On 8/30/2021 5:23 AM, WM wrote:
> Jim Burns schrieb am Sonntag, 29. August 2021 um 19:31:15 UTC+2:
>> On 8/29/2021 12:31 PM, WM wrote:
>>> Jim Burns schrieb
>>> am Sonntag, 29. August 2021 um 06:58:57 UTC+2:
>>>> In order for x to be next to 0,
>>>> there must be no point y between 0 and x,
>>>> there must be no point y in (0,x).
>>>> In other words, (0,x) must be a gap.
>>>
>>> You forget that dark points cannot be analyzed
>>> as individuals or ordered.
>> A gap with points in it is not a gap.
>> Please do not forget this.
>
> No, if there are point, there is no gap.

on the real line, there is always a point between two other points, so gaps cannot exist.

would you like a proof of that in Math ?

On Monday, August 30, 2021 at 7:22:52 AM UTC-7, Serg io wrote:
> On 8/30/2021 5:23 AM, WM wrote:
> > Jim Burns schrieb am Sonntag, 29. August 2021 um 19:31:15 UTC+2:
> >> On 8/29/2021 12:31 PM, WM wrote:
> >>> Jim Burns schrieb
> >>> am Sonntag, 29. August 2021 um 06:58:57 UTC+2:
> >>>> In order for x to be next to 0,
> >>>> there must be no point y between 0 and x,
> >>>> there must be no point y in (0,x).
> >>>> In other words, (0,x) must be a gap.
> >>>
> >>> You forget that dark points cannot be analyzed
> >>> as individuals or ordered.
> >> A gap with points in it is not a gap.
> >> Please do not forget this.
> >
> > No, if there are point, there is no gap.
> on the real line, there is always a point between two other points, so gaps cannot exist.
>
> would you like a proof of that in Math ?

I separate those two concerns as "on" the line or "in" a line.

Having two models of real numbers require they be kept separate,
in terms of their values as real numbers, with respect to values
they have in a common system, like integers or rationals.

On Monday, August 30, 2021 at 4:22:52 PM UTC+2, Serg io wrote:

> on the real line, there is always a point between two other points, so gaps cannot exist.
>
> would you like a proof of that in Math ?

Depends on what we consider as a "gap". Does a missing point already constitute a gap?

For Rudin it does.

See: https://math.stackexchange.com/questions/3650200/gaps-or-holes-in-rational-number-system

In this sense the fact that there is always a rational number between two rational numbers, does not exclude that there are certain "gaps".

Or let's just consider IR\{0}. Here we still have: "there is always a point between two other points". But...

Of course, if we define a "gap" as an interval [a, b] (or (a, b) or (a, b), or [a, b)) for some real numbers a, b with b > a, then you're right.

On 8/30/2021 9:43 AM, Ross A. Finlayson wrote:
> On Monday, August 30, 2021 at 7:22:52 AM UTC-7, Serg io wrote:
>> On 8/30/2021 5:23 AM, WM wrote:
>>> Jim Burns schrieb am Sonntag, 29. August 2021 um 19:31:15 UTC+2:
>>>> On 8/29/2021 12:31 PM, WM wrote:
>>>>> Jim Burns schrieb
>>>>> am Sonntag, 29. August 2021 um 06:58:57 UTC+2:
>>>>>> In order for x to be next to 0,
>>>>>> there must be no point y between 0 and x,
>>>>>> there must be no point y in (0,x).
>>>>>> In other words, (0,x) must be a gap.
>>>>>
>>>>> You forget that dark points cannot be analyzed
>>>>> as individuals or ordered.
>>>> A gap with points in it is not a gap.
>>>> Please do not forget this.
>>>
>>> No, if there are point, there is no gap.
>> on the real line, there is always a point between two other points, so gaps cannot exist.
>>
>> would you like a proof of that in Math ?
>
> I separate those two concerns as "on" the line or "in" a line.
>
> Having two models of real numbers require they be kept separate,
> in terms of their values as real numbers, with respect to values
> they have in a common system, like integers or rationals.
>

I don't see the distinction there between in or on, I'm rusty, what is the difference ?

I see integers, rationals, irrationals, counting numbers, natural numbers, positive/negitive, odd/even as subsets of the real line, and the real line
is a subset of the complex plain.

