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).

Regards, WM

On 8/31/2021 7:04 AM, WM wrote:
> Jim Burns schrieb am Montag, 30. August 2021 um 21:44:51 UTC+2:
>> On 8/30/2021 6:23 AM, WM wrote:
>
>>> A defined point or a definable point cannot be next to zero.
>> I will call a point d with such a 1/k between d and 0
>> an _Archimedean_ point.
>
>> An Archimedean point cannot be next to zero.
>
> Yes.
>>
>> In the standard line, all points are Archimedean.
>
> But in the standard line no point is next to another point. Hence there is a gap between 0 and 1/k. This gap can be reduced but cannot made disappear.
>
>> No points are next to zero.
>
> Then there are no points next to zero. What is where no points are but further points can appear?
>
>>>> What is there that can be next to zero
>>>> without there being a gap between it and zero?
>>>
>>> Dark points.
>> If d is a point next to 0, then there are no points
>> between 0 and d. That's what "next" means.
>
> Yes. But if d cannot be determined, then it could be any of the aleph_0 dark points being there between 0 and the next defined point.

wrong.
1 If 0 and d are "next to" each other, there are no points in between 0 and d, no matter what d is.
2 If there are points between 0 and d, then 0 and d are not "next to" each other

you can't weasel out of this fact.

>>
>> If there are no points between 0 and d,
>> then that is a gap. That's what "gap" means.
>
> No, gap is the space between 0 and the smallest defined real number - unless there are dark points between 0 and the smallest defined real number.

so how do you determine there are dark points between 0 and the smallest defined real number ?

>
> Regards, WM
>

Serg io schrieb am Dienstag, 31. August 2021 um 15:27:13 UTC+2:
> On 8/31/2021 7:04 AM, WM wrote:

> > No, gap is the space between 0 and the smallest defined real number - unless there are dark points between 0 and the smallest defined real number.
> so how do you determine there are dark points between 0 and the smallest defined real number ?

Either there is something, or there is nothing. The something would not be definable. So it is dark. The nothing would be a gap.

Regards, WM

On 8/31/2021 6:53 AM, WM wrote:
> Jim Burns schrieb am Montag, 30. August 2021 um 21:12:06 UTC+2:
>> On 8/30/2021 5:36 AM, WM wrote:
>>> Jim Burns schrieb
>>> am Sonntag, 29. August 2021 um 18:43:59 UTC+2:
>>
>>>>> Arithmetic is consistent.
>>>>
>>>> Arithmetic says
>>>> for each positive rational p, p > 0 ,
>>>> a unique positive rational q, q > 1,
>>>> exists such that q = p + 1.
>>>
>>> Arithmetic would remain consistent in potential infinity
>>> without any dark numbers.
>> Arithmetic is consistent. We agree.
>> Arithmetic says that, using the condition q = p + 1,
>> the numbers p > 0 and q > 1 can be paired one-to-one.
>>> This proves that arithmetic concerns only definable numbers.
>>> Although we cannot know about the properties of dark individuals,
>>> q = p + 1 is correct for definable and dark numbers, at least
>>> collectively, because we have the geometric argument:
>>> Translation invariance. The interval (0, 1] is identical to
>>> the interval (1, 2] except the added 1.
>> Using the condition q = p + 1,
>> the numbers p > 0 and q > 1 can be paired one-to-one.
>
> The definable numbers.
>
>>>> Arithmetic says
>>>> for each positive rational q, q > 1 ,
>>>> a unique positive rational r, 1 > r > 0,
>>>> exists such that r = 1/q.
>>>
>>> Yes, but this can hold only for the definable rationals.
>> For thread-historical reasons, I am only talking _here_ about
>> positive rationals. However, this holds for all non-zero
>> real numbers. Obviously.
>
> Obviously.
>>
>> And there is a geometric argument that,
>> using the condition r = 1/q,
>> the numbers q > 1 and r, 0 < r < 1, can be paired
>> one-to-one.
>
> Sorry, your nice argument is geometric but it is based on similarity. Similarity is not identity.

no, it is a mapping, one to one.

