On 8/28/2021 2:01 PM, Greg Cunt wrote:
> On Saturday, August 28, 2021 at 8:49:58 PM UTC+2, WM wrote:
>> GC wrote:
>>>
>>> If this were true than [a, a] = {a} would have to be /a/; but that's not the case in the context of ZF(C).
>>>
>> Who cares?
>
> Anyone who pretends to talk about "set theory".
>

WM and his Pretend Math.

On Saturday, August 28, 2021 at 12:17:07 PM UTC-7, Serg io wrote:
> On 8/28/2021 2:01 PM, Greg Cunt wrote:
> > On Saturday, August 28, 2021 at 8:49:58 PM UTC+2, WM wrote:
> >> GC wrote:
> >>>
> >>> If this were true than [a, a] = {a} would have to be /a/; but that's not the case in the context of ZF(C).
> >>>
> >> Who cares?
> >
> > Anyone who pretends to talk about "set theory".
> >
> WM and his Pretend Math.

They say ants are mind-controlled by pheromones.

What's the point of bringing the Church into it?

Talk about set theory? I know set theory.

WM must acknowledge that the solution is me,
what rather that he doesn't, and until he does, won't.

Rather of course that I wrote a modern apologetics in foundations.

To read!

Or, thank you, I have a modern apologetics in foundations already.

On Saturday, August 28, 2021 at 3:47:34 PM UTC-3, WM wrote:

> > It is not true that there must be a gap next to zero, or a point next to zero.
> Either there is a definable point next to zero or there is no definable point next to zero. Do you agree?
There is neither a point ("definable" or not) nor a gap next to zero.

--
William Hughes.

"me" doesn't post so much here anymore. He was quite
fun when he defended Dan's f(x) quantification of f.

R.I.P.

Ross A. Finlayson schrieb am Samstag, 28. August 2021 um 21:28:51 UTC+2:
> WM must acknowledge that the solution is me,

On Saturday, August 28, 2021 at 10:20:24 PM UTC+2, Mostowski Collapse wrote:

> "me" doesn't post so much here anymore. He was quite
> fun when he defended Dan's f(x) quantification of f.

Geh scheißen, Du Depp.

Aber d a z u bist Du wohl auch zu blöde.

On Saturday, August 28, 2021 at 9:12:30 PM UTC+2, WM wrote:

> "The point next to zero" is already a definition.

No, it's not. It's saublöder Scheißdreck, Mückenheim.

> 1/n is a /definable/ unit fraction if n is the last element of a FISON. Equivalently 1/n is a /definable/ unit fraction if n has a decimal representation.

Great. That implies that that 1/n is a definable unit fraction for each and every natural number n.

In other words, EACH AND`EVERY unit fraction is definable.

Thanks for clarification.

On Saturday, August 28, 2021 at 12:01:59 PM UTC-7, WM wrote:
> FredJeffries schrieb am Samstag, 28. August 2021 um 03:31:01 UTC+2:
>
> > The same 'arguments' are offered as twenty years ago. NOTHING has been accomplished. No one has learned ANYTHING over these tens of thousands of posts.
> That is wrong. I have learnt a lot over the years, laid down here: https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf. For instance I have realized only in 2015 that Cantor indexes the different unit intervals irregularly. I have never found any literature noting or explaining let alone defending this. Meanwhile I have found many colleagues suspecting Cantor's enumeration. That's a lot beeing accomplished.

Q.E.D.

I rest my case.

On Sunday, August 29, 2021 at 12:20:12 AM UTC+2, FredJeffries wrote:

> Cantor indexes the different unit intervals irregularly.

Look, dumbo, "Cantor" indexes the fractions in such a way that EACH AND EVERY fraction is "indexed" by unique natural number. That's all that counts.

See: https://en.wikipedia.org/wiki/Pairing_function

Your problems with this simple fact stem from a psychosis.

See: https://en.wikipedia.org/wiki/Delusional_disorder

On Saturday, August 28, 2021 at 9:01:59 PM UTC+2, WM wrote:

> Cantor indexes the different unit intervals irregularly.

Look, dumbo, "Cantor" indexes the fractions in such a way that EACH AND EVERY fraction is "indexed" by a unique natural number. That's all that counts here.

See: https://en.wikipedia.org/wiki/Pairing_function

Your problems with this simple fact stem from a psychosis.

