From Newsgroup: comp.theory

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
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From Newsgroup: comp.theory

Mikko <mikko.levanto@iki.fi> writes:

On 2025-08-25 02:53:17 +0000, Mr Flibble said:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

That you can't see infinitesimals does not mean that they don't exist.
It only means that if they exist they are too small to be seen.

Oh? Perhaps I'm taking a joke too seriously?

It's not a matter of being seen. Infinitesimals, in the way I
presume MF is using the term, are number-like entities. They're
included in some mathematical models and not in others. For example,
the standard "real" numbers do not include infinitesimals.

Of course the fact that they're called "real", like the word
"imaginary", doesn't say anything about their validity or existence.

You can't see the number 42, but it definitely exists within most
mathematical models.

I don't know what MF means by "Infinitesimals don't exist".
I'm not actually that curious what he means; my question was more
of a challenge to his claim.

I see another response from him in this thread in which seems to
be implicitly using the rules of real arithmetic to demonstrate
that the existence of infinitesimals would be inconsistent.
That's true enough, but it only demonstrates that infinitesimals
don't exist in the real numbers. I still don't know what he means by "Infinitesimals don't exist".
--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
void Void(void) { Void(); } /* The recursive call of the void */
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From Newsgroup: comp.theory

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

Take 2, 2-ary points that are not equal to each other. Typed out in the newsreader sorry for any typos:

p0 = (-1, 0);
p1 = (1, 0);
pdif = p1 - p0; // differential

k0 = p0 + pdif / 2; // mid point
k1 = k0 + pdif / 4; // 3/4 point
k2 = p0 + pdif * (1 / sqrt(2)); // .707... point
...


So, there are an infinite number of points on the line segment from p0
and p1. Right? p0 + pdif * (0...1)?
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

Take 2, 2-ary points that are not equal to each other. Typed out in the newsreader sorry for any typos:

p0 = (-1, 0);
p1 = (1, 0);
pdif = p1 - p0; // differential

k0 = p0 + pdif / 2; // mid point
k1 = k0 + pdif / 4; // 3/4 point
k2 = p0 + pdif * (1 / sqrt(2));-a // .707... point
...

So, there are an infinite number of points on the line segment from p0
and p1. Right? p0 + pdif * (0...1)?

--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
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From Newsgroup: comp.theory

On 2025-08-26, Mr Flibble <flibble@red-dwarf.jmc.corp> wrote:

On Tue, 26 Aug 2025 12:40:05 +0300, Mikko wrote:

On 2025-08-25 02:53:17 +0000, Mr Flibble said:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

That you can't see infinitesimals does not mean that they don't exist.
It only means that if they exist they are too small to be seen.

False, there is always a number smaller than the value of a claimed infinitesimal.

Therefore, an infinitesimal has to be something which is not
(that kind of) a number. A positive infinitesimal is something outside of
the real numbers which is closer to zero than any positive real.

You can just postulate the existence of such a thing as an axiom,
and see where that goes.
--
TXR Programming Language: http://nongnu.org/txr
Cygnal: Cygwin Native Application Library: http://kylheku.com/cygnal
Mastodon: @Kazinator@mstdn.ca
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From Newsgroup: comp.theory

On 2025-08-26, olcott <polcott333@gmail.com> wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

I don't think that's how it works.

Note [0, 1] and [0, 2] contain the same number of real numbers;
and two line segments of different lengths contain the same number of points. --
TXR Programming Language: http://nongnu.org/txr
Cygnal: Cygwin Native Application Library: http://kylheku.com/cygnal
Mastodon: @Kazinator@mstdn.ca
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From Newsgroup: comp.theory

On 8/26/2025 4:36 PM, Kaz Kylheku wrote:

On 2025-08-26, olcott <polcott333@gmail.com> wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem >>>>> proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

I don't think that's how it works.

Note [0, 1] and [0, 2] contain the same number of real numbers;
and two line segments of different lengths contain the same number of points.

That ignores the difference of length of these
two line segments: [0.0, 1.0] - [0.0, 1.0)
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On Tue, 26 Aug 2025 13:21:16 -0700, Chris M. Thomasson wrote:

On 8/26/2025 10:29 AM, Mr Flibble wrote:

On Tue, 26 Aug 2025 12:40:05 +0300, Mikko wrote:

On 2025-08-25 02:53:17 +0000, Mr Flibble said:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

That you can't see infinitesimals does not mean that they don't exist.
It only means that if they exist they are too small to be seen.

False, there is always a number smaller than the value of a claimed
infinitesimal.

Indeed. .1, .01, .001, .001, .0001, ...

limit 0.

We can see the steps, and we know its limit is 0 wrt infinity. Looking
at it all the way down, we will never see an iteration as perfectly
equal to its limit... Fair enough?

I was talking about infinitesimals not limits: they are different concepts
(in fact limits supplanted infinitesimals in 19th century calculus).

/Flibble
--
meet ever shorter deadlines, known as "beat the clock"
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From Newsgroup: comp.theory

On 27/08/2025 00:53, Mr Flibble wrote:

I was talking about infinitesimals not limits: they are different concepts (in fact limits supplanted infinitesimals in 19th century calculus).

Indeed, but infinitesimals came back in the 20thC, in the forms
of (a) non-standard analysis, (b) surreal numbers, and (c) [combinatorial] games. Before people write complete nonsense on the subject, I suggest
that at least they should wiki those topics. They aren't particularly difficult, and they're very interesting [at least to mathematicians].
--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Spindler
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From Newsgroup: comp.theory

On 8/26/25 1:29 PM, Mr Flibble wrote:

On Tue, 26 Aug 2025 12:40:05 +0300, Mikko wrote:

On 2025-08-25 02:53:17 +0000, Mr Flibble said:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

That you can't see infinitesimals does not mean that they don't exist.
It only means that if they exist they are too small to be seen.

False, there is always a number smaller than the value of a claimed infinitesimal.

/Flibble

Are you sure?

What value is smaller that delta in the number system of infinitesimals,
where all infinitesimals are multiples of delta.

They work sort of like the integers, being spaced apart.


Just as there is no integer between 0 and 1, there is no infinitesimal
between 0 and delta.

Note, once you introduce the term infinitesimal, you are allowing the
space they are defined in.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On Tue, 26 Aug 2025 21:49:06 -0400, Richard Damon wrote:

On 8/26/25 1:29 PM, Mr Flibble wrote:

On Tue, 26 Aug 2025 12:40:05 +0300, Mikko wrote:

On 2025-08-25 02:53:17 +0000, Mr Flibble said:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

That you can't see infinitesimals does not mean that they don't exist.
It only means that if they exist they are too small to be seen.

False, there is always a number smaller than the value of a claimed
infinitesimal.

/Flibble

Are you sure?

What value is smaller that delta in the number system of infinitesimals, where all infinitesimals are multiples of delta.

They work sort of like the integers, being spaced apart.

Just as there is no integer between 0 and 1, there is no infinitesimal between 0 and delta.