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:
> > > > William schrieb am Sonntag, 29. August 2021 um 20:09:41 UTC+2:
> > > > > On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:
> > > > >
> > > > > > What is next to zero? Note that "nothing" is synonymous to "gap".
> > > > > Nope
> > > > > A gap next to 0 is an interval [0,x] (note x must be a real number
> > > > The end of the gap is a real number x
> > > and a "gap", like a set, is not something that changes.
> > The collection of definable points is the complement.
> Nope, the set of points that you can write down plus the points that you cannot write down is the complement.

No. The gap is what you cannot define or write down.

Regards, WM

Greg Cunt schrieb am Montag, 30. August 2021 um 15:15:31 UTC+2:
> On Monday, August 30, 2021 at 2:30:49 PM UTC+2, WM wrote:
> > Greg Cunt schrieb am Montag, 30. August 2021 um 14:07:35 UTC+2:
> > >
> > > The problem with these sentences/claims is that THERE IS NO present king of France. (Concerning your orignal claim: There is no first unit fraction [encoutered].)
> > >
> > There is a first unit fraction encounterd, depending on the scale, the velocity, the attentiveness, etc.
> What you mean is "realized" not "encounterd". So your "cursor" is a compexe structure with inner workings? (How do unit fractions and/or rational numbers affect this mechanism, btw.? I mean, how does it recognize unit fractions?)

We recognize whether the cursor crosses a fractions.
>
> Still there's no first unit fraction. THAT was the original topic,

The topic is what can be realized, not what matheologians believe.

Regards, WM

On 8/30/2021 11:52 AM, WM wrote:
> 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:
>>>>> William schrieb am Sonntag, 29. August 2021 um 20:09:41 UTC+2:
>>>>>> On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:
>>>>>>
>>>>>>> What is next to zero? Note that "nothing" is synonymous to "gap".
>>>>>> Nope
>>>>>> A gap next to 0 is an interval [0,x] (note x must be a real number
>>>>> The end of the gap is a real number x
>>>> and a "gap", like a set, is not something that changes.
>>> The collection of definable points is the complement.
>> Nope, the set of points that you can write down plus the points that you cannot write down is the complement.
>
> No. The gap is what you cannot define or write down.
>
> Regards, WM
>

if you cannot define it, nor write it down ?

Is it made of imaginatorium ?

On 8/30/2021 11:57 AM, Serg io wrote:
> On 8/30/2021 11:52 AM, WM wrote:
>> 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:
>>>>>> William schrieb am Sonntag, 29. August 2021 um 20:09:41 UTC+2:
>>>>>>> On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:
>>>>>>>
>>>>>>>> What is next to zero? Note that "nothing" is synonymous to "gap".
>>>>>>> Nope
>>>>>>> A gap next to 0 is an interval [0,x] (note x must be a real number
>>>>>> The end of the gap is a real number x
>>>>> and a "gap", like a set, is not something that changes.
>>>> The collection of definable points is the complement.
>>> Nope, the set of points that you can write down plus the points that you cannot write down is the complement.
>>
>> No. The gap is what you cannot define or write down.
>>
>> Regards, WM
>>
>
> if you cannot define it, nor write it down ?
>
> Is it made of imaginatorium ?
>

to say there is a gap, you must define the endpoints.

if you define the endpoints, anyone can show there is a point in the gap

therefore, I predict that WM will never go down that path of being able to define a gap.

On 8/30/2021 11:57 AM, WM wrote:
> Greg Cunt schrieb am Montag, 30. August 2021 um 15:15:31 UTC+2:
>> On Monday, August 30, 2021 at 2:30:49 PM UTC+2, WM wrote:
>>> Greg Cunt schrieb am Montag, 30. August 2021 um 14:07:35 UTC+2:
>>>>
>>>> The problem with these sentences/claims is that THERE IS NO present king of France. (Concerning your orignal claim: There is no first unit fraction [encoutered].)
>>>>
>>> There is a first unit fraction encounterd, depending on the scale, the velocity, the attentiveness, etc.
>> What you mean is "realized" not "encounterd". So your "cursor" is a compexe structure with inner workings? (How do unit fractions and/or rational numbers affect this mechanism, btw.? I mean, how does it recognize unit fractions?)
>
> We recognize whether the cursor crosses a fractions.

the cursor identifies ALL numbers as it moves across.