See: https://en.wikipedia.org/wiki/Delusional_disorder

On 8/28/2021 2:00 PM, WM wrote:
> FredJeffries schrieb am Samstag, 28. August 2021 um 03:31:01 UTC+2:
>
>> The same 'arguments' are offered as twenty years ago. NOTHING has been accomplished. No one has learned ANYTHING over these tens of thousands of posts.
>
> That is wrong. I have learnt a lot over the years, laid down here: https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf.

that paper has not been peer reviewed.

For instance I have realized only in 2015 that Cantor indexes the different unit intervals irregularly.

you put "intervals" on it to create your perceived problem, but there isn't any at all.

your "intervals" fail.

> I have never found any literature noting or explaining let alone defending this.

because it is RED HERRING.

> Meanwhile I have found many colleagues suspecting Cantor's enumeration.

your many colleagues are dim witted.

Cantor's enumeration is so simple and easy, and You cannot understand it.

> That's a lot beeing accomplished.
>
> Regards, WM
>

On 8/28/2021 6:02 PM, Greg Cunt wrote:
> On Saturday, August 28, 2021 at 9:01:59 PM UTC+2, WM wrote:
>
>> Cantor indexes the different unit intervals irregularly.
>
> Look, dumbo, "Cantor" indexes the fractions in such a way that EACH AND EVERY fraction is "indexed" by a unique natural number. That's all that counts here.
>
> See: https://en.wikipedia.org/wiki/Pairing_function
>
> Your problems with this simple fact stem from a psychosis.
>
> See: https://en.wikipedia.org/wiki/Delusional_disorder
>

Cantor enumeration is extremely simple and clear. WM flails against it to no avail or purpose, demonstrating incompetence, which makes people think he
never took Algebra.

Infinite series are very well known, WM partitions them at k into finite and infinite, then WM gets totally lost with his "next to", "dark",
"definable", etc etc, using only his imagination, unsupported by facts, unsupported by math, or derivation in math.

At least he avoids insulting people, like other trolls.

On 8/28/2021 2:47 PM, WM wrote:
> William schrieb
> am Freitag, 27. August 2021 um 23:32:34 UTC+2:

>> Correct. There is not a gap next to zero.
>
> The alternative is a point.

If x is the point next to zero,
then there is a gap, the open segment (0,x).

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.

Consider the unit fractions 1/k.
There is no open segment (0,1/k) in which
there are no unit fractions (see 1/(k+1)).
There is no (0,1/k) which is a gap.
No 1/k is next to 0.

( We can't say
( | if there is no 1/k next to 0,
( | then there is no gap.
( We can say the same of -1 as we say of 0.
( There is no 1/k next to -1.
( There is no (-1,1/k) without a unit fraction.
( But there is a gap. (-1,0) has no unit fractions.
( ( We can say
( | if there is no gap,
( | then there is no 1/k next to 0.

Also,
| if there is no next point, our choices are
| there are no points, as at an end, and
| there are infinitely-many points, each with a point between.

>> There is a point or not a point next to 0.
>> This is a consequence of bivalent logic.
>
> So it is.
>
>> There is no a point next to 0.
>> It is not true that there must be a gap next to zero,
>> or a point next to zero.
>
> Either there is a definable point next to zero
> or there is no definable point next to zero.
> Do you agree?

Either there is a definable point _and no dark point_ next to zero,
or there is _no_ definable point _and no dark point_ next to zero.

----
Suppose that a dark point d exists next to 0.
If any 1/k were between 0 and d, d would not be next to 0.
So, for each 1/k, d is between 0 and 1/k.

Consider the interval (0,1/7).
For any point x in (0,1/7), if we repeat x 7 times,
x+x+x+x+x+x+x will not meet or pass the point 1.

Consider the interval (0,1/k).
For any point x in (0,1/k), if we repeat x k times,
x+x+x+...+x+x+x will not meet or pass the point 1.

d is in (0,1/k).
If we repeat d k times,
d+d+d+...+d+d+d will not meet or pass the point 1.

This is true of _any finite j_
d is in (0,1/j).
Repeating d _any finite number of times_ will not
meet or pass the point 1.

Therefore,
this assumed dark point d does not have the Archimedean
property.
https://en.wikipedia.org/wiki/Archimedean_property

The standard kind of lines, from Archimedes at least,
have the Archimedean property.
Those lines do not have any dark d next to 0.

Do you agree?