Note, once you introduce the term infinitesimal, you are allowing the
space they are defined in.

I am talking about real numbers but of course you know this as you are a disingenuous f--k.

/Flibble
--
meet ever shorter deadlines, known as "beat the clock"
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From Newsgroup: comp.theory

On 8/26/25 9:52 PM, Mr Flibble wrote:

On Tue, 26 Aug 2025 21:49:06 -0400, Richard Damon wrote:

On 8/26/25 1:29 PM, Mr Flibble wrote:

On Tue, 26 Aug 2025 12:40:05 +0300, Mikko wrote:

On 2025-08-25 02:53:17 +0000, Mr Flibble said:

Much like infinitesimals, Olcott's refutations of the Halting Problem >>>>> proofs don't exist.

That you can't see infinitesimals does not mean that they don't exist. >>>> It only means that if they exist they are too small to be seen.

False, there is always a number smaller than the value of a claimed
infinitesimal.

/Flibble

Are you sure?

What value is smaller that delta in the number system of infinitesimals,
where all infinitesimals are multiples of delta.

They work sort of like the integers, being spaced apart.


Just as there is no integer between 0 and 1, there is no infinitesimal
between 0 and delta.

Note, once you introduce the term infinitesimal, you are allowing the
space they are defined in.

I am talking about real numbers but of course you know this as you are a disingenuous f--k.

/Flibble

Then the word "infintesimal" isn't defined, so you can't talk about them.

Or are you saying someone can say that 0.5 doesn't exist, because they
were talking about the integers WITHOUT SAYING SO,

Yes, infinitesimals don't exist in the Real Number System, but then so
doesn't infinity, or sqrt(-1), both of which are also concepts that
exist in extensions of the system.

The, the statement "Infintesimals don't exist" without a clear
indication that you mean to work in the Real Number system is just a lie.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 2025-08-26, olcott <polcott333@gmail.com> wrote:

On 8/26/2025 4:36 PM, Kaz Kylheku wrote:

On 2025-08-26, olcott <polcott333@gmail.com> wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem >>>>>> proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

I don't think that's how it works.

Note [0, 1] and [0, 2] contain the same number of real numbers;
and two line segments of different lengths contain the same number of points.

That ignores the difference of length of these
two line segments: [0.0, 1.0] - [0.0, 1.0)

Yes it does.

The length being different by an infinitesimal is a different concept
from the wrong idea that we are taking away one geometric point or one
real number (point on the number line).

An infinitesimal is smaller than any real number.

If the symbol +| represents the positive infinitesimal, then suppose
suppose we have the range [0, +|]. That range cannot be big enough
to contain a real number other than 0, because that would mean that
+| is at least as big as some nonzero positive real number.

Or something; given your track record with halting this is
likely pointless to debate. I'm as unprepared for this as you are,
and that likely makes one of us who will ever admit it.
--
TXR Programming Language: http://nongnu.org/txr
Cygnal: Cygwin Native Application Library: http://kylheku.com/cygnal
Mastodon: @Kazinator@mstdn.ca
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From Newsgroup: comp.theory

On 2025-08-27, Mr Flibble <flibble@red-dwarf.jmc.corp> wrote:

On Tue, 26 Aug 2025 21:49:06 -0400, Richard Damon wrote:

Note, once you introduce the term infinitesimal, you are allowing the
space they are defined in.

I am talking about real numbers but of course you know this as you are a disingenuous f--k.

Now you're just farting and lighting a match, while trying to use subtle psychological tricks to make everyone believe you're gaslighting.

If only everyone would pay attention to what you're SAYING instead
of obsessively focusing on assuming that you must be wrong ...
--
TXR Programming Language: http://nongnu.org/txr
Cygnal: Cygwin Native Application Library: http://kylheku.com/cygnal
Mastodon: @Kazinator@mstdn.ca
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From Newsgroup: comp.theory

On 2025-08-26 20:30:51 +0000, olcott said:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

No, it is not. It is a singlet. A singlet is not infinitesimal.
--
Mikko

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/27/2025 12:31 AM, Kaz Kylheku wrote:

On 2025-08-26, olcott <polcott333@gmail.com> wrote:

On 8/26/2025 4:36 PM, Kaz Kylheku wrote:

On 2025-08-26, olcott <polcott333@gmail.com> wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem >>>>>>> proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

I don't think that's how it works.

Note [0, 1] and [0, 2] contain the same number of real numbers;
and two line segments of different lengths contain the same number of points.

That ignores the difference of length of these
two line segments: [0.0, 1.0] - [0.0, 1.0)

Yes it does.

The length being different by an infinitesimal is a different concept
from the wrong idea that we are taking away one geometric point or one
real number (point on the number line).

An infinitesimal is smaller than any real number.

If the symbol +| represents the positive infinitesimal, then suppose
suppose we have the range [0, +|]. That range cannot be big enough
to contain a real number other than 0, because that would mean that
+| is at least as big as some nonzero positive real number.

Or something; given your track record with halting this is
likely pointless to debate. I'm as unprepared for this as you are,
and that likely makes one of us who will ever admit it.

The difference of length of these two
line segments: [0.0, 1.0] - [0.0, 1.0)
Seems to concretely show an example of an infinitesimal.

The difference of length of these two
line segments: [0.0, 1.0] - (0.0, 1.0)
Seems to concretely show an example of two infinitesimals.

That is as far as I ever got with this.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On Wed, 2025-08-27 at 08:39 -0500, olcott wrote:

On 8/27/2025 12:31 AM, Kaz Kylheku wrote:

On 2025-08-26, olcott <polcott333@gmail.com> wrote:

On 8/26/2025 4:36 PM, Kaz Kylheku wrote:

On 2025-08-26, olcott <polcott333@gmail.com> wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

I don't think that's how it works.

Note [0, 1] and [0, 2] contain the same number of real numbers;
and two line segments of different lengths contain the same number of points.

That ignores the difference of length of these
two line segments: [0.0, 1.0] - [0.0, 1.0)

Yes it does.

The length being different by an infinitesimal is a different concept
from the wrong idea that we are taking away one geometric point or one
real number (point on the number line).

An infinitesimal is smaller than any real number.

If the symbol +| represents the positive infinitesimal, then suppose suppose we have the range [0, +|]. That range cannot be big enough
to contain a real number other than 0, because that would mean that
+| is at least as big as some nonzero positive real number.

Or something; given your track record with halting this is
likely pointless to debate. I'm as unprepared for this as you are,
and that likely makes one of us who will ever admit it.

The difference of length of these two
line segments: [0.0, 1.0] - [0.0, 1.0)
Seems to concretely show an example of an infinitesimal.

The difference of length of these two
line segments: [0.0, 1.0] - (0.0, 1.0)
Seems to concretely show an example of two infinitesimals.

That is as far as I ever got with this.

