On 9/1/2021 2:33 PM, WM wrote:
> FromTheRafters schrieb am Mittwoch, 1. September 2021 um 20:56:35 UTC+2:
>> WM pretended :
>>> William schrieb am Mittwoch, 1. September 2021 um 03:13:24 UTC+2:
>>>> On Tuesday, August 31, 2021 at 5:44:38 PM UTC-3, WM wrote:
>>>>> William schrieb am Dienstag, 31. August 2021 um 16:51:07 UTC+2:
>>>>>> On Tuesday, August 31, 2021 at 10:13:33 AM UTC-3, WM wrote:
>>>>>>> William schrieb am Dienstag, 31. August 2021 um 15:08:58 UTC+2:
>>>>>>>> On Tuesday, August 31, 2021 at 9:17:24 AM UTC-3, WM wrote:
>>>>>>>>> William schrieb am Montag, 30. August 2021 um 20:03:10 UTC+2:
>>>>>>>>> A "gap" is something that can contain an unchanging real number (not at
>>>>>>>>> "dark real number" whatever that is).
>>>>>>>>>> A gap is an interval [0,x].
>>>>>>>>> Yes
>>>>>>>> Indeed, and x is an unchanging real number (not a "dark real number"
>>>>>>>> whatever that is).
>>>>>>> x is either the numerical value, then it is unchanging, but the end of
>>>>>>> the interval is changing, or x is the end oft the interval, then it is
>>>>>>> changing (like the largest known prime number).
>>>>>> Piffle.
>>>>>> A "gap" does not change.
>>>>> That would be deplorable for everybody who has gaps in his knowledge. Try
>>>>> it. It can change.
>>>>>
>>>> A "gap" that changes is not capable of containing a real number
>>>
>>> Take the interval [0, 1/n] for n in |N. It changes and contains many real
>>> numbers for every n.
>> It is a sequence of intervals, one for every natural number.
>
> This sequence can be interpreted as the description of a timewise changing interval. Each of these intervals contains real numbers since no definable 1/n is 0.
>
> Regards, WM
>

no, you use the wrong notation. If you want timewise, then use the right notation.
And lose the "definable", it identifies the entire sentence as suspect for bad math.

On 9/1/2021 2:47 PM, WM wrote:
> Jim Burns schrieb am Mittwoch, 1. September 2021 um 18:40:47 UTC+2:
>> On 9/1/2021 8:36 AM, WM wrote:
>
>>> But every finite set of endsegments you can describe has an
>>> infinite intersection. The infinite set has an empty
>>> intersection, but you cannot describe most of its endsegments.
>>> Same with FISONs and points in aline.
>> I can _make a true claim_ about each end segment.
>> Same with FISONs and points in a line.
>> It seems that this is not enough for you.
>> You want some interaction with each of them.
>
> You cannot make a true claim about each endsegment. You only clnclude it from some about which you can make true claims.
>
>>> I describe collectively the number of points in a unit
>>> interval. I find that all unit intervals have same number
>>> of points. You find that all intervals have same number of
>>> points. Therefore, what you call same is not same.
>> What I call same is there is at least one way to match
>> the elements, one-to-one. I don't know how I could have been
>> more clear about that.
>
> There is no way. It would raise a contradiction.
>>
>> Another meaning of "same" is to match one unmatched to the
>> next natural number until there are no more unmatched.
>> In this instance, that cannot be what you are I or anyone
>> means by "same". There is no last point in any interval.
>
> There is a last positive point before zero. But it is dark.

no, there is no "last positive point". We have proved that.

>
> Regards, WM
>

On Wednesday, September 1, 2021 at 9:43:26 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Mittwoch, 1. September 2021 um 16:59:48 UTC+2:
> >
> > Hint: If we are talking about "changing so-and-so" in math its only a "figure of speech". Mathematical objects don't (actually) "change".
> >
> Never heard of fluxions?

Sure, dumbo:

Bishop George Berkeley, a prominent philosopher of the time, slammed Newton's fluxions in his essay The Analyst, published in 1734.Berkeley refused to believe that they were accurate because of the use of the infinitesimal {\displaystyle o}o. He did not believe it could be ignored and pointed out that if it was zero, the consequence would be division by zero. Berkeley referred to them as "ghosts of departed quantities", a statement which unnerved mathematicians of the time and led to the eventual disuse of infinitesimals in calculus.
(Wikipedia)

Mucke, you really are FULL OF SHIT!