William schrieb am Samstag, 28. August 2021 um 22:01:23 UTC+2:
> On Saturday, August 28, 2021 at 3:47:34 PM UTC-3, WM wrote:
>
> > > It is not true that there must be a gap next to zero, or a point next to zero.
> > Either there is a definable point next to zero or there is no definable point next to zero. Do you agree?
> There is neither a point ("definable" or not) nor a gap next to zero.

What is next to zero? Note that "nothing" is synonymous to "gap".

Regards, WM

Greg Cunt schrieb am Samstag, 28. August 2021 um 23:25:48 UTC+2:
> On Saturday, August 28, 2021 at 9:12:30 PM UTC+2, WM wrote:
>
> > "The point next to zero" is already a definition.
> No, it's not.

Everybody knows what is meant like with the Grossone. There is nothimng between the point next to zero and zero. There is nothing between the Grossone and omega.
>
> > 1/n is a /definable/ unit fraction if n is the last element of a FISON. Equivalently 1/n is a /definable/ unit fraction if n has a decimal representation.
>
> Great. That implies that that 1/n is a definable unit fraction for each and every natural number n.
>
> In other words, EACH AND`EVERY unit fraction is definable.

Not quite right. Every definable unit fraction is definable. But every definable unit fraction has aleph_0 successors, mostly undefinable, among them the smallest unit fraction.

Regards, WM

On Sunday, August 29, 2021 at 6:13:35 PM UTC+2, WM wrote:
> Greg Cunt schrieb am Samstag, 28. August 2021 um 23:25:48 UTC+2:
> > On Saturday, August 28, 2021 at 9:12:30 PM UTC+2, WM wrote:
> > >
> > > "The point next to zero" is already a definition.
> > >
> > No, it's not.
> >
> <bla bla> the Grossone <bla bla>

We don't talk about the Grossone here, crank.

> There is nothing between the point next to zero and <bla>

There is no "point next to zero", dumbo.

Hence the phrase "the point next to zero" is NONSENSICAL.

You know, just like the phrase "the round square".

__________________________________________

> > > 1/n is a /definable/ unit fraction if n is the last element of a FISON. Equivalently 1/n is a /definable/ unit fraction if n has a decimal representation. [WM]
> >
> > Great. That implies that that 1/n is a definable unit fraction for each and every natural number n.
> >
> > In other words, EACH AND VERY unit fraction is definable.
> >
> Every definable unit fraction is definable.

Guess so. You know each and every even number is even!

On the other hand, from your definition(s) we got:

|| EACH AND VERY unit fraction is definable. ||

Great result. Thanks for the carification!

On Sunday, August 29, 2021 at 9:13:35 AM UTC-7, WM wrote:
> Greg Cunt schrieb am Samstag, 28. August 2021 um 23:25:48 UTC+2:
> > On Saturday, August 28, 2021 at 9:12:30 PM UTC+2, WM wrote:
> >
> > > "The point next to zero" is already a definition.
> > No, it's not.
> Everybody knows what is meant like with the Grossone. There is nothimng between the point next to zero and zero. There is nothing between the Grossone and omega.
> >
> > > 1/n is a /definable/ unit fraction if n is the last element of a FISON. Equivalently 1/n is a /definable/ unit fraction if n has a decimal representation.
> >
> > Great. That implies that that 1/n is a definable unit fraction for each and every natural number n.
> >
> > In other words, EACH AND`EVERY unit fraction is definable.
> Not quite right. Every definable unit fraction is definable. But every definable unit fraction has aleph_0 successors, mostly undefinable, among them the smallest unit fraction.
>
> Regards, WM

There is only one Grossone it's Yaroslav Sergeyev's.

"Smallest unit fraction", here is that in their resulting iota-multiples,
that multiples of fractions make lesser fractions or odd fractions or wholes,
I write a similar term in "iota-multiples".

With omega, I suppose as an ordinal, and there are varieties and
von Neumann's in sets will do as a model, or ordinals, there is
also for example Aleph_1, the cardinality of the cardinal that
is the cardinal of the natural numbers what usually the ordinals model,
numbers and counting most directly as in set theory's constants.

The, "point next to zero", has about as much definition, as "point at infinity".

The "point at infinity", has _all_ the definition of the point at infinity.

With only knowing that the numbers go to it, "there is no last and only next".

Really it's old hat and the best arguments for infinity and infinitesimals
subsume definition, in terms of that in the dialectic they're direct. (In effect.)

I think as many people have "integer plus non-integer part" as "reduced fraction".