>>
>> Given the co-linear points 0, 1, Q, with Q beyond 1,
>> construct two similar right triangles Q0C and C0R
>> with corresponding sides of lengths q, 1, (don't care)
>> and 1, r, (don't care).
>>
>> Because similar triangles, r/1 = 1/q.
>>
>> Construction.
>>
>> Find the midpoint between points 0 and Q.
>> Label the midpoint Q'.
>>
>> Draw a circle centered Q' of radius q/2.
>> Label the circle G.
>> G will meet points 0 and Q.
>>
>> Draw a circle centered at point 0 of radius 1.
>> Label the circle H.
>> H will meet point 1 and intersect G at two points.
>> Label the two points where H and G intersect C,C'.
>>
>> Draw the line segment between points C and C'.
>> Label the segment V.
>> V will intersect the segment between points 0 and 1.
>> Label the point of intersection R.
>>
>> The angle QC0 is a right triangle
>> ( because
>
> because Thales proved it
>>
>> Segment V is perpendicular to segment 01
>> ( similar triangles.
>>
>> Angle CR0 is a right triangle.
>>
>> Triangles Q0C and C0R are similar.
>>
>> Therefore r = 1/q.
>>
>
>>> Otherwise half of all rationals were in (0, 1],
>>> but by adding 1 to every fractions we would find
>>> half of all rationals in (1, 2].
>>> This would show arithmetic inconsistent.
>> This shows that [1, +inf) is Dedekind infinite.
>> It shows that each element of [1, +inf) can be matched
>> one-to-one to each element of (1,2], a proper subset.
>>
>> It's arithmetic.
>> You said arithmetic is consistent.
>
> It is. If it showed equal number of fractions in (0 1] and (1, oo) it would be inconsistent. Because equal number in a set and its proper subset is inconsistent. Your argument, slightly modified could show equal numbers in all unit intervals and in all pairs of unit intervals and in all triples of unit intervals, and in all intervals. That is obviously not the same sharp equalness as is proved by translation invariance but it is simply the property of potential infinity. All infinite sets are equal, namely infinite (when we refrain from the silly diagonal argument).
>
>> Did you mean "unless I (WM) don't want that result"?
>
> No, I mean unless it is inconsistent.
>
> Regards, WM
>

On Monday, August 30, 2021 at 12:21:37 PM UTC-7, Jim Burns wrote:
> On 8/30/2021 12:57 PM, Serg io wrote:
> > On 8/30/2021 11:52 AM, WM wrote:
>
> >> No. The gap is what you cannot define or write down.
> > if you cannot define it, nor write it down ?
> > Is it made of imaginatorium ?
> <quote>
>
> Our revels now are ended. These our actors,
> As I foretold you, were all spirits, and
> Are melted into air, into thin air:
> And like the baseless fabric of this vision,
> The cloud-capp'd tow'rs, the gorgeous palaces,
> The solemn temples, the great globe itself,
> Yea, all which it inherit, shall dissolve,
> And, like this insubstantial pageant faded,
> Leave not a rack behind. *We are such stuff*
> *As dreams are made on* and our little life
> Is rounded with a sleep.
>
> </quote>
>
> Prospero, "The Tempest", Wm. Shakespeare

Tower of rain....

Sticks and stones /
may break my bones /
and your bones. /
.....

The tower of rain is a metaphor for the cumulative hierarchy.

Metaphor fails of course except the strong metonymy.
(The strong platonist's strong metonymy.) In such a theory
all the theories exist one of which is real.

On 8/31/2021 8:35 AM, WM wrote:
> Serg io schrieb am Dienstag, 31. August 2021 um 15:27:13 UTC+2:
>> On 8/31/2021 7:04 AM, WM wrote:
>
>>> No, gap is the space between 0 and the smallest defined real number - unless there are dark points between 0 and the smallest defined real number.
>> so how do you determine there are dark points between 0 and the smallest defined real number ?
>
> Either there is something, or there is nothing. The something would not be definable. So it is dark. The nothing would be a gap.
>
> Regards, WM
>

no. I wish you could understand math proofs, and algebra, but you do not.

1. It is easy proof that no number, defined or not, can be specified to be the "smallest real number".
So there is no "smallest real number"

2. It is easy proof that no gap, defined or not, can be specified between 0 and the smallest defined real number.
So there is no "gap"

you prefer to believe your imagination, instead of using Math. Math is not one of your skills.

If you want the proofs, I can put them up. Is is simply the average of the endpoints to show there are numbers in whatever gap you specify, or there is
a number smaller than any number you assume to be smallest.

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. x is the end of the interval and is an unchanging real number (not a "dark real number" whatever that is).

--
William Hughes

WM explained on 8/31/2021 :
> 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.

We have that already, without a need for dark numbers.