William schrieb am Mittwoch, 1. September 2021 um 23:01:49 UTC+2:
> On Wednesday, September 1, 2021 at 4:40:34 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 1. September 2021 um 16:37:00 UTC+2:
>
> > > And what [ every element of the collection of all intervals [0,1/n] ] contains is not a real number as a real number does not change.
> > Every definable interval [0, 1/n] contains real numbers.
> Also every interval [0,1/n] where you cannot write down n contains real numbers.

But they cannot be written to.

> So no interval [0,1/n] is a gap

Then there are numbers that cannot be written down.

> (note there is no element m, of |N_F where [0,1/m] is something that changes) A gap that gets smaller when you want in to is nonsense.

Try to diminish (0, 1/10^10000]. It is possible, but the remaining gap contains aleph_0 dark numbers.

Regards, WM

William schrieb am Mittwoch, 1. September 2021 um 23:07:01 UTC+2:
> On Wednesday, September 1, 2021 at 4:48:02 PM UTC-3, WM wrote:
>
> > There is a last positive point before zero. But it is dark.
> >
> So whatever it is it is not a number of the form 1/n which n an element of |N_F, nor is it a real number ("dark real numbers" whatever they are are not real numbers)

All real numbers which you call real numbers have an infinite number of points between themselves and zero. These points are not what you call real number, because then all could be removed as individuals and nothing would remain.

Regards, WM

Jim Burns schrieb am Donnerstag, 2. September 2021 um 01:35:49 UTC+2:
> On 9/1/2021 3:47 PM, WM wrote:

> > You cannot make a true claim about each endsegment.
> An end segment is an end segment.
> And I can say what an end segment is.

That is the general description.
>
> An end segment is a non-empty set which is closed upwards.

Every endsegment that you can individually define is infinite and has an infinite intersection with all its predecessors. Only endsegments which you cannot individually define, every infinite set for instance, can have an empty intersection.

Regards, WM

On Thursday, September 2, 2021 at 4:04:55 PM UTC-3, WM wrote:
> William schrieb am Mittwoch, 1. September 2021 um 23:01:49 UTC+2:
> > On Wednesday, September 1, 2021 at 4:40:34 PM UTC-3, WM wrote:
> > > William schrieb am Mittwoch, 1. September 2021 um 16:37:00 UTC+2:
> >
> > > > And what [ every element of the collection of all intervals [0,1/n] ] contains is not a real number as a real number does not change.
> > > Every definable interval [0, 1/n] contains real numbers.
> > Also every interval [0,1/n] where you cannot write down n contains real numbers.
> But they cannot be written to.

So What? They are still real numbers and an interval which contains real numbers is not a gap.

> > So no interval [0,1/n] is a gap
> Then there are numbers that cannot be written down.

Indeed some numbers do not have the property "Can be written down". So What?

--
William Hughes

On 9/2/2021 2:08 PM, WM wrote:
> William schrieb am Mittwoch, 1. September 2021 um 23:07:01 UTC+2:
>> On Wednesday, September 1, 2021 at 4:48:02 PM UTC-3, WM wrote:
>>
>>> There is a last positive point before zero. But it is dark.
>>>
>> So whatever it is it is not a number of the form 1/n which n an element of |N_F, nor is it a real number ("dark real numbers" whatever they are are not real numbers)
>
> All real numbers which you call real numbers have an infinite number of points between themselves and zero. These points are not what you call real number, because then all could be removed as individuals and nothing would remain.
>
> Regards, WM
>
>

1. "All real numbers which you call real numbers have an infinite number of points between themselves and zero."

--if by points you mean numbers, that is true. Also between any two real numbers there is an infinite number of numbers.

2. "These points are not what you call real number",

wrong on 3 counts;

--actually a point is a location specified by a number.
--no one backed up a dump truck and unloaded an infinite amount of "point dirt" into your number gap, that is fiction.
--they are real numbers

3. "because then all could be removed as individuals and nothing would remain."

