The smallest number is the point on the number line
specified by LHS of this interval (0,1].

Alternatively it is element s of the set of Real numbers:
∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))

This smallest possible number is necessarily indivisible.

Because set subtractions of [0,1] - [0,1) = the point specifying 1.0
we can see that the width of a single geometric point is not actually
zero, instead it has an infinitesimally small width.

The line segments specified by the above pair of intervals differ in
length by the a single geometric point.

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

On Thu, 24 Mar 2022 18:44:32 -0500
olcott <NoOne@NoWhere.com> wrote:

> The smallest number is the point on the number line
> specified by LHS of this interval (0,1].
>
> Alternatively it is element s of the set of Real numbers:
> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>
> This smallest possible number is necessarily indivisible.
>
> Because set subtractions of [0,1] - [0,1) = the point specifying 1.0
> we can see that the width of a single geometric point is not actually
> zero, instead it has an infinitesimally small width.
>
> The line segments specified by the above pair of intervals differ in
> length by the a single geometric point.
You still need to define exactly what "infinitesimally small" actually
means but I wouldn't bother: a "geometric point" has no width.

/Flibble

On 3/24/22 7:44 PM, olcott wrote:
> The smallest number is the point on the number line
> specified by LHS of this interval (0,1].
>
> Alternatively it is element s of the set of Real numbers:
> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>
> This smallest possible number is necessarily indivisible.
>
> Because set subtractions of [0,1] - [0,1) = the point specifying 1.0
> we can see that the width of a single geometric point is not actually
> zero, instead it has an infinitesimally small width.
>
> The line segments specified by the above pair of intervals differ in
> length by the a single geometric point.
>

Such a number, just doesn't exist.

ALL real numbers can be divided by any other real number (except 0) and
when the divisior isn't 1, or the dividend 0, you get a new number.

Thus EVERY number s > 0 has a smaller number s/2 closer to zero then this.

This is part of the Density Property of the Reals.

You are basically showing that you do not understand how Math actually
works.

Note, the Reals do NOT contain any 'infintesimal' numbers, those are
other number systems, but the points on the line are shown to map 1:1
with the set of The Reals.

FAIL.

On Thu, 24 Mar 2022 18:44:32 -0500
olcott <NoOne@NoWhere.com> wrote:

> The smallest number is the point on the number line
> specified by LHS of this interval (0,1].
>
> Alternatively it is element s of the set of Real numbers:
> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>
> This smallest possible number is necessarily indivisible.
>
> Because set subtractions of [0,1] - [0,1) = the point specifying 1.0
> we can see that the width of a single geometric point is not actually
> zero, instead it has an infinitesimally small width.
>
> The line segments specified by the above pair of intervals differ in
> length by the a single geometric point.
>

By your logic:

(1) The interval [0,1) contains an infinite number of geometrical points
and
(2) Geometrical points have infinitesimally small width
ergo
(3) The interval [0,1) has infinite length

(3) is obviously false so are you claiming that (1) is incorrect and
the interval [0,1) contains a FINITE number of geometrical points?

You are inventing shit up as you go along: you are ignoring both logic
and accepted mathematical axioms.

/Flibble

On 3/24/2022 6:57 PM, Mr Flibble wrote:
> On Thu, 24 Mar 2022 18:44:32 -0500
> olcott <NoOne@NoWhere.com> wrote:
>
>> The smallest number is the point on the number line
>> specified by LHS of this interval (0,1].
>>
>> Alternatively it is element s of the set of Real numbers:
>> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>>
>> This smallest possible number is necessarily indivisible.
>>
>> Because set subtractions of [0,1] - [0,1) = the point specifying 1.0
>> we can see that the width of a single geometric point is not actually
>> zero, instead it has an infinitesimally small width.
>>
>> The line segments specified by the above pair of intervals differ in
>> length by the a single geometric point.
>
> You still need to define exactly what "infinitesimally small" actually
> means but I wouldn't bother: a "geometric point" has no width.
>
> /Flibble
>

If a geometric point actually had no width then [0,1] would not have one
more point that comes after the last point of this interval [0,1).

