number of people is 7,985,359,035,
> or
> the number of people is not 7,985,359,035.
>
> One or the other, not both.
>
>
> Consider a particle with trajectory 𝒓(𝑡)
>
> At each point in time 𝑡, the particle has
> a single position 𝒓(𝑡).
> Its position _at that time_ does not vary.
>
> How we express things which vary, such as
> positions of particles, is by pairing
> a unique position with each point in time.

I used to describe that as "immediate" vis-a-vis, "deferred",
or, the "eventual", where the event is completion,
of the course-of-passage, which is the literature's term, of the ordinals,
what result a compact ordinal, with an infinite descending chain.

You describe classical motion or d for distance or p for position,
usual terms, but our continuum mechanics, has particle/wave duality,
we needn't get into that, but it's just as well present the ordinal's,
as the integer part of the reals in a model, the course-of-passage
according to classical motion, through the non-integer part.

From zero to one, for the set of instances, for as often as your clock
can stop, it's well-ordered.

So: well-order the reals.

Of course there's about five thousand posts where I've put
"well-order the reals", and done so.

And this is one right here.

Gus Gassmann schrieb am Mittwoch, 2. November 2022 um 12:16:37 UTC+1:
> On Wednesday, 2 November 2022 at 06:55:50 UTC-3, WM wrote:
> > Jim Burns schrieb am Dienstag, 1. November 2022 um 19:15:26 UTC+1:
> [...]
> > > A counter-example to
> > > | All successors have successors.
> > is [*NOT*] this. ℕ \ 1 \ 2 \ 3 \ ... = { }.
>
> You are constantly using unjustified quantifier swaps

No, I use only justified steps. Try to understand them.

The set ℕ of all natural numbers can be exhausted by subtracting collectively all its elements ℕ \ {1, 2, 3, ...} = Ø.
But it cannot be exhausted by subtracting only individual elements that have names (like 7 or any fixed substitute for n).
∀n ∈ ℕ: ℕ \ {1, 2, 3, ..., n} =/= Ø.

Almost all natural numbers are undefinable.
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo
|{1, 2, 3, ..., n}| < ℵo.

Regards, WM

Jim Burns schrieb am Donnerstag, 3. November 2022 um 02:28:00 UTC+1:
> On 11/2/2022 5:53 AM, WM wrote:

> > If
> > only elements with infinitely many
> > successors are collected,
> > then
> > the successors are missing.
> No.

Yes. The successors are not collected. This is how we proceed. You cannot hinder that procedure. If you cannot understand then there is no reason to continue this discussion.

> Any element j of ℕ with only finitely
> many eventual-successors in ℕ
> has eventual-successors missing from ℕ

They are missing from ℕ_def. They are dark. Their successors and their number of successors cannot be determined.

Regards, WM

Jim Burns schrieb am Donnerstag, 3. November 2022 um 02:28:00 UTC+1:
> On 11/2/2022 5:53 AM, WM wrote:

> > Variables vary.
> If we vary a variable --
> within the scope of a particular use --
> we are committing the fallacy of equivocation.

No, that's their character.

> > Like the number of people.
> > The values they are taking are constants.
> Either
> the number of people is 7,985,359,035,
> or
> the number of people is not 7,985,359,035.
>
> One or the other, not both.

The number of people can be a function of time.
>
> Consider a particle with trajectory 𝒓(𝑡)
>
> At each point in time 𝑡, the particle has
> a single position 𝒓(𝑡).
> Its position _at that time_ does not vary.

But its position during the movement does vary.
>
> How we express things which vary, such as
> positions of particles, is by pairing
> a unique position with each point in time.

The number of people has two meanings. In mathematics many words have more than one meaning.