3 questions;

--what fictional construct are you using for removal ?
--how do you know nothing would remain
--can you express your thought in Math ?

On 9/2/2021 3:26 PM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 2. September 2021 um 01:35:49 UTC+2:
>> On 9/1/2021 3:47 PM, WM wrote:

>>> You cannot make a true claim about each endsegment.
>>
>> An end segment is an end segment.
>> And I can say what an end segment is.
>
> That is the general description.

The general description is a true claim about each
end segment.

>> An end segment is a non-empty set which is closed upwards.
>
> Every endsegment that you can individually define

Every end segment that I can individually define
is an end segment.
Every end segment that I cannot individually define
is an end segment.
Whatever "individually define" means.

An end segment is a non-empty set which is closed upwards.
This is a true claim about each end segment.

> Every endsegment that you can individually define is infinite
> and has an infinite intersection with all its predecessors.
> Only endsegments which you cannot individually define,
> every infinite set for instance, can have an empty intersection.

An end segment of FISONable naturals is a set of FISONable
naturals which is non-empty and closed upwards.

k is a FISONable natural iff
a steppable set {0,...,k} of FISONable naturals exists
in which, for all adjacent i,j, j = i+1.

For each FISONable natural k, an end segment E(k+1) of
FISONable naturals exists which k is NOT in.

| Let _Ends_ be an arbitrary collection of end segments of
| FISONable naturals.
| Let INT(Ends) be its intersection.
| | If INT(Ends) is NOT empty,
| | then a FISONable natural m is in INT(Ends),
| | and m is NOT in end segment E(m+1) of FISONable naturals,
| | and INT(Ends) is NOT a subset of E(m+1),
| | and E(m+1) is NOT in the collection Ends,
| | and Ends is NOT the collection of all end segments of
| FISONable naturals.
| | In short,
| if INT(Ends) is NOT empty,
| then Ends is NOT the collection of all end segments of
| FISONable naturals.
| | That's true iff
|| if Ends is the collection of all end segments of
|| FISONable naturals,
|| then INT(Ends) is empty.

tisdag 31 augusti 2021 kl. 14:11:50 UTC+2 skrev WM:
> FromTheRafters schrieb am Dienstag, 31. August 2021 um 00:01:45 UTC+2:
> > WM formulated the question :
> > > William schrieb am Montag, 30. August 2021 um 15:24:04 UTC+2:
> > >> On Monday, August 30, 2021 at 7:52:58 AM UTC-3, WM wrote:
> > >>> William schrieb am Sonntag, 29. August 2021 um 22:35:47 UTC+2:
> > >>>> On Sunday, August 29, 2021 at 4:43:07 PM UTC-3, WM wrote:
>
> > > No. The gap is what you cannot define or write down.
> > So a 'gap' is a mathematical term for something which cannot be
> > defined?
> I confused dark and gap. My argument is correctly stated here:
> Either there is nothing between the smallest defined real number and zero. Then there is a gap. Or there are dark numbers. They fill the gap and make the real line continuous.
>
> Regards, WM
Your terms "Dark" "definable" etc are all meaningless!

On Friday, September 3, 2021 at 7:06:41 AM UTC+2, zelos...@gmail.com wrote:

> Your terms "Dark" "definable" etc are all meaningless!

Yes, but they are quite important for him - in the context of his delusion.

https://en.wikipedia.org/wiki/Delusion

William schrieb am Donnerstag, 2. September 2021 um 21:44:44 UTC+2:
> On Thursday, September 2, 2021 at 4:08:37 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 1. September 2021 um 23:07:01 UTC+2:
> > > On Wednesday, September 1, 2021 at 4:48:02 PM UTC-3, WM wrote:
> > >
> > > > There is a last positive point before zero. But it is dark.
> > > >
> > > So whatever it is it is not a number of the form 1/n which n an element of |N_F, nor is it a real number ("dark real numbers" whatever they are are not real numbers)
> > All real numbers which you call real numbers have an infinite number of
> real numbers between themselves and zero. They are real numbers not "dark real numbers" (whatever they are).

Then they all could be removed as individuals and nothing would remain. But that is impossible.

