On 9/10/2022 11:23 AM, WM wrote:
> Jim Burns schrieb am Freitag,
> 9. September 2022 um 22:30:02 UTC+2:

>> I don't know if this is current WM-dogma,
>> but, at one time, WM did not accept
>> that the empty set ∅ existed.
>
> I am not the only one who is in doubt.

Consider the set ℕ of natural numbers.

We claim that,
if
some predicate P(k) exists
such that
P(k) has a truth value for each k in ℕ,
then
some subset {k ∈ ℕ : P(k)} of ℕ exists
such that
{k ∈ ℕ : P(k)} contains all and only
such k in ℕ for which P(k) is true.

When we make that claim,
we have considered the possibility that
there are no k in ℕ for which P(k) is true.
Perhaps P(k) claims that k is between 2 and 3.

In that case, we would say
{ k ∈ ℕ : 2 < k < 3 } = ∅

Suppose someone (you, for instance) denies
that ∅ exists.
Something that _does not_ arise from
the non-existence of ∅
is the existence of naturals between 2 and 3.

The most that would arise is that we would need
a different way to say the same thing.
~Ej ∈ { k ∈ ℕ : 2 < k < 3 }
or
~Ej ∈ ℕ : 2 < j < 3
or
something else that you _would_
allow yourself to read.

However,
even if you refused to read everything we write
(a plausible possibility, for you)
there will still not be any natural numbers
between 2 and 3.

Denying ∅ doesn't change facts,
only how facts are expressed..

Jim Burns schrieb am Samstag, 10. September 2022 um 20:17:42 UTC+2:

> In that case, we would say
> { k ∈ ℕ : 2 < k < 3 } = ∅
>
> Suppose someone (you, for instance) denies
> that ∅ exists.

I do not deny it. But I am in doubt that the empty set ∅ will grow to something different when we include it in curly brackets. Some aspects support it, others deny it.

> Denying ∅ doesn't change facts,
> only how facts are expressed.

In arithmetic without zero that is similar. There zero has to be replaced by "nothing".

Regards, WM

Fritz Feldhase schrieb am Samstag, 10. September 2022 um 19:14:41 UTC+2:
> On Saturday, September 10, 2022 at 5:23:06 PM UTC+2, WM wrote:
> > Jim Burns schrieb am Freitag, 9. September 2022 um 22:30:02 UTC+2:
> >
> > > I don't know if this is current WM-dogma,
> > > but, at one time, WM did not accept
> > > that the empty set ∅ existed.
> >
> > I am not the only one who is in doubt.
> NO ONE -except you- *is* in doubt,

"I increasingly doubt the justifiability of the 'null set'." [E. Zermelo, letter to A. Fraenkel (9 May 1921)]

Regards, WM

On Saturday, September 10, 2022 at 9:12:25 PM UTC+2, WM wrote:

> I am in doubt that the empty set ∅ will grow to something different when we include it in curly brackets.

Great. After all IT WON'T "grow to something different when we include it in curly brackets".

You know, sets dont't grow or shrink.

But the set containing the empty set as its sole element differs from the empty set (since the empty set dies NOT contain any set as element).

Hint: card({}) = 0 while card({{}}) = 1.

> Some aspects support it, others deny it.

Nonsense,

> In arithmetic without zero that is similar. There zero has to be replaced by "nothing".

You are a complete idiot, Mückenheim!

No, zero is not "nothing".

Gus Gassmann schrieb am Samstag, 10. September 2022 um 17:54:43 UTC+2:
> On Saturday, 10 September 2022 at 12:23:06 UTC-3, WM wrote:
> [...]
> > I think it is only the expression for nothing. Nothing is named the empty set.
> What you call it is ultimately irrelevant. It is important that no natural number is element of the intersection over all end segments.It is equally important (and easier) that there is no natural number that is both even and odd.I call each intersection the empty set, because it does not matter how the intersection is formed. If it contains no element, then it contains no element.

All definable endsegments are infinite and have infinite intersection together.
If there is an empty intersection, then there must be other endegments producing it. The definable endsegments will not be sufficient.

If you claim that no natnumber can remain, then this holds also for the endsegments.

If you claim that only infinite endsegments are existing, then you have to show the first one reducing the infinite set.