On 8/31/2021 7:53 AM, WM wrote:
> Jim Burns schrieb
> am Montag, 30. August 2021 um 21:12:06 UTC+2:

>> And there is a geometric argument that,
>> using the condition r = 1/q,
>> the numbers q > 1 and r, 0 < r < 1, can be paired
>> one-to-one.
>
> Sorry, your nice argument is geometric but it is based on
> similarity. Similarity is not identity.

https://math.stackexchange.com/questions/2371677/aaa-similarity-theorem
| | Def:
| Two triangles are called similar if and only if
| one can be scaled into the other.
| | AAA Theorem:
| If two triangles have the same angles if and only if
| they are similar.

>> Given the co-linear points 0, 1, Q, with Q beyond 1,
>> construct two similar right triangles Q0C and C0R
>> with corresponding sides of lengths q, 1, (don't care)
>> and 1, r, (don't care).
>>
>> Because similar triangles, r/1 = 1/q.

>>>> Arithmetic says
>>>> for each positive rational q, q > 1 ,
>>>> a unique positive rational r, 1 > r > 0,
>>>> exists such that r = 1/q.
>>>
>>> Yes, but this can hold only for the definable rationals.

It holds for geometric points on geometric lines.

Geometrically, this can be understood as a claim that
_similar_ triangles Q0C and C0R exist with the point Q
a distance q from 0 and the point R a distance r from 0,
and r = 1/q.

>>> Otherwise half of all rationals were in (0, 1],
>>> but by adding 1 to every fractions we would find
>>> half of all rationals in (1, 2].
>>> This would show arithmetic inconsistent.
>>
>> This shows that [1, +inf) is Dedekind infinite.
>> It shows that each element of [1, +inf) can be matched
>> one-to-one to each element of (1,2], a proper subset.
>>
>> It's arithmetic.
>> You said arithmetic is consistent.
>
> It is.
> If it showed equal number of fractions in (0 1] and (1, oo)
> it would be inconsistent.

Arithmetic shows an equal number of points, equal number of
rationals, equal number of fractions in intervals (0,1] and
[1, +inf).

What's next for you? Give up arithmetic?

> Because equal number in a set and its proper subset
> is inconsistent.

Let's examine your claim geometrically.

Consider four points B,C,B',C' which determine two
_parallel_ lines BC and B'C', with B' and C' closer together
than B and C. Imagine the lines running left-to-right,
with BC below B'C'.

Points B,B' and points C,C' determine two lines which
intersect at a point above line BC.
Label their intersection A.

We'll show there is a bijection between each point
of segment B'C' and a proper subset of B'C'.

Given a point X in segment BC, there is a line through
X and point A intersecting segment B'C' at point Y.
X, Y and A are co-linear.

Given a point Y in segment B'C', there is a line through
Y and point A intersecting segment BC at point X.
X, Y and A are co-linear.

The relationship
| Point X and point Y are co-linear with point A
is a one-to-one matching. It is a bijection between BC and B'C'.

Given point C' and line BB', there is a line through C'
parallel to BB'. It will intersect segment BC.
Label their intersection D.
Note that segment BD is a proper subset of segment BC.

Given a point Y in segment B'C' and segment BB', there is a line
parallel to BB'. It will intersect BD at point Z.

Given a point Z in segment BD and segment BB', there is a line
parallel to BB'. It will intersect B'C' at point Y.

The relationship
| Points Y and Z determine a line parallel to BB'
is a one-to-one matching. It is a bijection between B'C' and BD.
BD is a proper subset of BC.

To be absolutely clear, we can combine the bijections between
BC and B'C' and between B'C' and BD to get a bijection
between BC and BD, a proper subset of BC.
We get the relationahsip
| Points X and Z are such that a point Y exists in B'C'
| such that X, Y and point A are co-linear, and
| segment Y is parallel to segment BB'.

> Because equal number in a set and its proper subset
> is inconsistent.

No.

If what you (WM) want to do is to refrain from describing
or reasoning about things involving infinity, you might be able
to do that. However, that would leave you with a choice
between not describing anything involving infinity
(including geometry) or describing them incorrectly.

> Your argument, slightly modified could show equal numbers
> in all unit intervals and in all pairs of unit intervals and
> in all triples of unit intervals, and in all intervals.
> That is obviously not the same sharp equalness as is proved
> by translation invariance but it is simply the property of
> potential infinity.

Being able to match one-to-one the points of any interval
to any other interval is a property of these intervals.

> All infinite sets are equal, namely infinite
> (when we refrain from the silly diagonal argument).

The different sizes (cardinalities) of infinities is a separate
issue. Here, I have been discussing an infinite set being the
same cardinality as one of its proper subsets.