Regards, WM

On Thursday, 3 November 2022 at 07:57:56 UTC-3, WM wrote:
> Gus Gassmann schrieb am Mittwoch, 2. November 2022 um 12:16:37 UTC+1:
> > On Wednesday, 2 November 2022 at 06:55:50 UTC-3, WM wrote:
> > > Jim Burns schrieb am Dienstag, 1. November 2022 um 19:15:26 UTC+1:
> > [...]
> > > > A counter-example to
> > > > | All successors have successors.
> > > is [*NOT*] this. ℕ \ 1 \ 2 \ 3 \ ... = { }.
> >
> > You are constantly using unjustified quantifier swaps
> No, I use only justified steps. Try to understand them.
>
> The set ℕ of all natural numbers can be exhausted by subtracting collectively all its elements ℕ \ {1, 2, 3, ...} = Ø.
> But it cannot be exhausted by subtracting only individual elements that have names (like 7 or any fixed substitute for n).

Yes, so? No finite set is infinite. No process of finite length has infinitely many steps. Big deal! You are swapping quantifiers left and write and you are too fucking stupid to recognize it. There is no sense in pursuing this further. EOD

On Thursday, November 3, 2022 at 12:21:59 PM UTC+1, Gus Gassmann wrote:

> There is no sense in pursuing this further. EOD

"Die Botschaft hör ich wohl, allein mir fehlt der Glaube." (Johann Wolfgang von Goethe)

On Thursday, November 3, 2022 at 12:03:49 PM UTC+1, WM wrote:

> If you cannot understand then there is no reason to continue this discussion.

Actually, there never has been a reason (sic!) to "continue this discussion".

:-)

On Thursday, November 3, 2022 at 12:16:27 PM UTC+1, WM wrote:

> In [muckenmath] many words have more than one meaning.

Indeed! Actually, most of them have potentially infinitely many meanings.

On 11/3/2022 7:03 AM, WM wrote:
> Jim Burns schrieb am Donnerstag,
> 3. November 2022 um 02:28:00 UTC+1:
>> On 11/2/2022 5:53 AM, WM wrote:

>>> If
>>> only elements with infinitely many
>>> successors are collected,
>>> then
>>> the successors are missing.
>>
>> No.
>
> Yes.
> The successors are not collected.

Anything with fewer than all eventual-successors
is not ℕ

Let j ∈ ℕ and
let 𝐸(j) be the set of all eventual-successors of j

If k is an eventual-successor of j
then k⁺⁺ is an eventual-successor of j
k ∈ 𝐸(j) ⟹ k⁺⁺ ∈ 𝐸(j)

| Assume that FISON 𝐹 exists such that
| all of 𝐸(j) which is in ℕ is in FISON 𝐹
| 𝐸(j)∩ℕ ⊆ 𝐹₀⋮⌝
| | m is last in 𝐹 𝐹⌝ = m
| m⁺⁺ ∉ 𝐹
| Let 𝐹⁺⁺ be the FISON immediately after 𝐹
| 𝐹⁺⁺ = 𝐹⊕⟨m⁺⁺⟩
| | m⁺⁺ ∉ 𝐹
| m⁺⁺ ∉ 𝐸(j)∩ℕ
| Also
| j⁺⁺ ∈ 𝐸(j)∩ℕ
| | Because 𝐹⁺⁺ is a FISON
| k,k⁺⁺ exist between j⁺⁺ and m⁺⁺
| such that
| k ∈ 𝐸(j)∩ℕ and k⁺⁺ ∉ 𝐸(j)∩ℕ
| | k ∈ 𝐸(j)∩ℕ
| k ∈ 𝐸(j)
| k ∈ 𝐸(j) ⟹ k⁺⁺ ∈ 𝐸(j)
| k⁺⁺ ∈ 𝐸(j)
| | k⁺⁺ ∉ 𝐸(j)∩ℕ
| k⁺⁺ ∉ 𝐸(j) ∨ k⁺⁺ ∉ ℕ
| k⁺⁺ ∈ 𝐸(j)
| k⁺⁺ ∉ ℕ
| | k⁺⁺ ∈ 𝐸(j) ∧ k⁺⁺ ∉ ℕ
| 𝐸(j) ⊈ ℕ

Therefore,
if FISON 𝐹 exists such that 𝐸(j)∩ℕ ⊆ 𝐹₀⋮⌝
then 𝐸(j) ⊈ ℕ

>>> If
>>> only elements with infinitely many
>>> successors are collected,
>>> then
>>> the successors are missing.