Regards, WM

On Saturday, September 10, 2022 at 9:14:33 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Samstag, 10. September 2022 um 19:14:41 UTC+2:
> > On Saturday, September 10, 2022 at 5:23:06 PM UTC+2, WM wrote:
> > > Jim Burns schrieb am Freitag, 9. September 2022 um 22:30:02 UTC+2:
> > > >
> > > > I don't know if this is current WM-dogma, but, at one time, WM did not accept
> > > > that the empty set ∅ existed.
> > > >
> > > I am not the only one who is in doubt.
> > >
> > NO ONE -except you- *is* in doubt,
> >
> "I increasingly doubt the justifiability of the 'null set'." [E. Zermelo, letter to A. Fraenkel (9 May 1921)]

So you think Zermelo *is* in doubt today, 2022? Huh?!

Man, you've really lost ALL your brain cells.

On Saturday, September 10, 2022 at 9:19:19 PM UTC+2, Fritz Feldhase wrote:
> On Saturday, September 10, 2022 at 9:12:25 PM UTC+2, WM wrote:
> >
> > I am in doubt that the empty set ∅ will grow to something different when we include it in curly brackets.
> >
> Great. After all IT WON'T "grow to something different when we include it in curly brackets".
>
> You know, sets dont't grow or shrink.
>
> But the set containing the empty set as its sole element differs from the empty set (since the empty set does NOT contain any set as element).
>
> Hint: card({}) = 0 while card({{}}) = 1.
>>
> > Some aspects support it, others deny it.
> >
> Nonsense.
> >
> > In arithmetic without zero that is similar. There zero has to be replaced by "nothing".
> >
> You are a complete idiot, Mückenheim!
>
> No, zero is not "nothing".

Hint: card({0, 1, 2}) = 3, while card({1, 2}) = 2. Hence 0 is NOT "nothing", dumbo.

Jim Burns schrieb am Samstag, 10. September 2022 um 00:27:51 UTC+2:

> The set ⋃𝓕 of things-in-FISONs is infinite.

Nonsense.

> Suppose that ⋃𝓕 is "not really infinite".

That is proved since all FISONs have infinite endsegments.
That does not change by forming the union.

> >> ∅ is next[2] after all end segments
> >
> > This is my last answer to you,
> > if your stupidity is not aible to
> > recognize your mistake.
> > Only one element can be lost per step.
> > The empty set cannot follow immediately
> > upon infinite sets.
> There is no set 𝐵 closer than ∅

So you deny that ∀k ∈ ℕ: E(k+1) = E(k) \ {k} holds for all natural numbers?

> What would make 𝐵 closer?

The contents of the infinite endsegment is infinite. Part of it is finite. Alas it is impossible to describe it. That proves dark numbers.
>
Regards, WM

On 9/10/2022 3:20 PM, WM wrote:

> All definable endsegments are infinite

Yes.

> and have infinite intersection together.

No.
No set 𝐵 is between ∅ and all infinite
end segments of ⋃𝓕, like ⋃𝓕 and E(i+1)

if
⋃𝓕 ⊃ 𝐵 ⊃ ∅
then
i ∈ 𝐵 -> i ∈ ⋃𝓕
and
not E(i+1) ⊃ 𝐵 ⊃ ∅

> If there is an empty intersection,
> then there must be other endegments producing it.
> The definable endsegments will not be sufficient.

Each element in a (definable) end segment
is not in another (definable) end segment.
That is sufficient for their intersection
to be empty.

> If you claim that only infinite endsegments
> are existing, then you have to show the first one
> reducing the infinite set.

There is no jump from an infinite end segment
to anywhere after all infinite end segments,
because
each 'delete-least' step
which starts at an infinite end segment
ends at an infinite end segment.

On 9/10/2022 2:20 PM, WM wrote:
> Gus Gassmann schrieb am Samstag, 10. September 2022 um 17:54:43 UTC+2:
>> On Saturday, 10 September 2022 at 12:23:06 UTC-3, WM wrote:
>> [...]
>>> I think it is only the expression for nothing. Nothing is named the empty set.
>> What you call it is ultimately irrelevant. It is important that no natural number is element of the intersection over all end segments.It is equally important (and easier) that there is no natural number that is both even and odd.I call each intersection the empty set, because it does not matter how the intersection is formed. If it contains no element, then it contains no element.
>
> All definable endsegments are infinite and have infinite intersection together.