>> Did you mean "unless I (WM) don't want that result"?
>
> No, I mean unless it is inconsistent.

The points of a segment BC can be matched one-to-one
to the points a segment BD, a proper subset of BC.
Does that mean geometry is inconsistent?

On 8/31/2021 8:04 AM, WM wrote:
> Jim Burns schrieb am Montag, 30. August 2021 um 21:44:51 UTC+2:

>> If there are no points between 0 and d,
>> then that is a gap. That's what "gap" means.
>
> No, gap is the space between 0 and the smallest defined
> real number

....the smallest real number.
I already excluded all non-real-number real numbers.

> - unless there are dark points between 0 and the smallest
> defined real number.

....the smallest real number.
I already excluded all non-real-number real numbers.

We can describe one of infinitely-many individuals in terms
that are true no matter which individual is referred to,
then we can reason truth-preserving-ly to further claims about
one of the same infinitely-many individuals.

If we have some description of what makes a number _definable_
(like, steppable 0,...,k with adjacent i, j = i+1), then
we can do that for infinitely-many definable numbers.

However, we can't do that for _defined_ numbers.
The kind of information needed would range far outside
what numbers are and are not.

We don't know if, in the future, Thanos collects all the
Infinity Stones, snaps his fingers and turns all of Those
Who Define Numbers to dust, or if the Avengers defeat him.
We can't reason about that.

But, whether a definable number is defined or not,
it has a steppable 0,...,k with adjacent i, j = i+1.
Thanos won't be able to change that. We can reason
about that without peeking into the future.

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.

Regards, WM

FromTheRafters schrieb am Dienstag, 31. August 2021 um 17:52:54 UTC+2:
> WM explained on 8/31/2021 :

> > 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.
> We have that already, without a need for dark numbers.

You have two rational numbers such that nothing can be inserted between them? I doubt that.

Regards, WM

Jim Burns schrieb am Dienstag, 31. August 2021 um 21:09:42 UTC+2:
> On 8/31/2021 7:53 AM, WM wrote:

> > Sorry, your nice argument is geometric but it is based on
> > similarity. Similarity is not identity.

> Geometrically, this can be understood as a claim that
> _similar_ triangles Q0C and C0R exist with the point Q
> a distance q from 0 and the point R a distance r from 0,
> and r = 1/q.

For every definable point Q it is correct, like for every definable n there is a FISON.

> Arithmetic shows an equal number of points, equal number of
> rationals, equal number of fractions in intervals (0,1] and
> [1, +inf).
>
No, that is too obviously nonsense, to be arithmetic. It is matheology.

Arithmetic and geometry, the pilars of mathematics, prove same number of points in every unit interval and a difference to every other interval.

Regards, WM

On 8/31/2021 3:44 PM, 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.
>
> Regards, WM
>

Some Brains are Infested with Dark Numbers...

On Tuesday, 31 August 2021 at 17:44:38 UTC-3, WM wrote:
[...]
> That would be deplorable for everybody who has gaps in his knowledge. Try it. It can change.

Unfortunately, Muckenheim is now at the stage of his life where his memory gaps are increasing. This is not a change to be happy about.

On 8/31/2021 4:53 PM, WM wrote:
> Jim Burns schrieb
> am Dienstag, 31. August 2021 um 21:09:42 UTC+2:
>> On 8/31/2021 7:53 AM, WM wrote:

>>> Sorry, your nice argument is geometric but it is based on
>>> similarity. Similarity is not identity.
>
>> Geometrically, this can be understood as a claim that
>> _similar_ triangles Q0C and C0R exist with the point Q
>> a distance q from 0 and the point R a distance r from 0,
>> and r = 1/q.
>
> For every definable point Q it is correct,
> like for every definable n there is a FISON.

We decide to reason about points in a line.
We describe a point: some point 1/k is between it and 0.
We reason from that description.

We decide to reason about the numbers we count with.
We describe a number: it has a FISON.
We reason from the description.

Call them what you like, it won't change what we are describing
and reasoning about: _these_ points _these_ numbers.

What you're calling a contradiction is only you describing
_something else_ Points not Archimedean. Numbers not FISONable.

>> Arithmetic shows an equal number of points, equal number of
>> rationals, equal number of fractions in intervals (0,1] and
>> [1, +inf).
>
> No, that is too obviously nonsense, to be arithmetic.
> It is matheology.

It is _division_ It is an _arithmetic operation_
For each x,y, if ~(y=0), there is a unique z, z = x/y.