Regards, WM

William schrieb am Donnerstag, 2. September 2021 um 21:47:04 UTC+2:
> On Thursday, September 2, 2021 at 4:04:55 PM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 1. September 2021 um 23:01:49 UTC+2:
> > > On Wednesday, September 1, 2021 at 4:40:34 PM UTC-3, WM wrote:
> > > > William schrieb am Mittwoch, 1. September 2021 um 16:37:00 UTC+2:
> > >
> > > > > And what [ every element of the collection of all intervals [0,1/n] ] contains is not a real number as a real number does not change.
> > > > Every definable interval [0, 1/n] contains real numbers.
> > > Also every interval [0,1/n] where you cannot write down n contains real numbers.
> > But they cannot be written to.
> So What? They are still real numbers and an interval which contains real numbers is not a gap.

Correct.

> > > So no interval [0,1/n] is a gap
> > Then there are numbers that cannot be written down.
> Indeed some numbers do not have the property "Can be written down".

I call them dark.

Regards, WM

Jim Burns schrieb am Freitag, 3. September 2021 um 02:40:50 UTC+2:

> Every end segment that I can individually define
> is an end segment.

and has an infinite intersection with all other endsegments.

> Every end segment that I cannot individually define of that kind.
> is an end segment.

But it is not necessarily infinite.

Regards, WM

On Friday, September 3, 2021 at 2:19:57 PM UTC-3, WM wrote:
> William schrieb am Donnerstag, 2. September 2021 um 21:47:04 UTC+2:
> > On Thursday, September 2, 2021 at 4:04:55 PM UTC-3, WM wrote:
> > > William schrieb am Mittwoch, 1. September 2021 um 23:01:49 UTC+2:
> > > > On Wednesday, September 1, 2021 at 4:40:34 PM UTC-3, WM wrote:
> > > > > William schrieb am Mittwoch, 1. September 2021 um 16:37:00 UTC+2:
> > > >
> > > > > > And what [ every element of the collection of all intervals [0,1/n] ] contains is not a real number as a real number does not change.
> > > > > Every definable interval [0, 1/n] contains real numbers.
> > > > Also every interval [0,1/n] where you cannot write down n contains real numbers.
> > > But they cannot be written to.
> > So What? They are still real numbers and an interval which contains real numbers is not a gap.
> Correct.
> > > > So no interval [0,1/n] is a gap
> > > Then there are numbers that cannot be written down.
> > Indeed some numbers do not have the property "Can be written down".
> I call them dark.

Nope. To you saying that something is "dark" means a lot more that "Cannot be written down"

--
William Hughes

On 9/3/2021 1:22 PM, WM wrote:
> Jim Burns schrieb
> am Freitag, 3. September 2021 um 02:40:50 UTC+2:

>> Every end segment that I can individually define
>> is an end segment.
>
> and has an infinite intersection with all other endsegments.

| Let _Ends_ be an arbitrary collection of end segments of
| FISONable naturals.
| Let INT(Ends) be its intersection.

| In short,
| if INT(Ends) is NOT empty,
| then Ends is NOT the collection of all end segments of
| FISONable naturals.

>> Every end segment that I cannot individually define of that kind.
>> is an end segment.

Is there a reason you are misquoting me, beyond your desire
to as obnoxious as possible?

JB:
>> Every end segment that I can individually define
>> is an end segment.
>> Every end segment that I cannot individually define
>> is an end segment.
>> Whatever "individually define" means.
>>
>> An end segment is a non-empty set which is closed upwards.
>> This is a true claim about each end segment.

> But it is not necessarily infinite.

An end segment of FISONable naturals is necessarily infinite.

Let E(k) be and end segment of FISONable naturals {k,k+1,...}.
Let {k,...,m} be an initial segment of E(k) ending at m.
m+1 is in E(k) and not in {k,...,m}.
E(k) is infinite.

Also,
| Let _Ends_ be an arbitrary collection of end segments of
| FISONable naturals.
| Let INT(Ends) be its intersection.

| In short,
| if INT(Ends) is NOT empty,
| then Ends is NOT the collection of all end segments of
| FISONable naturals.