Wrong.

> If there is an empty intersection, then there must be other endegments producing it. The definable endsegments will not be sufficient.

Wrong.

>
> If you claim that no natnumber can remain, then this holds also for the endsegments.

Wrong.

>
> If you claim that only infinite endsegments are existing, then you have to show the first one reducing the infinite set.

Wrong.

>
> Regards, WM
>

100% wrong.

On 9/10/2022 2:12 PM, WM wrote:
> Jim Burns schrieb am Samstag, 10. September 2022 um 20:17:42 UTC+2:
>
>> In that case, we would say
>> { k ∈ ℕ : 2 < k < 3 } = ∅
>>
>> Suppose someone (you, for instance) denies
>> that ∅ exists.
>
> I do not deny it. But I am in doubt that the empty set ∅ will grow to something different when we include it in curly brackets. Some aspects support it, others deny it.

that is your personal opinion, and it is not supported by facts.

>
>> Denying ∅ doesn't change facts,
>> only how facts are expressed.
>
> In arithmetic without zero that is similar. There zero has to be replaced by "nothing".

still wrong. Oversimplify is your problem

>
> Regards, WM

On Saturday, September 10, 2022 at 9:20:41 PM UTC+2, WM wrote:

> If there is an empty intersection, then there must be [...] endegments producing it.

Fascinating, Mückenheim!

Yeah, any "selection" of infinitely many endsegments will "produce" an empty intersection! (For example the set of all endsegments will do!)

How would you know?

Fritz Feldhase schrieb am Samstag, 10. September 2022 um 06:07:40 UTC+2:
> On Friday, September 9, 2022 at 10:10:14 PM UTC+2, WM wrote:
> >
> > The empty set cannot follow immediately upon infinite sets.
>
> The set {E_n : n e IN} u {{}} is certainly ordered the following way (by the subset relation):
>
> E_1 > E_2 > E_3 > ... > {}.

If all endsegments are infinite, then there is a discontinuity, a jump from more than one element to 0 elements. This is wrong because a jump is prohobited by
∀k ∈ ℕ: E(k+1) = E(k) \ {k} .

Regards, WM

Fritz Feldhase schrieb am Samstag, 10. September 2022 um 06:17:39 UTC+2:
> On Saturday, September 10, 2022 at 12:40:53 AM UTC+2, FromTheRafters wrote:
>
> > All natural numbers less than or equal to n cannot be anything else but
> > finite.
> Should read: "The set of all natural numbers less than or equal to n cannot be anything else but finite."

Should read: For all n ∈ ℕ: The set of all natural numbers less than or equal to n cannot be anything else but finite. The set of all definable natural numbers cannot be anything else but finite.

Regards, WM

JVR schrieb am Samstag, 10. September 2022 um 09:40:17 UTC+2:
> In fact, matrices are not really defined.

11 Matrizen
Eine m × n-Matrix ist ein Raster aus m × n Koeffizienten, die in m Zeilen und n Spalten
angeordnet sind:
(aij)1≤i≤m,1≤j≤n (11.1)
Zur Abkürzung werden wir auch kurz (aij) schreiben,

Regards, WM

Gus Gassmann schrieb am Samstag, 10. September 2022 um 17:54:43 UTC+2:
> It is important that no natural number is element of the intersection over all end segments.

What do we conclude from this important fact?
The general construction formula for endsegments
∀k ∈ ℕ: E(k+1) = E(k) \ {k}
shows that no loss of elements can be larger than one element.
This proves the existence of dark endsegments because all visible endsegments are infinite.

Regards, WM

Jim Burns schrieb am Samstag, 10. September 2022 um 22:07:47 UTC+2:

> each 'delete-least' step
> which starts at an infinite end segment
> ends at an infinite end segment.

Each deletion concerns only one elements. If all endsegments were infinite but the deletion leads to the empty set, then the last step (the step that produces the empty set) would be larger than one element.

Regards, WM

On Saturday, September 10, 2022 at 10:33:50 PM UTC+2, WM wrote:
> Fritz Feldhase schrieb am Samstag, 10. September 2022 um 06:07:40 UTC+2:
> > On Friday, September 9, 2022 at 10:10:14 PM UTC+2, WM wrote:
> > >
> > > The empty set cannot follow immediately upon infinite sets.
> > >
> > The set {E_n : n e IN} u {{}} is certainly ordered the following way (by the subset relation):
> >
> > E_1 > E_2 > E_3 > ... > {}.
> >
> If all endsegments are infinite, then there is a discontinuity, a jump from more than one element to 0 elements.