>>
>> Still there's no first unit fraction. THAT was the original topic,
>
> The topic is what can be realized, not what matheologians believe.

wrong. a short Math Proof shows no number can be "first"

it is the same question as what number is next to infinity, none.

=> WM you are in a deep rut with this, it is so simple, how many more years can you go one about "next to" or "first" ?

>
> Regards, WM
>

On Monday, August 30, 2021 at 1:53:00 PM UTC-3, WM wrote:

> No. The gap is what you cannot define or write down.

Piffle.
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].

--
William Hughes

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.

>> 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.

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.

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 triangles 0Q'C and CQ'Q are isosceles, and
( angles C0Q, 0QC, QC0 sum to 180 degrees.

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.

( There's another construction, starting from 0, R, 1,
( R between 0 and 1, of the same two triangles.

> 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.
Did you mean "unless I (WM) don't want that result"?

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

On 8/30/2021 6:23 AM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 29. August 2021 um 19:31:15 UTC+2:
>> On 8/29/2021 12:31 PM, WM wrote:

>>> But either there is something next to 0 or nothing.
>>> This are the alternatives:
>>> A defined point or not a defined point.
>>
>> For each interval (0, d), there is a unit fraction 1/k
>> such that a steppable 0,...,k exists in which,
>> for adjacent i,j, j = i+1.
>
> 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.

https://en.wikipedia.org/wiki/Archimedean_property
| | The property, typically construed, states that
| given two positive numbers x and y,
| there is an integer n so that nx > y.

An Archimedean point cannot be next to zero.

In the standard line, all points are Archimedean.
No points are next to zero.

>> 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.

If there are no points between 0 and d,
then that is a gap. That's what "gap" means.

So, no. Obviously.

On 8/30/2021 6:56 AM, WM wrote:
> Jim Burns schrieb
> am Sonntag, 29. August 2021 um 22:50:17 UTC+2:
>> On 8/27/2021 7:27 AM, WM wrote:
>>> Jim Burns schrieb
>>> am Freitag, 27. August 2021 um 00:20:04 UTC+2:

>>>> Each unit fraction 1/k is not _next to_ 0
>>>> because 0 < 1/(k+1) < 1/k
>>>
>>> The cursor moving from 0 to 1 will encounter
>>> some unit fraction first.
>>
>> The cursor can only encounter a non-unit-fraction unit fraction
>> first. Any unit-fraction unit fraction will not be first.
>
> One of them must be first.
> If the cursor is at 1/17,
> then this will not be the first unit fraction encoutered.

How do you know it won't?
Because you know 1/18 and 1/34, and 1/131072 exist,
even though you have not _encountered_ them.

The same argument generalizes to each unit fraction 1/k.
1/k is not first, because 1/(k+1), 1/(2*k), 1/(2^k) exist.

We can reason about things which exist but which we do not
encounter. This is very powerful. You seem to think it's
a flaw, something to be eliminated.

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:
>>>>> William schrieb am Sonntag, 29. August 2021 um 20:09:41 UTC+2:
>>>>>> On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:
>>>>>>
>>>>>>> What is next to zero? Note that "nothing" is synonymous to "gap".
>>>>>> Nope
>>>>>> A gap next to 0 is an interval [0,x] (note x must be a real number
>>>>> The end of the gap is a real number x
>>>> and a "gap", like a set, is not something that changes.
>>> The collection of definable points is the complement.
>> Nope, the set of points that you can write down plus the points that you
>> cannot write down is the complement.
>
> No. The gap is what you cannot define or write down.

So a 'gap' is a mathematical term for something which cannot be
defined?