At last, you woke up at this problem. You are absolutely correct.
Typically, there are infinitely many infinities, thus infinitesimals
For example:
123...
345...
967... // Infinitely many such numbers. 1/n gets as many infinitesimals.
If infinite counting is invalid, don't use method of exhaustion, pi, then-a calculus, at least, is in trouble.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/27/2025 3:09 AM, Mikko wrote:

On 2025-08-26 20:30:51 +0000, olcott said:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

No, it is not. It is a singlet. A singlet is not infinitesimal.

I am stipulating that the difference in the
length of the two line segments is defined
to be infinitesimal.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/27/2025 6:39 AM, olcott wrote:

On 8/27/2025 12:31 AM, Kaz Kylheku wrote:

On 2025-08-26, olcott <polcott333@gmail.com> wrote:

On 8/26/2025 4:36 PM, Kaz Kylheku wrote:

On 2025-08-26, olcott <polcott333@gmail.com> wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting >>>>>>>> Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

I don't think that's how it works.

Note [0, 1] and [0, 2] contain the same number of real numbers;
and two line segments of different lengths contain the same number
of points.

That ignores the difference of length of these
two line segments: [0.0, 1.0] - [0.0, 1.0)

Yes it does.

The length being different by an infinitesimal is a different concept
from the wrong idea that we are taking away one geometric point or one
real number (point on the number line).

An infinitesimal is smaller than any real number.

If the symbol +| represents the positive infinitesimal, then suppose
suppose we have the range [0, +|]. That range cannot be big enough
to contain a real number other than 0, because that would mean that
+| is at least as big as some nonzero positive real number.

Or something; given your track record with halting this is
likely pointless to debate. I'm as unprepared for this as you are,
and that likely makes one of us who will ever admit it.

The difference of length of these two
line segments: [0.0, 1.0] - [0.0, 1.0)
Seems to concretely show an example of an infinitesimal.

The difference of length of these two
line segments: [0.0, 1.0] - (0.0, 1.0)
Seems to concretely show an example of two infinitesimals.

That is as far as I ever got with this.

.1, .01, .001, ...

Strives to get close to an infinitesimal... ;^)
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 2025-08-27 15:10:18 +0000, olcott said:

On 8/27/2025 3:09 AM, Mikko wrote:

On 2025-08-26 20:30:51 +0000, olcott said:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

No, it is not. It is a singlet. A singlet is not infinitesimal.

I am stipulating that the difference in the
length of the two line segments is defined
to be infinitesimal.

You weren't in your message quoted above.
--
Mikko

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 28/08/2025 07:40, Mikko wrote:

On 2025-08-27 15:10:18 +0000, olcott said:

On 8/27/2025 3:09 AM, Mikko wrote:

On 2025-08-26 20:30:51 +0000, olcott said:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the
Halting Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

No, it is not. It is a singlet. A singlet is not infinitesimal.

I am stipulating that the difference in the
length of the two line segments is defined
to be infinitesimal.

You weren't in your message quoted above.

The clue is in the word "stipulating".

Your chain is being yanked.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

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From Newsgroup: comp.theory

On 8/28/2025 1:40 AM, Mikko wrote:

On 2025-08-27 15:10:18 +0000, olcott said:

On 8/27/2025 3:09 AM, Mikko wrote:

On 2025-08-26 20:30:51 +0000, olcott said:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem >>>>> proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

No, it is not. It is a singlet. A singlet is not infinitesimal.

I am stipulating that the difference in the
length of the two line segments is defined
to be infinitesimal.

You weren't in your message quoted above.

This is a statement and not a question.
[0.0, 1.0] - [0.0, 1.0) == infinitesimal.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 28/08/2025 14:51, olcott wrote:

On 8/28/2025 1:40 AM, Mikko wrote:

On 2025-08-27 15:10:18 +0000, olcott said:

I am stipulating that the difference in the
length of the two line segments is defined
to be infinitesimal.

You weren't in your message quoted above.

This is a statement and not a question.
[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

It is /literally/ axiomatic that in "real" arithmetic and
geometry, there are no infinitesimals. This is the axiom of either
Archimedes or Eudoxus [take your pick], so was known to the ancient
Greeks. Also known to the ancient Greeks: a POINT is that which
has no parts, or which has no magnitude [Euclid, "Elements", book 1,
definition 1 in Todhunter's translation]. Note: /no/ magnitude,
not "infinitesimal" magnitude. So if you're going to "stipulate"
that the difference between two lengths, differing by one point,
is infinitesimal then those lengths can't be "real". What other
system of arithmetic and geometry are you stipulating? Enquiring
minds wish to know.

[There are infinitesimal hyper-reals, surreals and games
(amongst others), which are as real as any abstraction ever is.
But you owe it to your readers to explain what /you/ mean.]
--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Wolf
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From Newsgroup: comp.theory

On 8/28/2025 11:17 AM, Andy Walker wrote:

On 28/08/2025 14:51, olcott wrote:

On 8/28/2025 1:40 AM, Mikko wrote:

On 2025-08-27 15:10:18 +0000, olcott said:

I am stipulating that the difference in the
length of the two line segments is defined
to be infinitesimal.

You weren't in your message quoted above.

This is a statement and not a question.
[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

-a-a-a-aIt is /literally/ axiomatic that in "real" arithmetic and
geometry, there are no infinitesimals.

Sure and in this same way there are no dead
cats in ice cream cartons. "Real number" is
just a figure-of-speech name that does not
entail that numbers outside of the set of "reals"
are fake.

This is the axiom of either
Archimedes or Eudoxus [take your pick], so was known to the ancient
Greeks.-a Also known to the ancient Greeks:-a a POINT is that which
has no parts, or which has no magnitude [Euclid, "Elements", book 1, definition 1 in Todhunter's translation].-a Note:-a /no/ magnitude,
not "infinitesimal" magnitude.-a So if you're going to "stipulate"
that the difference between two lengths, differing by one point,
is infinitesimal then those lengths can't be "real".-a What other
system of arithmetic and geometry are you stipulating?-a Enquiring
minds wish to know.

-a-a-a-a[There are infinitesimal hyper-reals, surreals and games
(amongst others), which are as real as any abstraction ever is.
But you owe it to your readers to explain what /you/ mean.]

--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
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From Newsgroup: comp.theory

On Thu, 28 Aug 2025 08:51:58 -0500, olcott wrote:

On 8/28/2025 1:40 AM, Mikko wrote:

On 2025-08-27 15:10:18 +0000, olcott said:

On 8/27/2025 3:09 AM, Mikko wrote:

On 2025-08-26 20:30:51 +0000, olcott said:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting
Problem proofs don't exist.

/Flibble

The difference in the length of a line of these two line segments
using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

No, it is not. It is a singlet. A singlet is not infinitesimal.

I am stipulating that the difference in the length of the two line
segments is defined to be infinitesimal.

You weren't in your message quoted above.

This is a statement and not a question. [0.0, 1.0] - [0.0, 1.0) == infinitesimal.

For what ever real number you think results from that subtraction there
will always be a real number smaller than it ergo it is not an
infinitesimal. For real numbers there are no infinitesimals -- this is axiomatic.