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

On 3/24/2022 7:10 PM, Mr Flibble wrote:
> On Thu, 24 Mar 2022 18:44:32 -0500
> olcott <NoOne@NoWhere.com> wrote:
>
>> The smallest number is the point on the number line
>> specified by LHS of this interval (0,1].
>>
>> Alternatively it is element s of the set of Real numbers:
>> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>>
>> This smallest possible number is necessarily indivisible.
>>
>> Because set subtractions of [0,1] - [0,1) = the point specifying 1.0
>> we can see that the width of a single geometric point is not actually
>> zero, instead it has an infinitesimally small width.
>>
>> The line segments specified by the above pair of intervals differ in
>> length by the a single geometric point.
>>
>
> By your logic:
>
> (1) The interval [0,1) contains an infinite number of geometrical points
> and
> (2) Geometrical points have infinitesimally small width
> ergo
> (3) The interval [0,1) has infinite length
>
> (3) is obviously false so are you claiming that (1) is incorrect and
> the interval [0,1) contains a FINITE number of geometrical points?
>
> You are inventing shit up as you go along: you are ignoring both logic
> and accepted mathematical axioms.
>
> /Flibble
>

It is the case that [0,1] does have one more point than [0,1) and this
point does come after the last point in [0,1).

We know that on the basis of the meaning of intervals.

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

Crank Peter Olcott wrote:
> On 3/24/2022 6:57 PM, Mr Flibble wrote:
>> On Thu, 24 Mar 2022 18:44:32 -0500
>> olcott <NoOne@NoWhere.com> wrote:
>>
>>> The smallest number is the point on the number line
>>> specified by LHS of this interval (0,1].
>>>
>>> Alternatively it is element s of the set of Real numbers:
>>> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>>>
>>> This smallest possible number is necessarily indivisible.
>>>
>>> Because set subtractions of [0,1] - [0,1) = the point specifying 1.0
>>> we can see that the width of a single geometric point is not actually
>>> zero, instead it has an infinitesimally small width.
>>>
>>> The line segments specified by the above pair of intervals differ in
>>> length by the a single geometric point.
>> You still need to define exactly what "infinitesimally small" actually
>> means but I wouldn't bother: a "geometric point" has no width.
>>
>> /Flibble
>>
>
> If a geometric point actually had no width then [0,1]

not "no width", width = 0.

Are you denying that the distance between a point and itself
is not 0? Amazing!

Python wrote:
> Crank Peter Olcott wrote:
....
>> If a geometric point actually had no width then [0,1]
>
> not "no width", width = 0.
>
> Are you denying that the distance between a point and itself
> is not 0? Amazing!

typo: *is 0*

On 3/24/2022 9:09 PM, Python wrote:
> Crank Peter Olcott wrote:
>> On 3/24/2022 6:57 PM, Mr Flibble wrote:
>>> On Thu, 24 Mar 2022 18:44:32 -0500
>>> olcott <NoOne@NoWhere.com> wrote:
>>>
>>>> The smallest number is the point on the number line
>>>> specified by LHS of this interval (0,1].
>>>>
>>>> Alternatively it is element s of the set of Real numbers:
>>>> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>>>>
>>>> This smallest possible number is necessarily indivisible.
>>>>
>>>> Because set subtractions of [0,1] - [0,1) = the point specifying 1.0
>>>> we can see that the width of a single geometric point is not actually
>>>> zero, instead it has an infinitesimally small width.
>>>>
>>>> The line segments specified by the above pair of intervals differ in
>>>> length by the a single geometric point.
>>> You still need to define exactly what "infinitesimally small" actually
>>> means but I wouldn't bother: a "geometric point" has no width.
>>>
>>> /Flibble
>>>
>>
>> If a geometric point actually had no width then [0,1]
>
> not "no width", width = 0.
>
> Are you denying that the distance between a point and itself
> is not 0? Amazing!

I am asserting that the length of the line segment specified by the
interval [0,1] is one geometric point longer than the length of the line
segment specified by the interval [0,1).

I am also asserting that this is self-evidently true on the basis of the
meaning of intervals.

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

olcott <NoOne@NoWhere.com> writes:
[...]
> I am asserting that the length of the line segment specified by the
> interval [0,1] is one geometric point longer than the length of the
> line segment specified by the interval [0,1).

The length of a line segment is not measured in geometric points.

> I am also asserting that this is self-evidently true on the basis of
> the meaning of intervals.