William schrieb am Freitag, 3. September 2021 um 19:28:11 UTC+2:
> On Friday, September 3, 2021 at 2:19:57 PM UTC-3, WM wrote:

> > > Indeed some numbers do not have the property "Can be written down".
> > I call them dark.
> Nope. To you saying that something is "dark" means a lot more that "Cannot be written down"

A dark number cannot be identified by any means. It cannot be handled as a single individual. Its neighbours (in case of unit fractions) cannot be found. Do you attach more or less or other properties to numbers which cannot be written down?

Regards, WM

William schrieb am Freitag, 3. September 2021 um 19:34:14 UTC+2:
> On Friday, September 3, 2021 at 2:14:17 PM UTC-3, WM wrote:
> > William schrieb am Donnerstag, 2. September 2021 um 21:44:44 UTC+2:
> > > On Thursday, September 2, 2021 at 4:08:37 PM UTC-3, WM wrote:
> > > > William schrieb am Mittwoch, 1. September 2021 um 23:07:01 UTC+2:
> > > > > On Wednesday, September 1, 2021 at 4:48:02 PM UTC-3, WM wrote:
> > > > >
> > > > > > There is a last positive point before zero. But it is dark.
> > > > > >
> > > > > So whatever it is it is not a number of the form 1/n which n an element of |N_F, nor is it a real number ("dark real numbers" whatever they are are not real numbers)
> > > > All real numbers which you call real numbers have an infinite number of
> > > real numbers between themselves and zero. They are real numbers not "dark real numbers" (whatever they are).
> > Then they all could be removed as individuals and nothing would remain
> Indeed, if you remove all real numbers (and every real number has the property that it can be distinguished)

No. Then you could remove all real numbers individually such that nothing remains. Then necessarily a last one would be removed.

Regards, WM

Jim Burns schrieb am Freitag, 3. September 2021 um 22:04:18 UTC+2:
> On 9/3/2021 1:22 PM, WM wrote:
> > Jim Burns schrieb
> > am Freitag, 3. September 2021 um 02:40:50 UTC+2:
>
> >> Every end segment that I can individually define
> >> is an end segment.
> >
> > and has an infinite intersection with all other endsegments.
> | Let _Ends_ be an arbitrary collection of end segments of
> | FISONable naturals.
> | Let INT(Ends) be its intersection.
> | In short,
> | if INT(Ends) is NOT empty,
> | then Ends is NOT the collection of all end segments of
> | FISONable naturals.

The intersection cannot be empty without an empty endsegment because of inclusion monotony.

Regards, WM

On 9/3/2021 4:47 PM, WM wrote:
> Jim Burns schrieb
> am Freitag, 3. September 2021 um 22:04:18 UTC+2:
>> On 9/3/2021 1:22 PM, WM wrote:
>>> Jim Burns schrieb
>>> am Freitag, 3. September 2021 um 02:40:50 UTC+2:

>>>> Every end segment that I can individually define
>>>> is an end segment.
>>>
>>> and has an infinite intersection with all other endsegments.
>>
>> | Let _Ends_ be an arbitrary collection of end segments of
>> | FISONable naturals.
>> | Let INT(Ends) be its intersection.
>> | In short,
>> | if INT(Ends) is NOT empty,
>> | then Ends is NOT the collection of all end segments of
>> | FISONable naturals.
>
> The intersection cannot be empty without an empty endsegment
> because of inclusion monotony.

Why?

----
What is the first claim you object to?

(i)
If INT(Ends) is NOT empty,
then a FISONable natural m is in INT(Ends).

(ii)
If m is a FISONable natural,
then a FISONable-natural end segment E(m+1) exists
such that m is not-in E(m+1).

(iii)
If m is in INT(Ends) and not-in E(m+1),
then INT(Ends) is not a subset of E(m+1).

(iv)
If INT(Ends) is not a subset of E(m+1),
then FISONable-natural end segment E(m+1) is not a member of
FISONable-natural end-segment collection Ends.

(v)
If FISONable-natural end segment E(m+1) is not a member of
FISONable-natural end-segment collection Ends,
then Ends is NOT the collection of all FISONable-natural
end segments.

(vi)
By i, ii, iii, iv, v,
if INT(Ends) is NOT empty,
then Ends is NOT the collection of all FISONable-natural
end segments.