/Flibble
--
meet ever shorter deadlines, known as "beat the clock"
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From Newsgroup: comp.theory

On 28/08/2025 17:44, olcott wrote:

[...] "Real number" is
just-a a figure-of-speech name that does not
entail that numbers outside of the set of "reals"
are fake.

No-one said "fake". People are claiming that infinitesimals
do [or do not] exist. That depends on the universe of discourse, in
the same way that words such as "bonjour" or "herr" exist in some
languages but not [natively] in others. So if you [or others] make
dogmatic statements about such words, there's no point discussing them
unless that universe is known and agreed. Absent a context, maths is
usually discussed in terms of the "standard" real numbers, Euclidean
geometry, etc., with standard results in algebra, calculus and so on.
So, your unqualified "stipulation" was, quite simply, wrong; there
are no infinitesimal lengths. In other contexts, less usual, you could
be correct or you could again be wrong. So I ask again: what context
are you assuming for your stipulation? If you're going to invent your
own mathematical theories, the rest of us need to know what the basis
is for them; otherwise your claims are quite literally nonsense.
--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Wolf
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/28/2025 1:34 PM, Andy Walker wrote:

On 28/08/2025 17:44, olcott wrote:

[...] "Real number" is
just-a a figure-of-speech name that does not
entail that numbers outside of the set of "reals"
are fake.

-a-a-a-aNo-one said "fake".-a People are claiming that infinitesimals
do [or do not] exist.-a That depends on the universe of discourse, in
the same way that words such as "bonjour" or "herr" exist in some
languages but not [natively] in others.-a So if you [or others] make
dogmatic statements about such words, there's no point discussing them
unless that universe is known and agreed.

Good so far.

Absent a context, maths is
usually discussed in terms of the "standard" real numbers, Euclidean geometry, etc., with standard results in algebra, calculus and so on.
So, your unqualified "stipulation" was, quite simply, wrong;

Not at all really.
I defined one concrete instance of an infinitesimal.

there
are no infinitesimal lengths.

I proved that assumption is false.
There is a difference in the length of the two
specified line segments otherwise they would
be exactly one-and-the-same line segment.

In other contexts, less usual, you could
be correct or you could again be wrong.-a So I ask again:-a what context
are you assuming for your stipulation?

In what way is the notion of an infinitesimal length not incoherent?
[0.0, 1.0] - [0.0, 1,0) is this way.

If you're going to invent your
own mathematical theories, the rest of us need to know what the basis
is for them;-a otherwise your claims are quite literally nonsense.

--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
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From Newsgroup: comp.theory

On 28/08/2025 20:30, olcott wrote:
[I wrote:]

Absent a context, maths is
usually discussed in terms of the "standard" real numbers, Euclidean
geometry, etc., with standard results in algebra, calculus and so on.
So, your unqualified "stipulation" was, quite simply, wrong;

Not at all really.
I defined one concrete instance of an infinitesimal.

You /claimed/ to have it, but your claim directly contradicts one
of the axioms. So your claim is incorrect /in standard mathematics/. It
/may/ be correct in other contexts where the axioms are different. You
do surely understand that axioms are true /by definition/, so that if
something contradicts an axiom it is incorrect /in that context/? Maths
has different contexts, just as do both natural and computer languages;
you can't expect to speak Swahili in France and be widely understood, you
can't feed Pascal source to a C compiler and expect it to work, and you
can't expect maths to work unchanged irrespective of the axioms used.

-athere
are no infinitesimal lengths.

I proved that assumption is false.

You didn't /prove/ anything; you asserted it, without the context
that is needed.

There is a difference in the length of the two
specified line segments otherwise they would
be exactly one-and-the-same line segment.

No, in standard mathematics they are /exactly/ the same /length/; recall that by Euclid, book 1, definition 1, the one point by which they
differ has /no/ magnitude [not infinitesimal magnitude]. "Same length"
is not the same as "exactly the same". In other contexts [such as combinatorial game theory] your claim /could/ be correct.

-aIn other contexts, less usual, you could
be correct or you could again be wrong.-a So I ask again:-a what context
are you assuming for your stipulation?

In what way is the notion of an infinitesimal length not incoherent?

No-one said it was "incoherent". But it's not part of standard mathematics. You can't use it unless you supply a context in which the Archimedean axiom does not apply. Archimedes and Euclid have better
provenance than you, so /by default/ their axioms trump yours, esp as
yours have so far gone unstated.

You can't treat mathematics in a cavalier way or it will come back
and bite you. I've explained this at least three times recently; I'm
happy to answer questions, but a dogmatic unfounded claim without a clear statement of context is not a question. So if you make such a claim, you
may assume that my answer is to repeat these three articles; there won't
be other responses on my part.
--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Wolf
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From Newsgroup: comp.theory

On 8/28/2025 3:44 PM, Andy Walker wrote:

On 28/08/2025 20:30, olcott wrote:
[I wrote:]

Absent a context, maths is
usually discussed in terms of the "standard" real numbers, Euclidean
geometry, etc., with standard results in algebra, calculus and so on.
So, your unqualified "stipulation" was, quite simply, wrong;

Not at all really.
I defined one concrete instance of an infinitesimal.

-a-a-a-aYou /claimed/ to have it, but your claim directly contradicts one
of the axioms.-a So your claim is incorrect /in standard mathematics/.-a It /may/ be correct in other contexts where the axioms are different.-a You
do surely understand that axioms are true /by definition/, so that if something contradicts an axiom it is incorrect /in that context/?-a Maths
has different contexts, just as do both natural and computer languages;
you can't expect to speak Swahili in France and be widely understood, you can't feed Pascal source to a C compiler and expect it to work, and you
can't expect maths to work unchanged irrespective of the axioms used.

-athere
are no infinitesimal lengths.

I proved that assumption is false.

-a-a-a-aYou didn't /prove/ anything;-a you asserted it, without the context that is needed.

There is a difference in the length of the two
specified line segments otherwise they would
be exactly one-and-the-same line segment.

-a-a-a-aNo, in standard mathematics they are /exactly/ the same /length/; recall that by Euclid, book 1, definition 1, the one point by which they differ has /no/ magnitude [not infinitesimal magnitude].-a "Same length"
is not the same as "exactly the same".-a In other contexts [such as combinatorial game theory] your claim /could/ be correct.

-aIn other contexts, less usual, you could
be correct or you could again be wrong.-a So I ask again:-a what context >>> are you assuming for your stipulation?

In what way is the notion of an infinitesimal length not incoherent?

-a-a-a-aNo-one said it was "incoherent".-a But it's not part of standard mathematics.

*Yes new ideas are not part of any existing standard*

There *is* a difference in the length of these two line
segments: [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

This seems to overrule the idea in Geometry that
a geometric point has no size at all because the
difference in length is one geometric point.

You can't use it unless you supply a context in which the
Archimedean axiom does not apply.-a Archimedes and Euclid have better provenance than you, so /by default/ their axioms trump yours, esp as
yours have so far gone unstated.