I.e., because each point in the lattice is the origin, their number
system is seamless and smooth, besides that for one origin there
is projection as over 1, in space.

On Sunday, August 29, 2021 at 9:23:52 AM UTC-7, Greg Cunt wrote:
> On Sunday, August 29, 2021 at 6:13:35 PM UTC+2, WM wrote:
> > Greg Cunt schrieb am Samstag, 28. August 2021 um 23:25:48 UTC+2:
> > > On Saturday, August 28, 2021 at 9:12:30 PM UTC+2, WM wrote:
> > > >
> > > > "The point next to zero" is already a definition.
> > > >
> > > No, it's not.
> > >
> > <bla bla> the Grossone <bla bla>
>
> We don't talk about the Grossone here, crank.
>
> > There is nothing between the point next to zero and <bla>
>
> There is no "point next to zero", dumbo.
>
> Hence the phrase "the point next to zero" is NONSENSICAL.
>
> You know, just like the phrase "the round square".
>
> __________________________________________
>
> > > > 1/n is a /definable/ unit fraction if n is the last element of a FISON. Equivalently 1/n is a /definable/ unit fraction if n has a decimal representation. [WM]
> > >
> > > Great. That implies that that 1/n is a definable unit fraction for each and every natural number n.
> > >
> > > In other words, EACH AND VERY unit fraction is definable.
> > >
> > Every definable unit fraction is definable.
> Guess so. You know each and every even number is even!
>
> On the other hand, from your definition(s) we got:
>
> || EACH AND VERY unit fraction is definable. ||
>
> Great result. Thanks for the carification!

Excuse me is that your real name?

Wouldn't want to necessarily offend you,
or, everybody else, or, the entire medium.
First though I'd offend you.

Jim Burns schrieb am Sonntag, 29. August 2021 um 06:58:57 UTC+2:
> On 8/28/2021 2:47 PM, WM wrote:

> 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. There are ℵo dark points in the interval (0, 1/n) where n is the largest known or defined natural number. This status is not permanent.
>
> Consider the unit fractions 1/k.
> There is no open segment (0,1/k) in which
> there are no unit fractions (see 1/(k+1)).
> There is no (0,1/k) which is a gap.
> No 1/k is next to 0.

No definable 1/k is next to 0.

But either there is something next to 0 or nothing.

This are the alternatives: A defined point or not a defined point.

> Either there is a definable point _and no dark point_ next to zero,
> or there is _no_ definable point _and no dark point_ next to zero.

Then no point is next to zero. Other objects are not available. "Nothing" is next to zero, i.e., there is a gap-
>
> Suppose that a dark point d exists next to 0.
> If any 1/k were between 0 and d, d would not be next to 0.

For dark 1/k there is no contradiction. Dark points are not individually selectable and have no order.

> Therefore,
> this assumed dark point d does not have the Archimedean
> property.
> https://en.wikipedia.org/wiki/Archimedean_property

At least it cannot be proven.
>
> The standard kind of lines, from Archimedes at least,
> have the Archimedean property.

For definable points it holds.

> Those lines do not have any dark d next to 0.
>
> Do you agree?

It is possible to refrain from dark numbers. But then there is a gap next to zero. A very small gap. Hardly visible to ordinary mathematicians.

Regards, WM

Greg Cunt schrieb am Sonntag, 29. August 2021 um 18:23:52 UTC+2:
> On Sunday, August 29, 2021 at 6:13:35 PM UTC+2, WM wrote:

> > There is nothing between the point next to zero and <bla>
>
> There is no "point next to zero",

Then there is a gap. Then the real line is discontinuous. That is possible.

Regards, WM

Ross A. Finlayson schrieb am Sonntag, 29. August 2021 um 18:24:24 UTC+2:
> On Sunday, August 29, 2021 at 9:13:35 AM UTC-7, WM wrote:

> There is only one Grossone it's Yaroslav Sergeyev's.
>
Yes.

> The, "point next to zero", has about as much definition, as "point at infinity".

No, point next to omega would be the analogy.

Regards, WM

On 8/27/2021 7:14 AM, WM wrote:
> Jim Burns schrieb
> am Donnerstag, 26. August 2021 um 23:17:48 UTC+2:
>> On 8/26/2021 4:52 PM, WM wrote:

>>>> I can prove it exists, that thing I say exists.
>>>
>>> In an inconsistent theory everything can be proved.
>>
>> If you can prove arithmetic is inconsistent, go ahead.
>
> 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 says
for each positive rational q, q > 1 ,
a unique positive rational r, 1 > r > 0,
exists such that r = 1/q.