(vii)
By vi,
if Ends is the collection of all FISONable-natural
end segments, then INT(Ends) is empty.

On Friday, September 3, 2021 at 5:38:46 PM UTC-3, WM wrote:
> William schrieb am Freitag, 3. September 2021 um 19:28:11 UTC+2:
> > On Friday, September 3, 2021 at 2:19:57 PM UTC-3, WM wrote:
>
> > > > Indeed some numbers do not have the property "Can be written down".
> > > I call them dark.
> > Nope. To you saying that something is "dark" means a lot more that "Cannot be written down"
> A dark number cannot be identified by any means. It cannot be handled as a single individual.

However, a number that does not have the property "Can be written down" still has the property "can be handled a a single individual'. "Dark" to you means a lot more than does not have the property "Can be written down".

--
William Hughes

On Thursday, September 2, 2021 at 3:04:12 AM UTC-7, Greg Cunt wrote:
> On Wednesday, September 1, 2021 at 9:43:26 PM UTC+2, WM wrote:
> > Greg Cunt schrieb am Mittwoch, 1. September 2021 um 16:59:48 UTC+2:
> > >
> > > Hint: If we are talking about "changing so-and-so" in math its only a "figure of speech". Mathematical objects don't (actually) "change".
> > >
> > Never heard of fluxions?
> Sure, dumbo:
>
> Bishop George Berkeley, a prominent philosopher of the time, slammed Newton's fluxions in his essay The Analyst, published in 1734.Berkeley refused to believe that they were accurate because of the use of the infinitesimal {\displaystyle o}o. He did not believe it could be ignored and pointed out that if it was zero, the consequence would be division by zero. Berkeley referred to them as "ghosts of departed quantities", a statement which unnerved mathematicians of the time and led to the eventual disuse of infinitesimals in calculus.
> (Wikipedia)
>
> Mucke, you really are FULL OF SHIT!

Nah, these days it's known that a) Berkeley's "rejections" were tongue-in-cheek, and
in places interpreted wrongly just like "hobgoblin of little minds", b) there are modern
justifications of infinitesimals and by many after the infinite divisibility of classical
geometry those have always been around, and c) boring no-nothings back-and-forth.

There are lots of modern emplacements of restoring standard infinitesimals as about
the differential, and without a lot of the extraneuous machinery of the ultrafilters,
to arrive at countable continuous domains in various models of real numbers satisfying
the more than just one definition of continuity.

Or, Duns Scotus predates and obviates Berkeley. How could he have obsoleted the
later itinerant layman in terms of mathematics? Because truth is enduring.

On Friday, September 3, 2021 at 7:22:58 PM UTC+2, WM wrote:
> Jim Burns schrieb am Freitag, 3. September 2021 um 02:40:50 UTC+2:
> >
> > Every end segment [....] is an end segment.
>
> But it is not necessarily infinite.

Shut up, you sillly asshole full of shit!

Each and every every end segment is INFINITE by definition.

Hint: An e IN: card({m e IN: m >= n} = aleph_0.

On Friday, September 3, 2021 at 10:38:46 PM UTC+2, WM wrote:

> A dark number cannot be identified by any means. It cannot be handled as a single individual. Its neighbours [...] cannot be found.

FASCINATING, they seem to be instances of WIMPs.

You know: "A decade after the dark matter problem was established in the 1970s, WIMPs were suggested as a potential solution to the issue. Athough the existence of WIMPs in nature is still hypothetical, it would resolve a number of astrophysical and cosmological problems related to dark matter."

Same in mathematics, I guess!

Source: https://en.wikipedia.org/wiki/Weakly_interacting_massive_particles

On Friday, September 3, 2021 at 10:47:48 PM UTC+2, WM wrote:

> The intersection [of the set of all endsegments] cannot be empty without an empty endsegment

This claim is simply wrong. In the context of set theory the intersection of the set of all endsegments *is* empty, but *no* endsegment is empty.

Hint: An e IN: n e {m e IN: m >= n}.

Hence your "argument" (lol) is bogus (to say the least):

> because of <bla bla>

Yeah, because you are a crank full of shit.