-a-a-a-aYou can't treat mathematics in a cavalier way or it will come back and bite you.-a I've explained this at least three times recently;-a I'm happy to answer questions, but a dogmatic unfounded claim without a clear statement of context is not a question.-a So if you make such a claim, you may assume that my answer is to repeat these three articles;-a there won't
be other responses on my part.

You seems to think of mathematics as a carefully
memorized set of rules. I look at it more broadly
as the process of determining what ideas are semantically
entailed by other ideas.

When we start with the interval notation and the vague
notion that infinitesimal means infinitely small then
we can see that the difference in length of these two
line segments [0.0,1.0] - [0.0,1.0) is infinitely small.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
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From Newsgroup: comp.theory

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line
segments: [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

There is not, at least in a context where points on a
line can correspond to real numbers.

If your statement is meant to be understood in the context of
hyperreal or surreal numbers, or something similar, say so.

I'll note again that the name "real" does not imply that the real
number system has a kind of existence that other number systems lack.
Real numbers happen to be used far more commonly than other systems.
All numbers are abstract concepts. But when someone talks about
entities like 0.0 and 1.0, it's reasonable to assume they're
referring to real numbers unless they explicitly establish a
different context.

This seems to overrule the idea in Geometry that
a geometric point has no size at all because the
difference in length is one geometric point.

It does not.

A geometric length is not a count of geometric points. If it were,
then the line segments [0.0,1.0] and [0.0,2.0] would have the same
length, since they have the same cardinality when viewed as sets of
points. The length of a line segment is a real number that is the
Euclidean distance between its endpoints. (It might be something
else in a different system, but we're assuming Euclidean geometry
and real numbers until you say differently.)

The two line segments [0.0,1.0] and [0.0,1.0) differ in that one
of them includes the point 1.0 and the other does not. They do
not differ in *length*. They differ in which points they include,
but they do not differ in *how many* points they include (infinities
don't behave like finite real numbers; aleph-1 + 0 = aleph-1).

If you disagree, please provide a rigorous definition of "length"
for a line segment.

And if that doesn't convince you, I *stipulate* that you're wrong.
Or is "stipulate" a magic word only when you use it?

[...]
--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
void Void(void) { Void(); } /* The recursive call of the void */
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From Newsgroup: comp.theory

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line
segments: [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

There is not, at least in a context where points on a
line can correspond to real numbers.

If your statement is meant to be understood in the context of
hyperreal or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the
basis of defining two line segments that differ in
length by one geometric point. THAT IS 100% OF THE WHOLE CONTEXT.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On Thu, 28 Aug 2025 19:05:41 -0500, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line segments:
[0.0,1.0] - [0.0,1.0) and this difference *is* infinitesimal.

There is not, at least in a context where points on a line can
correspond to real numbers.

If your statement is meant to be understood in the context of hyperreal
or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the basis of defining
two line segments that differ in length by one geometric point. THAT IS
100% OF THE WHOLE CONTEXT.

The "length" of a point is zero.

/Flibble
--
meet ever shorter deadlines, known as "beat the clock"
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From Newsgroup: comp.theory

On 8/26/2025 1:49 PM, olcott wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

Do you mean the difference between points p0 and p1? This difference is normalized within the line segment itself. It points in the direction of
p0 to p1. In the simple line code I showed.

Take 2, 2-ary points that are not equal to each other. Typed out in
the newsreader sorry for any typos:

p0 = (-1, 0);
p1 = (1, 0);
pdif = p1 - p0; // differential

k0 = p0 + pdif / 2; // mid point
k1 = k0 + pdif / 4; // 3/4 point
k2 = p0 + pdif * (1 / sqrt(2));-a // .707... point
...


So, there are an infinite number of points on the line segment from p0
and p1. Right? p0 + pdif * (0...1)?

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From Newsgroup: comp.theory

On 8/28/2025 5:14 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:49 PM, olcott wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem >>>>> proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

Do you mean the difference between points p0 and p1?

I should clarify that the difference is from p1 - p0 as in:

pdif = p1 - p0

This difference is
normalized within the line segment itself. It points in the direction of
p0 to p1. In the simple line code I showed.

Take 2, 2-ary points that are not equal to each other. Typed out in
the newsreader sorry for any typos:

p0 = (-1, 0);
p1 = (1, 0);
pdif = p1 - p0; // differential

k0 = p0 + pdif / 2; // mid point
k1 = k0 + pdif / 4; // 3/4 point
k2 = p0 + pdif * (1 / sqrt(2));-a // .707... point
...


So, there are an infinite number of points on the line segment from
p0 and p1. Right? p0 + pdif * (0...1)?

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From Newsgroup: comp.theory

On 29/08/2025 01:05, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line
segments:-a [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

There is not, at least in a context where points on a
line can correspond to real numbers.

If your statement is meant to be understood in the context of
hyperreal or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the
basis of defining two line segments that differ in
length by one geometric point. THAT IS 100% OF THE WHOLE CONTEXT.

The length of one geometric point is 0.

By your definition, then, the only infinitesimal is 0.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

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From Newsgroup: comp.theory

On 8/28/2025 7:14 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:49 PM, olcott wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem >>>>> proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

Do you mean the difference between points p0 and p1? This difference is normalized within the line segment itself. It points in the direction of
p0 to p1. In the simple line code I showed.

I am only referring to the above two line segments
and any attempt to change this subject will be ignored.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
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From Newsgroup: comp.theory

On 8/28/2025 7:10 PM, Mr Flibble wrote:

On Thu, 28 Aug 2025 19:05:41 -0500, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line segments:
[0.0,1.0] - [0.0,1.0) and this difference *is* infinitesimal.

There is not, at least in a context where points on a line can
correspond to real numbers.

If your statement is meant to be understood in the context of hyperreal
or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the basis of defining
two line segments that differ in length by one geometric point. THAT IS
100% OF THE WHOLE CONTEXT.

The "length" of a point is zero.

/Flibble

Apparently that has always been a misconception.
[0.0,1.0] - [0.0,1.0) = infinitesimal, thus not zero.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/28/2025 7:18 PM, Richard Heathfield wrote:

On 29/08/2025 01:05, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line
segments:-a [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

There is not, at least in a context where points on a
line can correspond to real numbers.

If your statement is meant to be understood in the context of
hyperreal or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the
basis of defining two line segments that differ in
length by one geometric point. THAT IS 100% OF THE WHOLE CONTEXT.

The length of one geometric point is 0.

By your definition, then, the only infinitesimal is 0.

[0.0,1.0] - [0.0,1.0) is different by one geometric point
and it is different, thus non zero.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

olcott <polcott333@gmail.com> writes:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line
segments: [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

There is not, at least in a context where points on a
line can correspond to real numbers.
If your statement is meant to be understood in the context of
hyperreal or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the
basis of defining two line segments that differ in
length by one geometric point. THAT IS 100% OF THE WHOLE CONTEXT.

You usually quote an entire article when you post a followup.
Here you've snipped the context in which I refute your claim.