Arithmetic says
for each positive rational r, 1 > r > 0,
for any positive rationals s,t,
a unique positive rational u, s < u < s+t,
exists such that u = s + t*r.

Therefore, arithmetic says
for each positive rational p, p > 0 ,
for any positive rationals s,t,
a unique positive rational u, s < u < s+t,
exists such that u = s + t/(p+1).

Similarly, arithmetic says
for any positive rationals s,t,
for each positive rational u, s < u < s+t,
a unique positive rational p, p > 0,
exists such that p = t/(u-s) - 1.

> Set theory is inconsistent because
> if we have an infinite matrix or plane,
> then its entries or points above the diagonal
> are precisely as many as below.

You're claiming that set theory is inconsistent
despite it *agreeing* with arithmetic.
Arithmetic, though, is consistent.
Show your work.

> This shows that Cantor's enumeration is incomplete,
> no bijection, no valid measure of infinite sets.

The justification for enumerations like Cantor's
_start with_ claims true of _any one_ of infinitely-many.
Such as, for each p, a unique q exists, q = p + 1.

If you decide we're only to look at naturals up to k,
or only rationals between s and s+t, you've altered
what the claims are about. This is equivocation.
It is a fallacy.

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

On Sunday, August 29, 2021 at 9:43:59 AM UTC-7, Jim Burns wrote:
> On 8/27/2021 7:14 AM, WM wrote:
> > Jim Burns schrieb
> > am Donnerstag, 26. August 2021 um 23:17:48 UTC+2:
> >> On 8/26/2021 4:52 PM, WM wrote:
>
> >>>> I can prove it exists, that thing I say exists.
> >>>
> >>> In an inconsistent theory everything can be proved.
> >>
> >> If you can prove arithmetic is inconsistent, go ahead.
> >
> > 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 says
> for each positive rational q, q > 1 ,
> a unique positive rational r, 1 > r > 0,
> exists such that r = 1/q.
>
> Arithmetic says
> for each positive rational r, 1 > r > 0,
> for any positive rationals s,t,
> a unique positive rational u, s < u < s+t,
> exists such that u = s + t*r.
>
> Therefore, arithmetic says
> for each positive rational p, p > 0 ,
> for any positive rationals s,t,
> a unique positive rational u, s < u < s+t,
> exists such that u = s + t/(p+1).
>
> Similarly, arithmetic says
> for any positive rationals s,t,
> for each positive rational u, s < u < s+t,
> a unique positive rational p, p > 0,
> exists such that p = t/(u-s) - 1.
>
> > Set theory is inconsistent because
> > if we have an infinite matrix or plane,
> > then its entries or points above the diagonal
> > are precisely as many as below.
>
> You're claiming that set theory is inconsistent
> despite it *agreeing* with arithmetic.
> Arithmetic, though, is consistent.
> Show your work.
>
> > This shows that Cantor's enumeration is incomplete,
> > no bijection, no valid measure of infinite sets.
>
> The justification for enumerations like Cantor's
> _start with_ claims true of _any one_ of infinitely-many.
> Such as, for each p, a unique q exists, q = p + 1.
>
> If you decide we're only to look at naturals up to k,
> or only rationals between s and s+t, you've altered
> what the claims are about. This is equivocation.
> It is a fallacy.
>
> https://en.wikipedia.org/wiki/Equivocation

Not me though....

Let's all agree to defend definition of the complete ordered field,
what is itself the pillar of the usual modern definition for analysis,
then also what "models of continuity", or from laws of large numbers,
are in effect, that when arguing and demonstrating the paradoxes of Zeno,
about continuity, motion, and time in the linear and infinitely divisible,
that while bearing up the pillar, is not toppled the dome.