No.
If
any element j has only finitely-many
eventual-successors in ℕ
𝐸(j)∩ℕ ⊆ 𝐹₀⋮⌝
then
eventual-successors of j are missing from ℕ
𝐸(j) ⊈ ℕ

> This is how we proceed.

How we proceed is from the knowledge that
a right triangle has three corners.

How we proceed is from the knowledge that
ℕ contains all eventual-successors of its elements.

If you decide that a right triangle can have _four_
corners, that's merely a refusal to talk about
what we're talking about,
even if you call it a right triangle.

If you decide that ℕ can be ℕ without containing
all eventual-successors of its elements,
that's merely a refusal to talk about what we're
talking about,
even if you call it ℕ

> You cannot hinder that procedure.
> If you cannot understand then
> there is no reason to continue this discussion.
>
>> Any element j of ℕ with only finitely
>> many eventual-successors in ℕ
>> has eventual-successors missing from ℕ
>
> They are missing from ℕ_def.
> They are dark.
> Their successors and their number of successors
> cannot be determined.

If ℕ is ℕ and j ∈ ℕ
then ℕ contains all eventual-successors of j
All eventual-successors of j are infinitely-many.

On 11/3/2022 3:57 AM, WM wrote:
> Gus Gassmann schrieb am Mittwoch, 2. November 2022 um 12:16:37 UTC+1:
>> On Wednesday, 2 November 2022 at 06:55:50 UTC-3, WM wrote:
>>> Jim Burns schrieb am Dienstag, 1. November 2022 um 19:15:26 UTC+1:
>> [...]
[...]
> Almost all natural numbers are undefinable.

Show me one natural number that cannot be defined... ;^)

On 11/3/2022 7:16 AM, WM wrote:
> Jim Burns schrieb am Donnerstag,
> 3. November 2022 um 02:28:00 UTC+1:
>> On 11/2/2022 5:53 AM, WM wrote:

>>> Variables vary.
>>
>> If we vary a variable --
>> within the scope of a particular use --
>> we are committing the fallacy of equivocation.
>
> No, that's their character.

A variable refers to _one_ of those which we are
describing and reasoning about.
We start with, and stay with, and end with
only claims about that _one_ which we know are
true, even without knowing _which_ one.

We can claim about a natural number that
it has a successor,
its successor is not 0,
its successor is not the successor of
any other natural number,
all without know _which_ natural number
"it" refers to.

That claim and a few other claims describe
_one_ natural number. We know they're true claims
because we know what we mean by "natural number".

If we extend those claims only in truth-preserving
way, we reason finitely about infinitely-many --
which, I have to say, is a pretty neat trick.

>>> Like the number of people.
>>> The values they are taking are constants.
>>
>> Either
>> the number of people is 7,985,359,035,
>> or
>> the number of people is not 7,985,359,035.
>> One or the other, not both.
>
> The number of people can be a function of time.

The number of people on the planet is a function.
The number of people in Augsburg is a function.
We do not permit one to morph into the other
when we discuss the number of people as a function
of time. That would be committing the fallacy of
equivocation.

>> Consider a particle with trajectory 𝒓(𝑡)
>>
>> At each point in time 𝑡, the particle has
>> a single position 𝒓(𝑡).
>> Its position _at that time_ does not vary.
>
> But its position during the movement does vary.

We pair each point of time with a unique
position. If we decide, no, we'll pair a
_different_ position with that point in time,
then that's a different trajectory.

_That trajectory_ doesn't vary how it pairs
time and space.

This allows us to describe a trajectory.
It is continuous. It moves according to these
forces. Physics.