On 8/30/2021 2:21 PM, 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
>

Shakesphere, a real genius

On 8/30/2021 5:01 PM, FromTheRafters wrote:
> 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:
>>>>>> William schrieb am Sonntag, 29. August 2021 um 20:09:41 UTC+2:
>>>>>>> On Sunday, August 29, 2021 at 1:09:02 PM UTC-3, WM wrote:
>>>>>>>> What is next to zero? Note that "nothing" is synonymous to "gap".
>>>>>>> Nope A gap next to 0 is an interval [0,x] (note x must be a real number
>>>>>> The end of the gap is a real number x
>>>>> and a "gap", like a set, is not something that changes.
>>>> The collection of definable points is the complement.
>>> Nope, the set of points that you can write down plus the points that you cannot write down is the complement.
>>
>> No. The gap is what you cannot define or write down.
>
> So a 'gap' is a mathematical term for something which cannot be defined?

if no one can define it, nore writ it down, then no one can think about it,

it has no characteristics at all, none.

only the three letters, gap means something something..., hey I'm drawing a BLANK here !

everybody is gap-less

måndag 30 augusti 2021 kl. 14:30:49 UTC+2 skrev WM:
> Greg Cunt schrieb am Montag, 30. August 2021 um 14:07:35 UTC+2:
>
> > > If the cursor is at 1/17, then this will not be the first unit fraction encoutered.
> > Well, this claim can be compared with Russell's "The present king of France is bald."
> Wrong. 1/17 does exist.
> > The problem with these sentences/claims is that THERE IS NO present king of France. (Concerning your orignal claim: There is no first unit fraction [encoutered].)
> There is a first unit fraction encounterd, depending on the scale, the velocity, the attentiveness, etc. But if having encounterede it, it will be possible to find even smaller unit fractions. Nevertheless ℵo unit fractions will remain dark.
>
> Regards, WM
"dark" is maeningless and there is no first from 0

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.
>
> 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

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.
>
> 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.

Regards, WM

Jim Burns schrieb am Montag, 30. August 2021 um 21:59:02 UTC+2:
> On 8/30/2021 6:56 AM, WM wrote:
> > Jim Burns schrieb
> > am Sonntag, 29. August 2021 um 22:50:17 UTC+2:
> >> On 8/27/2021 7:27 AM, WM wrote:
> >>> Jim Burns schrieb
> >>> am Freitag, 27. August 2021 um 00:20:04 UTC+2:
>
> >>>> Each unit fraction 1/k is not _next to_ 0
> >>>> because 0 < 1/(k+1) < 1/k
> >>>
> >>> The cursor moving from 0 to 1 will encounter
> >>> some unit fraction first.
> >>
> >> The cursor can only encounter a non-unit-fraction unit fraction
> >> first. Any unit-fraction unit fraction will not be first.
> >
> > One of them must be first.
> > If the cursor is at 1/17,
> > then this will not be the first unit fraction encoutered.
> How do you know it won't?
> Because you know 1/18 and 1/34, and 1/131072 exist,
> even though you have not _encountered_ them.
>
> The same argument generalizes to each unit fraction 1/k.
> 1/k is not first, because 1/(k+1), 1/(2*k), 1/(2^k) exist.

Of course. But if you put numbers in their places, you will see that the sequence does not end but every number put in place has aleph_0 successors.
>
> We can reason about things which exist but which we do not
> encounter. This is very powerful. You seem to think it's
> a flaw, something to be eliminated.

No, we know that no definable unit fraction can exist that is next to 0. Some dark numbers may become defined, but aleph_0 of them never will.

Regards, 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

William schrieb am Montag, 30. August 2021 um 20:03:10 UTC+2:
> On Monday, August 30, 2021 at 1:53:00 PM UTC-3, WM wrote:
>
> > No. The gap is what you cannot define or write down.
> Piffle.
> 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 the interval between 0 and the smallest defined real number, for instance. The gap contains nothing. But we can assume that there is no gap but dark numbers, some og which may become defined.

Regards, WM

On 8/31/2021 7:17 AM, WM wrote:
> William schrieb am Montag, 30. August 2021 um 20:03:10 UTC+2:
>> On Monday, August 30, 2021 at 1:53:00 PM UTC-3, WM wrote:
>>
>>> No. The gap is what you cannot define or write down.
>> Piffle.
>> 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 the interval between 0 and the smallest defined real number, for instance. The gap contains nothing. But we can assume that there is no gap but dark numbers, some og which may become defined.
>
> Regards, WM
>

So you talk about nothing, and the closer you look at it, the more you can't see it.

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).

--
William Hughes