On 8/29/2021 11:51 AM, Ross A. Finlayson wrote:
> On Sunday, August 29, 2021 at 9:43:59 AM UTC-7, Jim Burns wrote:
>> On 8/27/2021 7:14 AM, WM wrote:
>>> Jim Burns schrieb
>>> am Donnerstag, 26. August 2021 um 23:17:48 UTC+2:
>>>> On 8/26/2021 4:52 PM, WM wrote:
>>
>>>>>> I can prove it exists, that thing I say exists.
>>>>>
>>>>> In an inconsistent theory everything can be proved.
>>>>
>>>> If you can prove arithmetic is inconsistent, go ahead.
>>>
>>> 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 says
>> for each positive rational q, q > 1 ,
>> a unique positive rational r, 1 > r > 0,
>> exists such that r = 1/q.
>>
>> Arithmetic says
>> for each positive rational r, 1 > r > 0,
>> for any positive rationals s,t,
>> a unique positive rational u, s < u < s+t,
>> exists such that u = s + t*r.
>>
>> Therefore, arithmetic says
>> for each positive rational p, p > 0 ,
>> for any positive rationals s,t,
>> a unique positive rational u, s < u < s+t,
>> exists such that u = s + t/(p+1).
>>
>> Similarly, arithmetic says
>> for any positive rationals s,t,
>> for each positive rational u, s < u < s+t,
>> a unique positive rational p, p > 0,
>> exists such that p = t/(u-s) - 1.
>>
>>> Set theory is inconsistent because
>>> if we have an infinite matrix or plane,
>>> then its entries or points above the diagonal
>>> are precisely as many as below.
>>
>> You're claiming that set theory is inconsistent
>> despite it *agreeing* with arithmetic.
>> Arithmetic, though, is consistent.
>> Show your work.
>>
>>> This shows that Cantor's enumeration is incomplete,
>>> no bijection, no valid measure of infinite sets.
>>
>> The justification for enumerations like Cantor's
>> _start with_ claims true of _any one_ of infinitely-many.
>> Such as, for each p, a unique q exists, q = p + 1.
>>
>> If you decide we're only to look at naturals up to k,
>> or only rationals between s and s+t, you've altered
>> what the claims are about. This is equivocation.
>> It is a fallacy.
>>
>> https://en.wikipedia.org/wiki/Equivocation
>
> Not me though....
>
> Let's all agree to defend definition of the complete ordered field,
> what is itself the pillar of the usual modern definition for analysis,
> then also what "models of continuity", or from laws of large numbers,
> are in effect, that when arguing and demonstrating the paradoxes of Zeno,
> about continuity, motion, and time in the linear and infinitely divisible,
> that while bearing up the pillar, is not toppled the dome.
>

what was that ?? you use too many word sauce !

On 8/29/2021 12:08 PM, WM wrote:
> William schrieb
> am Samstag, 28. August 2021 um 22:01:23 UTC+2:
>> On Saturday, August 28, 2021 at 3:47:34 PM UTC-3,
>> WM wrote:

>>>> It is not true that there must be
>>>> a gap next to zero, or a point next to zero.
>>>
>>> Either there is a definable point next to zero or
>>> there is no definable point next to zero.
>>> Do you agree?
>>
>> There is neither a point ("definable" or not)
>> nor a gap next to zero.
>
> What is next to zero?
> Note that "nothing" is synonymous to "gap".

Note that "nothing" is NOT synonymous with "gap".

To borrow a term from art, a gap is _negative space_
It is an interval in which points are NOT.
https://en.wikipedia.org/wiki/Negative_space

In order for there to be a point next to zero,
there must be negative space next to zero, and then the point.
If there is no negative space, there is no next point.
If there is no negative space, there is no gap.

On Sunday, August 29, 2021 at 10:06:47 AM UTC-7, Jim Burns wrote:
> On 8/29/2021 12:08 PM, WM wrote:
> > William schrieb
> > am Samstag, 28. August 2021 um 22:01:23 UTC+2:
> >> On Saturday, August 28, 2021 at 3:47:34 PM UTC-3,
> >> WM wrote:
>
> >>>> It is not true that there must be
> >>>> a gap next to zero, or a point next to zero.
> >>>
> >>> Either there is a definable point next to zero or
> >>> there is no definable point next to zero.
> >>> Do you agree?
> >>
> >> There is neither a point ("definable" or not)
> >> nor a gap next to zero.
> >
> > What is next to zero?
> > Note that "nothing" is synonymous to "gap".
> Note that "nothing" is NOT synonymous with "gap".
>
> To borrow a term from art, a gap is _negative space_
> It is an interval in which points are NOT.
> https://en.wikipedia.org/wiki/Negative_space
>
> In order for there to be a point next to zero,
> there must be negative space next to zero, and then the point.
> If there is no negative space, there is no next point.
> If there is no negative space, there is no gap.

"The iota-values comprise cuts, of [0,1]."