Lengths do not differ by one point.

You appear to believe that the "length" of a line segment is defined
in terms of the number of points it contains, so that removing one
point changes the length. I explained, in text that you snipped,
that that's incorrect.

If you like, you can post a followup to my previous post in this
thread and respond to each point I made.

Or you can do anything else, and this will be my last post in this
thread. If you insist on being wrong *and* on refusing to address
any points I've made, I'm content to sit back and let you do that.
Of course this will deny you some of the attention that you appear
to crave.
--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
void Void(void) { Void(); } /* The recursive call of the void */
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 29/08/2025 02:46, olcott wrote:

On 8/28/2025 7:18 PM, Richard Heathfield wrote:

On 29/08/2025 01:05, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line
segments:-a [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

There is not, at least in a context where points on a
line can correspond to real numbers.

If your statement is meant to be understood in the context of
hyperreal or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the
basis of defining two line segments that differ in
length by one geometric point. THAT IS 100% OF THE WHOLE CONTEXT.

The length of one geometric point is 0.

By your definition, then, the only infinitesimal is 0.

[0.0,1.0] - [0.0,1.0) is different by one geometric point

Geometric points are 0-dimensional, so they have no size.

and it is different, thus non zero.

Okay; clearly we aren't working from the same axioms, so
discussion's rather pointless, wouldn't you say?

Good luck with your new mathematics.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/28/2025 8:59 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line
segments: [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

There is not, at least in a context where points on a
line can correspond to real numbers.
If your statement is meant to be understood in the context of
hyperreal or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the
basis of defining two line segments that differ in
length by one geometric point. THAT IS 100% OF THE WHOLE CONTEXT.

You usually quote an entire article when you post a followup.
Here you've snipped the context in which I refute your claim.

Lengths do not differ by one point.

These do differ by one point.
[0.0,1.0] - [0.0,1.0)
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/28/2025 9:04 PM, Richard Heathfield wrote:

On 29/08/2025 02:46, olcott wrote:

On 8/28/2025 7:18 PM, Richard Heathfield wrote:

On 29/08/2025 01:05, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line
segments:-a [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

There is not, at least in a context where points on a
line can correspond to real numbers.

If your statement is meant to be understood in the context of
hyperreal or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the
basis of defining two line segments that differ in
length by one geometric point. THAT IS 100% OF THE WHOLE CONTEXT.

The length of one geometric point is 0.

By your definition, then, the only infinitesimal is 0.

[0.0,1.0] - [0.0,1.0) is different by one geometric point

Geometric points are 0-dimensional, so they have no size.

Yes they are defined that way, yet that does
not account for the difference between the
length of these two line segments:
[0.0,1.0] - [0.0,1.0)

I think that the convention is to incorrectly
say that they are identical in length.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On Thu, 2025-08-28 at 22:23 -0500, olcott wrote:

On 8/28/2025 9:04 PM, Richard Heathfield wrote:

On 29/08/2025 02:46, olcott wrote:

On 8/28/2025 7:18 PM, Richard Heathfield wrote:

On 29/08/2025 01:05, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line segments:-a [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

There is not, at least in a context where points on a
line can correspond to real numbers.

If your statement is meant to be understood in the context of hyperreal or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the
basis of defining two line segments that differ in
length by one geometric point. THAT IS 100% OF THE WHOLE CONTEXT.

The length of one geometric point is 0.

By your definition, then, the only infinitesimal is 0.

[0.0,1.0] - [0.0,1.0) is different by one geometric point

Geometric points are 0-dimensional, so they have no size.

Yes they are defined that way, yet that does
not account for the difference between the
length of these two line segments:
[0.0,1.0] - [0.0,1.0)

I think that the convention is to incorrectly
say that they are identical in length.

Snipet from https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/download
.......
Proposition 3: Repeating decimals are irrational numbers.
Proof: Assume that the recurring decimal x, x<0, x has a recurring section S,
and is written in a base system as x=0.SreUSreeSrea...SreR. Then according to
the definition x= SreU/b +Sree/b-# +Srea/b-| +... +SreR/b^reR = qreU +qree +qrea +...+qreR
(breerao). x is composed of the continuous addition of infinite number of
non-zero rational numbers and the value of x is strictly increasing or
decreasing. From Theorem 2 and Axiom 2, if x can be expressed in the
form of p/q, then p and q are at least not finite numbers (non-natural
numbers). Therefore, x is not a rational number. And since a non-
rational number is an irrational number, the proposition is proved.
Note: Here is a brief explanation of common algebraic magic:
(1) x= 0.999...
(2) 10x= 9+x // 10x= 9.999...
(3) 9x=9
(4)x=1
Solution: There is no axiom or theorem that can prove (1)<=>(2).
(2) is one of the infinite interpretations of (1) (or (2) is
the definition of x,..., etc.). In short, (2),(1) cannot be
inferred from each other (no necessary relationship, or
still needs to be proved).
There are many other examples, e.g.:
1. If 0.999...= 1 holds, the unique prime factorization theorem for
positive integers will not hold:
0.999...= 999.../1000...= 9*(111...)/(5*2)... =1
<=> 3*3*(111...)= (5*2)*...
lhs integer contains the prime number 3, and the rhs integer cannot
contain 3.
2. Are interval [0,1),[0,1] equal? 0.999...ree[0,1)? What number is to
the left of 1 (infinitely close)? Can we define a number x that is
infinitely close to c, but not equal to c?
......
Proposition 8: There are infinitely many infinitely large numbers.
Proof: The proof can be obtained by combining Axiom 1 and Axiom 3. Proposition 9: Real numbers include infinitely large and infinitely small
numbers.
Proof: From the above analysis (Real numbers contain the concept of infinite
steps), 1-0.999... is an infinitely small number. And taking the
reciprocal of the infinite decimal, we can get an infinitely
large number. Therefore, under the premise that real number division
requires closure, real numbers must contain both infinitely large and
infinitely small numbers.
Proof2: x>0 <=> x/2 >0 always holds. Because real number introduces infinite
steps (e.g. circumference of a circle, limit, calculus).
So x>0 <=> x/2>0 <=> x/4>0 <=> ... <=> x/2^reR >0
<=> "infinitely small > 0"
(This is equivalent to showing that real numbers contain infinity and
infinitesimal).

....
-------------
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/28/2025 5:10 PM, Mr Flibble wrote:

On Thu, 28 Aug 2025 19:05:41 -0500, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line segments:
[0.0,1.0] - [0.0,1.0) and this difference *is* infinitesimal.

There is not, at least in a context where points on a line can
correspond to real numbers.

If your statement is meant to be understood in the context of hyperreal
or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the basis of defining
two line segments that differ in length by one geometric point. THAT IS
100% OF THE WHOLE CONTEXT.

The "length" of a point is zero.

What about... A point is a location. It's "length" can be its distance
from the origin?

