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.
On 8/26/2025 4:36 PM, Kaz Kylheku wrote:
On 2025-08-26, olcott <[email protected]> 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)
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.
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.
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.
On 8/28/2025 1:40 AM, Mikko wrote:
On 2025-08-27 15:10:18 +0000, olcott said:This is a statement and not a question.
I am stipulating that the difference in theYou weren't in your message quoted above.
length of the two line segments is defined
to be infinitesimal.
[0.0, 1.0] - [0.0, 1.0) == infinitesimal.
[...] "Real number" is
just a figure-of-speech name that does not
entail that numbers outside of the set of "reals"
are fake.
On 8/28/2025 1:40 AM, Mikko wrote:
On 2025-08-27 15:10:18 +0000, olcott said:This is a statement and not a question. [0.0, 1.0] - [0.0, 1.0) == infinitesimal.
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 HaltingThe difference in the length of a line of these two line segments
Problem proofs don't exist.
/Flibble
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.
Absent a context, maths isNot at all really.
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;
I defined one concrete instance of an infinitesimal.
You didn't /prove/ anything; you asserted it, without the contextthereI proved that assumption is false.
are no infinitesimal lengths.
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-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 betterIn other contexts, less usual, you couldIn what way is the notion of an infinitesimal length not incoherent?
be correct or you could again be wrong. So I ask again: what context
are you assuming for your stipulation?
On 8/28/2025 6:42 PM, Keith Thompson wrote:
olcott <[email protected]> writes:It proves that infinitesimals exist entirely on the basis of defining
[...]
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.
two line segments that differ in length by one geometric point. THAT IS
100% OF THE WHOLE CONTEXT.
On 8/28/2025 6:42 PM, Keith Thompson wrote:
olcott <[email protected]> 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.
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 <[email protected]> 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 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.
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.
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:This is a statement and not a question.
I am stipulating that the difference in theYou weren't in your message quoted above.
length of the two line segments is defined
to be infinitesimal.
[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.]
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 <[email protected]> writes:It proves that infinitesimals exist entirely on the basis of defining
[...]
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.
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.
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 <[email protected]> writes:It proves that infinitesimals exist entirely on the basis of defining
[...]
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.
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.
On 8/28/2025 8:59 PM, Keith Thompson wrote:
olcott <[email protected]> writes:These do differ by one point.
On 8/28/2025 6:42 PM, Keith Thompson wrote:
olcott <[email protected]> writes:
[...]
There *is* a difference in the length of these two lineThere is not, at least in a context where points on a
segments: [0.0,1.0] - [0.0,1.0) and this difference
*is* infinitesimal.
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.
[0.0,1.0] - [0.0,1.0)
On 8/28/2025 3:44 PM, Andy Walker wrote:Infinitesimals are not new and a part of nonstandard mathematics,
On 28/08/2025 20:30, olcott wrote:
*Yes new ideas are not part of any existing standard*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.
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.
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...
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