Is this what you're calling "matheology"?

> Arithmetic and geometry, the pilars of mathematics,
> prove same number of points in every unit interval
> and a difference to every other interval.

Consider two open intervals (a, a+b) and (c, c+d),
non-zero b and d.

For each point x in (a, a+b), a unique point y exists
in (c, c+d), and, for each point y in (c, c+d), a unique
point x exists in (a, a+b) such that d*(x-a) = b*(y-c).

You see before you any interval matched one-to-one
to any other interval -- even to a proper subset of the
first interval.

This is arithmetic, not set theory.
It is a thing that can happen with infinities,
parts being equal to wholes.

Where "set theory" comes into play
(AKA "variables with infinitely-many possible referents")
is it allows us to describe them and reason about them.
But "set theory" didn't put them there.
It took the dark away.

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.
>

Piffle
A "gap" that changes is not capable of containing a real number (not a "dark real number" whatever that is) that , like all real numbers, does not change.

--
William Hughes

WM submitted this idea :
> FromTheRafters schrieb am Dienstag, 31. August 2021 um 17:52:54 UTC+2:
>> WM explained on 8/31/2021 :
>
>>> 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.
>> We have that already, without a need for dark numbers.
>
> You have two rational numbers such that nothing can be inserted between them?
> I doubt that.

Non sequitur.

On Tuesday, August 31, 2021 at 3:13:33 PM UTC+2, WM wrote:

> like the largest known prime number

"the largest known prime number" is mumbo-jumbo

On Tuesday, August 31, 2021 at 10:47:26 PM UTC+2, WM wrote:
> FromTheRafters schrieb am Dienstag, 31. August 2021 um 17:52:54 UTC+2:
> > >
> > > [...] there are dark numbers. They fill the gap and make the real line continuous.
> > >
> > We have that already, without a need for dark numbers.
> >
> You have <bla bla bla>

It seems that you have lost you last marbles, Mückenheim.

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.

Regards, WM

On Wednesday, 1 September 2021 at 08:26:19 UTC-3, Greg Cunt wrote:
> On Tuesday, August 31, 2021 at 3:13:33 PM UTC+2, WM wrote:
>
> > like the largest known prime number
>
> "the largest known prime number" is mumbo-jumbo

I wouldn't be quite so harsh. Yes, there is a time dependence that should be made explicit, but we know (!) that the largest known prime at any moment is a Mersenne prime, and even with GIMPS there is a gap of at least three years (on average) until the next one is found. However, the rest of his contribution in that post *is* mumbo-jumbo.

On Wednesday, 1 September 2021 at 09:28:50 UTC-3, WM wrote:
> Take the interval [0, 1/n] for n in |N. It changes and contains many real numbers for every n.

My word, even HERE you manage to swap quantifiers improperly. You are a demented moron.

Jim Burns schrieb am Mittwoch, 1. September 2021 um 01:05:46 UTC+2:
> On 8/31/2021 4:53 PM, WM wrote:
> > Jim Burns schrieb
> > am Dienstag, 31. August 2021 um 21:09:42 UTC+2:
> >> On 8/31/2021 7:53 AM, WM wrote:
>
> >>> Sorry, your nice argument is geometric but it is based on
> >>> similarity. Similarity is not identity.
> >
> >> Geometrically, this can be understood as a claim that
> >> _similar_ triangles Q0C and C0R exist with the point Q
> >> a distance q from 0 and the point R a distance r from 0,
> >> and r = 1/q.
> >
> > For every definable point Q it is correct,
> > like for every definable n there is a FISON.
> We decide to reason about points in a line.
> We describe a point: some point 1/k is between it and 0.
> We reason from that description.
>
> We decide to reason about the numbers we count with.
> We describe a number: it has a FISON.

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.
>
> What you're calling a contradiction is only you describing
> _something else_

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.

> > Arithmetic and geometry, the pilars of mathematics,
> > prove same number of points in every unit interval
> > and a difference to every other interval.

Argument based upon definabiliyt of all elements of infinite sets deletd.

Regards, WM

On Wednesday, September 1, 2021 at 9:28:50 AM UTC-3, WM wrote:
> William schrieb am Mittwoch, 1. September 2021 um 03:13:24 UTC+2:

> > A "gap" that changes is not capable of containing a real number
> Take the interval [0, 1/n] for n in |N. It changes

Piffle. If it changes it is not an interval. And what it contains is not a real number as a real number does not change.

--
William Hughes