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/28/2025 9:08 PM, Chris M. Thomasson wrote:

On 8/28/2025 5:10 PM, Mr Flibble wrote:

On Thu, 28 Aug 2025 19:05:41 -0500, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line segments:
[0.0,1.0] - [0.0,1.0) and this difference *is* infinitesimal.

There is not, at least in a context where points on a line can
correspond to real numbers.

If your statement is meant to be understood in the context of hyperreal >>>> or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the basis of defining
two line segments that differ in length by one geometric point. THAT IS
100% OF THE WHOLE CONTEXT.

The "length" of a point is zero.

What about... A point is a location. It's "length" can be its distance
from the origin?

A point is a location. Now we can take said point and do other things.
Such as take its length from origin.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/28/2025 5:23 PM, olcott wrote:

On 8/28/2025 7:14 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:49 PM, olcott wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem >>>>>> proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

Do you mean the difference between points p0 and p1? This difference
is normalized within the line segment itself. It points in the
direction of p0 to p1. In the simple line code I showed.

I am only referring to the above two line segments
and any attempt to change this subject will be ignored.

A line segment is finite. However, there are infinite points on said line... --- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On Thu, 2025-08-28 at 21:08 -0700, Chris M. Thomasson wrote:

On 8/28/2025 5:10 PM, Mr Flibble wrote:

On Thu, 28 Aug 2025 19:05:41 -0500, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line segments: [0.0,1.0] - [0.0,1.0) and this difference *is* infinitesimal.

There is not, at least in a context where points on a line can correspond to real numbers.

If your statement is meant to be understood in the context of hyperreal or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the basis of defining
two line segments that differ in length by one geometric point. THAT IS 100% OF THE WHOLE CONTEXT.

The "length" of a point is zero.

What about... A point is a location. It's "length" can be its distance
from the origin?

Exactly !!!
Many people are confused with point/number and distance/interval.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 2025-08-28 13:51:58 +0000, olcott said:

On 8/28/2025 1:40 AM, Mikko wrote:

On 2025-08-27 15:10:18 +0000, olcott said:

On 8/27/2025 3:09 AM, Mikko wrote:

On 2025-08-26 20:30:51 +0000, olcott said:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem >>>>>> proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

No, it is not. It is a singlet. A singlet is not infinitesimal.

I am stipulating that the difference in the
length of the two line segments is defined
to be infinitesimal.

You weren't in your message quoted above.

This is a statement and not a question.
[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

That was understood. More specifically, it is a false statement.
--
Mikko

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 2025-08-28 16:17:57 +0000, Andy Walker said:

On 28/08/2025 14:51, olcott wrote:

On 8/28/2025 1:40 AM, Mikko wrote:

On 2025-08-27 15:10:18 +0000, olcott said:

I am stipulating that the difference in the
length of the two line segments is defined
to be infinitesimal.

You weren't in your message quoted above.

This is a statement and not a question.
[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

It is /literally/ axiomatic that in "real" arithmetic and
geometry, there are no infinitesimals. This is the axiom of either Archimedes or Eudoxus [take your pick], so was known to the ancient
Greeks. Also known to the ancient Greeks: a POINT is that which
has no parts, or which has no magnitude [Euclid, "Elements", book 1, definition 1 in Todhunter's translation]. Note: /no/ magnitude,
not "infinitesimal" magnitude. So if you're going to "stipulate"
that the difference between two lengths, differing by one point,
is infinitesimal then those lengths can't be "real". What other
system of arithmetic and geometry are you stipulating? Enquiring
minds wish to know.

[There are infinitesimal hyper-reals, surreals and games
(amongst others), which are as real as any abstraction ever is.
But you owe it to your readers to explain what /you/ mean.]

The problem is it is really hard to construct axioms that exclude
every model that has infinitesimals.
--
Mikko

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 2025-08-29 00:25:39 +0000, olcott said:

On 8/28/2025 7:10 PM, Mr Flibble wrote:

On Thu, 28 Aug 2025 19:05:41 -0500, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line segments:
[0.0,1.0] - [0.0,1.0) and this difference *is* infinitesimal.

There is not, at least in a context where points on a line can
correspond to real numbers.

If your statement is meant to be understood in the context of hyperreal >>>> or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the basis of defining
two line segments that differ in length by one geometric point. THAT IS
100% OF THE WHOLE CONTEXT.

The "length" of a point is zero.

/Flibble

Apparently that has always been a misconception.
[0.0,1.0] - [0.0,1.0) = infinitesimal, thus not zero.

Yes, that seems to be your misconception. The length of those
two intervals is 1.0. But they are different intervals as one
is closed and the other is not. Their difference is the closed
singlet interval [1.0, 1.0], the length of which is 0.
--
Mikko

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 2025-08-29 04:10:29 +0000, Chris M. Thomasson said:

On 8/28/2025 9:08 PM, Chris M. Thomasson wrote:

On 8/28/2025 5:10 PM, Mr Flibble wrote:

On Thu, 28 Aug 2025 19:05:41 -0500, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line segments:
[0.0,1.0] - [0.0,1.0) and this difference *is* infinitesimal.

There is not, at least in a context where points on a line can
correspond to real numbers.

If your statement is meant to be understood in the context of hyperreal >>>>> or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the basis of defining
two line segments that differ in length by one geometric point. THAT IS >>>> 100% OF THE WHOLE CONTEXT.

The "length" of a point is zero.

What about... A point is a location. It's "length" can be its distance
from the origin?

A point is a location. Now we can take said point and do other things.
Such as take its length from origin.

Length and distance are different words with different meanings. The
phrase "lenght from origin" is incorrect but "distance from origin"
is meaningful if an origin is specified.
--
Mikko

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 2025-08-29 03:21:13 +0000, olcott said:

On 8/28/2025 8:59 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line
segments: [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

There is not, at least in a context where points on a
line can correspond to real numbers.
If your statement is meant to be understood in the context of
hyperreal or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the
basis of defining two line segments that differ in
length by one geometric point. THAT IS 100% OF THE WHOLE CONTEXT.

You usually quote an entire article when you post a followup.
Here you've snipped the context in which I refute your claim.

Lengths do not differ by one point.

These do differ by one point.
[0.0,1.0] - [0.0,1.0)

Lengths cannot differ by one point because lengths are numbers and
differences of lengths are numbers but points are not numbers.
--
Mikko

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

Am Thu, 28 Aug 2025 16:01:44 -0500 schrieb olcott:

On 8/28/2025 3:44 PM, Andy Walker wrote:

On 28/08/2025 20:30, olcott wrote:

In what way is the notion of an infinitesimal length not incoherent?

-a-a-a-aNo-one said it was "incoherent".-a But it's not part of
-a-a-a-astandard mathematics.

*Yes new ideas are not part of any existing standard*

Infinitesimals are not new and a part of nonstandard mathematics,
such as the hyperreal numbers.
--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/28/2025 11:08 PM, Chris M. Thomasson wrote:

On 8/28/2025 5:10 PM, Mr Flibble wrote:

On Thu, 28 Aug 2025 19:05:41 -0500, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line segments:
[0.0,1.0] - [0.0,1.0) and this difference *is* infinitesimal.