You keep saying that, but you haven't told us what you think
the "meaning of intervals" is. Can you provide a mathematically
rigorous definition?

The intervals [0,1] and [0,1) are sets of real numbers, and can be
defined and understood without reference to geometry. If [0,1)
had a largest element, it would be possible to demonstrate it
without reference to geometry. In fact it's possible to prove
that it has no largest element without reference to geometry.
Treating real numbers as points on a number line can be useful for
informal understanding, but is not necessary when reasoning about
real numbers or sets of real numbers.

Again, in my posts on these threads I've been talking about sets
of real numbers -- and until recently I thought you were too.

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips
void Void(void) { Void(); } /* The recursive call of the void */

On 3/24/22 11:16 PM, olcott wrote:
> On 3/24/2022 9:09 PM, Python wrote:
>> Crank Peter Olcott wrote:
>>> On 3/24/2022 6:57 PM, Mr Flibble wrote:
>>>> On Thu, 24 Mar 2022 18:44:32 -0500
>>>> olcott <NoOne@NoWhere.com> wrote:
>>>>
>>>>> The smallest number is the point on the number line
>>>>> specified by LHS of this interval (0,1].
>>>>>
>>>>> Alternatively it is element s of the set of Real numbers:
>>>>> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>>>>>
>>>>> This smallest possible number is necessarily indivisible.
>>>>>
>>>>> Because set subtractions of [0,1] - [0,1) = the point specifying 1.0
>>>>> we can see that the width of a single geometric point is not actually
>>>>> zero, instead it has an infinitesimally small width.
>>>>>
>>>>> The line segments specified by the above pair of intervals differ in
>>>>> length by the a single geometric point.
>>>> You still need to define exactly what "infinitesimally small" actually
>>>> means but I wouldn't bother: a "geometric point" has no width.
>>>>
>>>> /Flibble
>>>>
>>>
>>> If a geometric point actually had no width then [0,1]
>>
>> not "no width", width = 0.
>>
>> Are you denying that the distance between a point and itself
>> is not 0? Amazing!
>
> I am asserting that the length of the line segment specified by the
> interval [0,1] is one geometric point longer than the length of the line
> segment specified by the interval [0,1).
>
> I am also asserting that this is self-evidently true on the basis of the
> meaning of intervals.
>

And the measure of one geometric point is 0, i.e. too small to be able
to measure.

Note also, as has been poited ot, 'Self Evident' isn't admissible in a
Formal Logic like this, only PROVABLE from the base axioms or previously
proven material.

You are working in the wrong branch of philosophy.

On 3/25/2022 12:23 AM, Keith Thompson wrote:
> olcott <NoOne@NoWhere.com> writes:
> [...]
>> I am asserting that the length of the line segment specified by the
>> interval [0,1] is one geometric point longer than the length of the
>> line segment specified by the interval [0,1).
>
> The length of a line segment is not measured in geometric points.
>

Not usually because no one ever noticed that the line segment specified
by the interval [0,1] is infinitesimally longer than the line segment
specified by the interval [0,1)

>> I am also asserting that this is self-evidently true on the basis of
>> the meaning of intervals.
>
> You keep saying that, but you haven't told us what you think
> the "meaning of intervals" is. Can you provide a mathematically
> rigorous definition?
>

[0,1] as one more point that [0,1) has appended to the end of its
ordered set of points.

> The intervals [0,1] and [0,1) are sets of real numbers, and can be

Which is bijective to points on a number line.

> defined and understood without reference to geometry. If [0,1)
> had a largest element, it would be possible to demonstrate it
> without reference to geometry.

That is a false assumption.

> In fact it's possible to prove
> that it has no largest element without reference to geometry.

This proof would be incorrect.

> Treating real numbers as points on a number line can be useful for
> informal understanding, but is not necessary when reasoning about
> real numbers or sets of real numbers.
>

[0,1] - [0,1) = a single point at 1.0

> Again, in my posts on these threads I've been talking about sets
> of real numbers -- and until recently I thought you were too.
>

They are bijective to points on a number line.

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

olcott <NoOne@NoWhere.com> writes:
> On 3/25/2022 12:23 AM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
[...]
>>> I am also asserting that this is self-evidently true on the basis of
>>> the meaning of intervals.
>> You keep saying that, but you haven't told us what you think
>> the "meaning of intervals" is. Can you provide a mathematically
>> rigorous definition?
>
> [0,1] as one more point that [0,1) has appended to the end of its
> ordered set of points.