>> How we express things which vary, such as
>> positions of particles, is by pairing
>> a unique position with each point in time.
>
> The number of people has two meanings.

| In logic, equivocation ("calling two different
| things by the same name") is an informal fallacy
| resulting from the use of a particular
| word/expression in multiple senses within
| an argument.
| https://en.wikipedia.org/wiki/Equivocation

Chris M. Thomasson was thinking very hard :
> On 11/3/2022 3:57 AM, WM wrote:
>> Gus Gassmann schrieb am Mittwoch, 2. November 2022 um 12:16:37 UTC+1:
>>> On Wednesday, 2 November 2022 at 06:55:50 UTC-3, WM wrote:
>>>> Jim Burns schrieb am Dienstag, 1. November 2022 um 19:15:26 UTC+1:
>>> [...]
> [...]
>> Almost all natural numbers are undefinable.
>
> Show me one natural number that cannot be defined... ;^)

A last one cannot be defined. :) Rather, a last one is defined as not
existing.

On 11/3/2022 1:38 PM, FromTheRafters wrote:
> Chris M. Thomasson was thinking very hard :
>> On 11/3/2022 3:57 AM, WM wrote:
>>> Gus Gassmann schrieb am Mittwoch, 2. November 2022 um 12:16:37 UTC+1:
>>>> On Wednesday, 2 November 2022 at 06:55:50 UTC-3, WM wrote:
>>>>> Jim Burns schrieb am Dienstag, 1. November 2022 um 19:15:26 UTC+1:
>>>> [...]
>> [...]
>>> Almost all natural numbers are undefinable.
>>
>> Show me one natural number that cannot be defined... ;^)
>
> A last one cannot be defined. :) Rather, a last one is defined as not
> existing.

LOL! It's sort of automatic. Any natural one can think of can one added
to it, 1 the natural unit of the naturals, so to speak. Hey everybody! I
thought of the largest natural. Not so fast! plus 1, its completely
automatic.

https://youtu.be/wgBgwd0ekIk

lol! ;^)

On 11/3/2022 1:35 PM, Jim Burns wrote:
> On 11/3/2022 7:16 AM, WM wrote:
>> Jim Burns schrieb am Donnerstag,
>> 3. November 2022 um 02:28:00 UTC+1:
>>> On 11/2/2022 5:53 AM, WM wrote:
>
>>>> Variables vary.
>>>
>>> If we vary a variable --
>>> within the scope of a particular use --
>>> we are committing the fallacy of equivocation.
>>
>> No, that's their character.
>
> A variable refers to _one_ of those which we are
> describing and reasoning about.
> We start with, and stay with, and end with
> only claims about that _one_ which we know are
> true, even without knowing _which_ one.
>
> We can claim about a natural number that
> it has a successor,
> its successor is not 0,
> its successor is not the successor of
> any other natural number,
> all without know _which_ natural number
> "it" refers to.
>
> That claim and a few other claims describe
> _one_ natural number. We know they're true claims
> because we know what we mean by "natural number".
>
> If we extend those claims only in truth-preserving
> way, we reason finitely about infinitely-many --
> which, I have to say, is a pretty neat trick.
>
>>>> Like the number of people.
>>>> The values they are taking are constants.
>>>
>>> Either
>>> the number of people is 7,985,359,035,
>>> or
>>> the number of people is not 7,985,359,035.
>>> One or the other, not both.
>>
>> The number of people can be a function of time.
>
> The number of people on the planet is a function.
> The number of people in Augsburg is a function.
> We do not permit one to morph into the other
> when we discuss the number of people as a function
> of time. That would be committing the fallacy of
> equivocation.
>
>>> Consider a particle with trajectory 𝒓(𝑡)
>>>
>>> At each point in time 𝑡, the particle has
>>> a single position 𝒓(𝑡).
>>> Its position _at that time_ does not vary.
>>
>> But its position during the movement does vary.
>
> We pair each point of time with a unique
> position. If we decide, no, we'll pair a
> _different_ position with that point in time,
> then that's a different trajectory.
>
> _That trajectory_ doesn't vary how it pairs
> time and space.
>
> This allows us to describe a trajectory.
> It is continuous. It moves according to these
> forces. Physics.
>
>>> How we express things which vary, such as
>>> positions of particles, is by pairing
>>> a unique position with each point in time.
>>
>> The number of people has two meanings.
>
> | In logic, equivocation ("calling two different
> | things by the same name") is an informal fallacy
> | resulting from the use of a particular
> | word/expression in multiple senses within
> | an argument.
> |
> https://en.wikipedia.org/wiki/Equivocation
>
>

Any natural number automatically has a successor that is one natural
unit away such that n + 1 is greater than n. The difference between n +
1 and n is always 1 as in (n + 1) - n = 1.