Nope, there is no jump.

Hint: For each and every ednsegment E there is an endsegment E' such that E > E' > {}.

Hence no jump, you silly crank.

On Saturday, September 10, 2022 at 10:57:52 PM UTC+2, WM wrote:

> it the deletion leads to the empty set

There is no "deletion" which "leads to the empty set", Du psychotisches Arschloch.

> then the [...] the step that produces the empty set would <bla>

There is no such step, Du hirnloser Affe!

Fuck off, Mückenheim!

On Saturday, 10 September 2022 at 16:20:41 UTC-3, WM wrote:
> Gus Gassmann schrieb am Samstag, 10. September 2022 um 17:54:43 UTC+2:
> > On Saturday, 10 September 2022 at 12:23:06 UTC-3, WM wrote:
> > [...]
> > > I think it is only the expression for nothing. Nothing is named the empty set.
> > What you call it is ultimately irrelevant. It is important that no natural number is element of the intersection over all end segments.It is equally important (and easier) that there is no natural number that is both even and odd.I call each intersection the empty set, because it does not matter how the intersection is formed. If it contains no element, then it contains no element.
> All definable endsegments are infinite and have infinite intersection together.

All natural numbers are not just definable, but defined. Therefore, all end segments are also defined (as well as definable). Since there are infinitely many of them, there intersection is empty.

The rest of your mind-numbingly stupid bullshit has been removed.

On Saturday, 10 September 2022 at 17:52:23 UTC-3, WM wrote:
> Gus Gassmann schrieb am Samstag, 10. September 2022 um 17:54:43 UTC+2:
> > It is important that no natural number is element of the intersection over all end segments.
> What do we conclude from this important fact?

That the intersection over all end segment is empty, you fucking moron.

I have once more removed the rest of your offensive bullshit.

On 9/10/2022 4:57 PM, WM wrote:
> Jim Burns schrieb am Samstag,
> 10. September 2022 um 22:07:47 UTC+2:

>> each 'delete-least' step
>> which starts at an infinite end segment
>> ends at an infinite end segment.
>
> Each deletion concerns only one elements.

And that determines which sets we are
talking about.

For each end segment,
there is a sequence of 'delete-least' steps
which ends at that end segment ==
for each split of the sequence, there are
end segments 𝐸,𝐸\{i} that jump the split.

For anything else,
there _isn't_ such a sequence.

> If all endsegments were infinite
> but the deletion leads to the empty set,

The deletions do not lead to the intersection
of all end segments (which = ∅)

There is no sequence of 'delete-least' steps
which ends at ∅
==
there is at least one split _without_
end segments 𝐸,𝐸\{i} that jump the split.

We know there is at least one split
without a jump because we can describe one.

We describe a split which,
for each 𝐸 before the split
𝐸\{i} is also before the split.
so there can't be 𝐸,𝐸\{i} with
𝐸 before and 𝐸\{i} after.

> then the last step
> (the step that produces the empty set)
> would be larger than one element.

Thus, in order to include the intersection of
all end segments, we would need to talk about
more than only sets at the end of sequences
of 'delete-least' steps.

WM <askasker48@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)

> Ben Bacarisse schrieb am Freitag, 9. September 2022 um 23:06:26 UTC+2:
>> WM <askas...@gmail.com> writes:
>> (AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
>> Unendlichen" at Hochschule Augsburg.)
>> > It is what we have been discussing in extensio. It is nonsense to
>> > claim the indexing of all fractions if most will be remaining not
>> > indexed forever.
>> My argument was entirely within WMaths. It used a potentially infinite
>> bijection between potentially infinite sets to define a potentially
>> infinite sequence of potentially infinite functions.
>
> There is no problem.

You accept my argument that, in WMaths, "the result" of the swaps
(suitably defined) is no Os in any finite position? I don't think so,
but maybe you have changed you mind.

> But I have been analyzing Cantor's completed infinity with a complete
> matrix of fractions.