Clearly you cannot provide a mathematically rigorous definition of
"interval". If you could, you would have done so by now.

Your claim relies on your faulty intuition about how intervals work,
but you can't even say what an interval is.

[...]

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips
void Void(void) { Void(); } /* The recursive call of the void */

On 3/25/2022 12:25 PM, Keith Thompson wrote:
> olcott <NoOne@NoWhere.com> writes:
>> On 3/25/2022 12:23 AM, Keith Thompson wrote:
>>> olcott <NoOne@NoWhere.com> writes:
> [...]
>>>> I am also asserting that this is self-evidently true on the basis of
>>>> the meaning of intervals.
>>> You keep saying that, but you haven't told us what you think
>>> the "meaning of intervals" is. Can you provide a mathematically
>>> rigorous definition?
>>
>> [0,1] as one more point that [0,1) has appended to the end of its
>> ordered set of points.
>
> Clearly you cannot provide a mathematically rigorous definition of
> "interval". If you could, you would have done so by now.
>

The smallest number is the point on the number line
specified by LHS of this interval (0,1].

Alternatively it is element s of the set of Real numbers:
∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))

> Your claim relies on your faulty intuition about how intervals work,
> but you can't even say what an interval is.

As long as we know there is a bijection between points on a number line
and reals as soon as we have uniquely identified a point this point
specifies its corresponding real.

--
Copyright 2022 Pete Olcott

"Talent hits a target no one else can hit;
Genius hits a target no one else can see."
Arthur Schopenhauer

On 3/25/2022 11:25 AM, Keith Thompson wrote:
> olcott <NoOne@NoWhere.com> writes:
>> On 3/25/2022 12:23 AM, Keith Thompson wrote:
>>> olcott <NoOne@NoWhere.com> writes:
> [...]
>>>> I am also asserting that this is self-evidently true on the basis of
>>>> the meaning of intervals.
>>> You keep saying that, but you haven't told us what you think
>>> the "meaning of intervals" is. Can you provide a mathematically
>>> rigorous definition?
>>
>> [0,1] as one more point that [0,1) has appended to the end of its
>> ordered set of points.
>
> Clearly you cannot provide a mathematically rigorous definition of
> "interval". If you could, you would have done so by now.
>
> Your claim relies on your faulty intuition about how intervals work,
> but you can't even say what an interval is.
Word of caution: Never issue a challenge to the dunce; his single
working capability is to find and regurgitate things using Google. Of
course he might try to reword it to serve trolling purposes whether he
understands what he's found or not. Than you are off to the races.
--
Jeff Barnett

On 3/25/2022 12:45 PM, Jeff Barnett wrote:
> On 3/25/2022 11:25 AM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>> On 3/25/2022 12:23 AM, Keith Thompson wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>> [...]
>>>>> I am also asserting that this is self-evidently true on the basis of
>>>>> the meaning of intervals.
>>>> You keep saying that, but you haven't told us what you think
>>>> the "meaning of intervals" is.  Can you provide a mathematically
>>>> rigorous definition?
>>>
>>> [0,1] as one more point that [0,1) has appended to the end of its
>>> ordered set of points.
>>
>> Clearly you cannot provide a mathematically rigorous definition of
>> "interval".  If you could, you would have done so by now.
>>
>> Your claim relies on your faulty intuition about how intervals work,
>> but you can't even say what an interval is.
> Word of caution: Never issue a challenge to the dunce; his single
> working capability is to find and regurgitate things using Google. Of
> course he might try to reword it to serve trolling purposes whether he
> understands what he's found or not. Than you are off to the races.

The smallest number is the point on the number line
specified by LHS of this interval (0,1].

Alternatively it is element s of the set of Real numbers:
∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))

If you find the above anywhere on Google it violates my copyright.