For instance, take 41.

41 + 1 = 42 and 42 > 41 where the difference is the natural unit of one.

Sound Kosher? Humm...

On Thursday, November 3, 2022 at 1:35:38 PM UTC-7, Jim Burns wrote:
> On 11/3/2022 7:16 AM, WM wrote:
> > Jim Burns schrieb am Donnerstag,
> > 3. November 2022 um 02:28:00 UTC+1:
> >> On 11/2/2022 5:53 AM, WM wrote:
>
> >>> Variables vary.
> >>
> >> If we vary a variable --
> >> within the scope of a particular use --
> >> we are committing the fallacy of equivocation.
> >
> > No, that's their character.
> A variable refers to _one_ of those which we are
> describing and reasoning about.
> We start with, and stay with, and end with
> only claims about that _one_ which we know are
> true, even without knowing _which_ one.
>
> We can claim about a natural number that
> it has a successor,
> its successor is not 0,
> its successor is not the successor of
> any other natural number,
> all without know _which_ natural number
> "it" refers to.
>
> That claim and a few other claims describe
> _one_ natural number. We know they're true claims
> because we know what we mean by "natural number".
>
> If we extend those claims only in truth-preserving
> way, we reason finitely about infinitely-many --
> which, I have to say, is a pretty neat trick.
> >>> Like the number of people.
> >>> The values they are taking are constants.
> >>
> >> Either
> >> the number of people is 7,985,359,035,
> >> or
> >> the number of people is not 7,985,359,035.
> >> One or the other, not both.
> >
> > The number of people can be a function of time.
> The number of people on the planet is a function.
> The number of people in Augsburg is a function.
> We do not permit one to morph into the other
> when we discuss the number of people as a function
> of time. That would be committing the fallacy of
> equivocation.
> >> Consider a particle with trajectory 𝒓(𝑡)
> >>
> >> At each point in time 𝑡, the particle has
> >> a single position 𝒓(𝑡).
> >> Its position _at that time_ does not vary.
> >
> > But its position during the movement does vary.
> We pair each point of time with a unique
> position. If we decide, no, we'll pair a
> _different_ position with that point in time,
> then that's a different trajectory.
>
> _That trajectory_ doesn't vary how it pairs
> time and space.
>
> This allows us to describe a trajectory.
> It is continuous. It moves according to these
> forces. Physics.
> >> How we express things which vary, such as
> >> positions of particles, is by pairing
> >> a unique position with each point in time.
> >
> > The number of people has two meanings.
> | In logic, equivocation ("calling two different
> | things by the same name") is an informal fallacy
> | resulting from the use of a particular
> | word/expression in multiple senses within
> | an argument.
> |
> https://en.wikipedia.org/wiki/Equivocation

Have you had any luck with: "a well-ordering, of the reals"?

I sure have.

There's one of them and it's line-drawing from zero to one,
and it's for after equipping set theory with a broader set theory,
making quite a full circle of "bigger mathematics".

My side of the fence is real tidy.

Also it goes all the way around your side.

Not that it's mine so much, but, I enjoy it.

Also it's a monism and a platonism and never bites itself.

Depeche Mode put it, something about a "Policy of Truth",
but I feel quite comfortable reading one off.

Then Stevie Ray Vaughan said something about a Crossfire,
but, that's your side of the fence.

Now Burns and I like you, at least you're not one of those
"material implication", nuts, I hope. Because they're getting schooled.