But you don't want to analyse the consequences of the swaps in potential
infinity.

>> It then used the
>> appropriate limit as defined in your book to obtain the result. Since
>> everything is potential, the only reasonable meaning for "the result" is
>> the limit produce by the potentially infinite sequence of swaps.
>
> Cantor has explicitly denied the potential infinity.

So much more interesting, then, that the Xs "disappear" in potential
infinity as well! You won't ever see how, but they do.

>> > ∀k ∈ ℕ : ∩{E(1), E(2), ..., E(k+1)} = ∩{E(1), E(2), ..., E(k)} \ {k}
>> >
>> > I assume that
>> > ∀k ∈ ℕ: E(k+1) = E(k) \ {k}
>> > holds for all natural numbers.
>> > Maybe you are not agreeing?
>> You've been saying this for years with no one disagreeing.
>
> Jim Birns has claimed that the empty set is next to the infinite. That
> means he is disagreeing.

His name is Burns. He does not disagree with what you write above. No
one does.

And you keep not answering this question:

In WMaths you have not yet been able to define set membership, equality
and difference (you say the definitions in your textbook are
simplifications for pedagogic reasons). Any progress on that?

I can see that it's obviously embarrassing that you don't know the
definitions of basic set operations in potential infinity, but you
should not keep cutting the question. If there is no progress, just say
so.

>> But, since you prefer to re-phrase my limit example in terms of
>> end segments, here's what you didn't know how to write:
>>
>> Given e_n(i) = [i >= n], then lim_{n->oo} e_n = z where z(i) = 0.
>
> If the limit is zero,

The limit, as per the definition in your textbook, is the zero function
(not zero).

> then before the limit there are finite
> endsegments. Of course not findable, they are dark.

The limit is as per your book. Are there dark objects in WMaths?

>> > The argument proves dark numbers, not individually definable but only
>> > complete in whatever limit.
>> If you were intellectually curious (and courageous), you'd try to
>> understand my argument -- the one you won't look
>
> I know it!

No, the argument about the Xs and Os is still a mystery to you.

> The limit! I have accepted that infinitely many elements
> which are present in every endsegmnet will disappear in the
> limit. Fine.

Great. You spent a lot of time denying that a few years ago. So that's
progress. You also denied that your book gives the method to determine
that lim_{n->oo} e_n = z.

But I think you are saying to contradictory things here: (1) if the
limit exists, there are dark numbers; (2) the limit exists in WMaths,
but WMaths has no dark numbers.

> There are no set limits in my book.

Wrong. Functions are sets (see page 18). Some sequences of functions
converge to a limit function. Examples are given (see page 202).

Why would say something so obviously and demonstrably wrong?

--
Ben.

On 9/10/2022 6:12 PM, Jim Burns wrote:
> On 9/10/2022 4:57 PM, WM wrote:

>> If all endsegments were infinite
>> but the deletion leads to the empty set,
>
> The deletions do not lead to the intersection
> of all end segments (which = ∅)

end segments ==
infinite end segments of UF

Also,
there is no set B, no matter what contents,
between ∅ and
all the infinite end segments of ⋃𝓕

WM <askasker48@gmail.com> writes:
(AKA Dr. Wolfgang Mückenheim or Mueckenheim who teaches "Geschichte des
Unendlichen" at Hochschule Augsburg.)

> Ben Bacarisse schrieb am Freitag, 9. September 2022 um 02:06:41 UTC+2:

>> What positions
>> have Os (you asked about positions originally) "in the limit" depends on
>> the pattern of swaps
>
> The pattern is given by Cantor.

You neither cited nor quoted Cantor saying anything about swaps. I
think you made that bit up yourself, but I would love to read how Cantor
defines the swaps and the result of an endless sequence of swaps.

>> Some sequences of indicator functions have
>> limit I(i) = 1 (i.e. no Os), others don't.
>
> Limits are irrelevant when we ask for indexing with natural numbers,
> i.e. in finite steps before the limit.

You asked about "the result" of an endless sequence of swaps. That can
not be revealed by the indicator function I_n for any n in N. Something
else is needed to make sense of the phrase "the result". I proposed the
limit of the sequence of indicator function. And, behold, with some
patters of swaps, the limit is the function I(i) = 1, meaning no Os in
any position, just as you claim is the case in set theory!

--
Ben.