There is not, at least in a context where points on a line can
correspond to real numbers.

If your statement is meant to be understood in the context of hyperreal >>>> or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the basis of defining
two line segments that differ in length by one geometric point. THAT IS
100% OF THE WHOLE CONTEXT.

The "length" of a point is zero.

What about... A point is a location. It's "length" can be its distance
from the origin?

That is the length of the line segment ending at the point.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/28/2025 11:11 PM, Chris M. Thomasson wrote:

On 8/28/2025 5:23 PM, olcott wrote:

On 8/28/2025 7:14 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:49 PM, olcott wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting
Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

Do you mean the difference between points p0 and p1? This difference
is normalized within the line segment itself. It points in the
direction of p0 to p1. In the simple line code I showed.

I am only referring to the above two line segments
and any attempt to change this subject will be ignored.

A line segment is finite. However, there are infinite points on said
line...

A line segment and a line are two entirely different things.
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/29/2025 6:30 AM, olcott wrote:

On 8/28/2025 11:11 PM, Chris M. Thomasson wrote:

On 8/28/2025 5:23 PM, olcott wrote:

On 8/28/2025 7:14 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:49 PM, olcott wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting >>>>>>>> Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

Do you mean the difference between points p0 and p1? This difference
is normalized within the line segment itself. It points in the
direction of p0 to p1. In the simple line code I showed.

I am only referring to the above two line segments
and any attempt to change this subject will be ignored.

A line segment is finite. However, there are infinite points on said
line...

A line segment and a line are two entirely different things.

We can take a line segment and make it extend via its differential off
into infinity...
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/29/2025 1:07 AM, Mikko wrote:

On 2025-08-29 04:10:29 +0000, Chris M. Thomasson said:

On 8/28/2025 9:08 PM, Chris M. Thomasson wrote:

On 8/28/2025 5:10 PM, Mr Flibble wrote:

On Thu, 28 Aug 2025 19:05:41 -0500, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line segments: >>>>>>> [0.0,1.0] - [0.0,1.0) and this difference *is* infinitesimal.

There is not, at least in a context where points on a line can
correspond to real numbers.

If your statement is meant to be understood in the context of
hyperreal
or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the basis of defining >>>>> two line segments that differ in length by one geometric point.
THAT IS
100% OF THE WHOLE CONTEXT.

The "length" of a point is zero.

What about... A point is a location. It's "length" can be its
distance from the origin?

A point is a location. Now we can take said point and do other things.
Such as take its length from origin.

Length and distance are different words with different meanings. The
phrase "lenght from origin" is incorrect but "distance from origin"
is meaningful if an origin is specified.

Okay. Distance from origin, say in a 4-ary vector space at point (0, 0,
0, 0) ?
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/29/2025 6:29 AM, olcott wrote:

On 8/28/2025 11:08 PM, Chris M. Thomasson wrote:

On 8/28/2025 5:10 PM, Mr Flibble wrote:

On Thu, 28 Aug 2025 19:05:41 -0500, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line segments:
[0.0,1.0] - [0.0,1.0) and this difference *is* infinitesimal.

There is not, at least in a context where points on a line can
correspond to real numbers.

If your statement is meant to be understood in the context of
hyperreal
or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the basis of defining
two line segments that differ in length by one geometric point. THAT IS >>>> 100% OF THE WHOLE CONTEXT.

The "length" of a point is zero.

What about... A point is a location. It's "length" can be its distance
from the origin?

That is the length of the line segment ending at the point.

Lets say origin is point (0, 0) for a 2-ary. For any point we get
sqrt(x^2 + y^2), right?
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 8/28/2025 7:04 PM, Richard Heathfield wrote:

On 29/08/2025 02:46, olcott wrote:

On 8/28/2025 7:18 PM, Richard Heathfield wrote:

On 29/08/2025 01:05, olcott wrote:

On 8/28/2025 6:42 PM, Keith Thompson wrote:

olcott <polcott333@gmail.com> writes:
[...]

There *is* a difference in the length of these two line
segments:-a [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.

There is not, at least in a context where points on a
line can correspond to real numbers.

If your statement is meant to be understood in the context of
hyperreal or surreal numbers, or something similar, say so.

It proves that infinitesimals exist entirely on the
basis of defining two line segments that differ in
length by one geometric point. THAT IS 100% OF THE WHOLE CONTEXT.

The length of one geometric point is 0.

By your definition, then, the only infinitesimal is 0.

[0.0,1.0] - [0.0,1.0) is different by one geometric point

Geometric points are 0-dimensional, so they have no size.

Nit pick, a geometric point can be n-ary. Say a point at (0, -1, 0, 1,
3)? It's a single location in a 5-ary space. An origin can be at (0, 0,
0, 0, 0).

and it is different, thus non zero.

Okay; clearly we aren't working from the same axioms, so discussion's
rather pointless, wouldn't you say?

Good luck with your new mathematics.

--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 2025-08-29 13:30:45 +0000, olcott said:

On 8/28/2025 11:11 PM, Chris M. Thomasson wrote:

On 8/28/2025 5:23 PM, olcott wrote:

On 8/28/2025 7:14 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:49 PM, olcott wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting Problem >>>>>>>> proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

Do you mean the difference between points p0 and p1? This difference is >>>> normalized within the line segment itself. It points in the direction >>>> of p0 to p1. In the simple line code I showed.

I am only referring to the above two line segments
and any attempt to change this subject will be ignored.

A line segment is finite. However, there are infinite points on said line...

A line segment and a line are two entirely different things.

No. A line contains line segments as parts.
--
Mikko

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From Newsgroup: comp.theory

On 8/29/25 12:11 AM, Chris M. Thomasson wrote:

On 8/28/2025 5:23 PM, olcott wrote:

On 8/28/2025 7:14 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:49 PM, olcott wrote:

On 8/26/2025 3:47 PM, Chris M. Thomasson wrote:

On 8/26/2025 1:30 PM, olcott wrote:

On 8/24/2025 9:53 PM, Mr Flibble wrote:

Much like infinitesimals, Olcott's refutations of the Halting
Problem
proofs don't exist.

/Flibble

The difference in the length of a line of these two
line segments using interval notation is infinitesimal.

[0.0, 1.0] - [0.0, 1.0) == infinitesimal.

AKA one geometric point of difference.

Do you mean the difference between points p0 and p1? This difference
is normalized within the line segment itself. It points in the
direction of p0 to p1. In the simple line code I showed.

I am only referring to the above two line segments
and any attempt to change this subject will be ignored.

A line segment is finite. However, there are infinite points on said
line...

He is refering to the diffence between a closed interval and a half-open
one.

Part of the problem is that "intervals" are not "line segments", but
just something closely related, and line segments, by there definition, classically include their end-points, so the half-open interval isn't a
line segment.

This is Olcotts typical not actually knowing the meaning of the terms he
is using, as he just guesses from a a quick glance at the field.
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