--
Copyright 2022 Pete Olcott

"Talent hits a target no one else can hit;
Genius hits a target no one else can see."
Arthur Schopenhauer

olcott <NoOne@NoWhere.com> writes:
> On 3/25/2022 12:25 PM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>> On 3/25/2022 12:23 AM, Keith Thompson wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>> [...]
>>>>> I am also asserting that this is self-evidently true on the basis of
>>>>> the meaning of intervals.
>>>> You keep saying that, but you haven't told us what you think
>>>> the "meaning of intervals" is. Can you provide a mathematically
>>>> rigorous definition?
>>>
>>> [0,1] as one more point that [0,1) has appended to the end of its
>>> ordered set of points.
>> Clearly you cannot provide a mathematically rigorous definition of
>> "interval". If you could, you would have done so by now.
>
> The smallest number is the point on the number line
> specified by LHS of this interval (0,1].
>
> Alternatively it is element s of the set of Real numbers:
> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>
>> Your claim relies on your faulty intuition about how intervals work,
>> but you can't even say what an interval is.
>
> As long as we know there is a bijection between points on a number
> line and reals as soon as we have uniquely identified a point this
> point specifies its corresponding real.

If you understood what an "interval" is, you would be able to provide a
definition for the word. You clearly don't, though you apparently think
you do. Your faulty understanding is leading you to commit serious
errors like believing that [0,1) has a last element and (0,1] has a
first element.

This Wikipedia article provides what seems to me to be a reasonable
definition. I encourage you to read it with an open mind.

https://en.wikipedia.org/wiki/Interval_(mathematics)

A quick and incomplete summary: an interval, in the current context, is
a set of real numbers. Understanding that also requires understanding
the terms "set" and "real number", and that the set of real numbers does
not include infinitesimals: https://en.wikipedia.org/wiki/Infinitesimal .

If at some point in the future you demonstrate that you understand what
an interval is, I *might* consider rejoining this discussion. Barring
that, I'm done for now. (I suspect some readers here will be relieved
to hear that.)

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips
void Void(void) { Void(); } /* The recursive call of the void */

On 3/25/2022 2:01 PM, Keith Thompson wrote:
> olcott <NoOne@NoWhere.com> writes:
>> On 3/25/2022 12:25 PM, Keith Thompson wrote:
>>> olcott <NoOne@NoWhere.com> writes:
>>>> On 3/25/2022 12:23 AM, Keith Thompson wrote:
>>>>> olcott <NoOne@NoWhere.com> writes:
>>> [...]
>>>>>> I am also asserting that this is self-evidently true on the basis of
>>>>>> the meaning of intervals.
>>>>> You keep saying that, but you haven't told us what you think
>>>>> the "meaning of intervals" is. Can you provide a mathematically
>>>>> rigorous definition?
>>>>
>>>> [0,1] as one more point that [0,1) has appended to the end of its
>>>> ordered set of points.
>>> Clearly you cannot provide a mathematically rigorous definition of
>>> "interval". If you could, you would have done so by now.
>>
>> The smallest number is the point on the number line
>> specified by LHS of this interval (0,1].
>>
>> Alternatively it is element s of the set of Real numbers:
>> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>>
>>> Your claim relies on your faulty intuition about how intervals work,
>>> but you can't even say what an interval is.
>>
>> As long as we know there is a bijection between points on a number
>> line and reals as soon as we have uniquely identified a point this
>> point specifies its corresponding real.
>
> If you understood what an "interval" is, you would be able to provide a
> definition for the word. You clearly don't, though you apparently think
> you do. Your faulty understanding is leading you to commit serious
> errors like believing that [0,1) has a last element and (0,1] has a
> first element.
>

It may be the case that it is conventionally understood that the last
element and the first element do not exist on the basis of the
difficulty of expressing these two numerically.

None-the-less we do know that s
∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
does correspond to the first element of (0,1]

We already have a rule that prohibits an element of the set of real
numbers in the denominator, 0 is not allowed in the denominator.

Now we have a new rule the prohibits an element of the set of real
numbers in the numerator, infinitesimal is not allowed in the numerator,
it is indivisible.

> This Wikipedia article provides what seems to me to be a reasonable
> definition. I encourage you to read it with an open mind.
>
> https://en.wikipedia.org/wiki/Interval_(mathematics)
>
> A quick and incomplete summary: an interval, in the current context, is
> a set of real numbers. Understanding that also requires understanding
> the terms "set" and "real number", and that the set of real numbers does
> not include infinitesimals: https://en.wikipedia.org/wiki/Infinitesimal .
>

Although because infinity is a process and thus not any sort of number
infinity is not a real number.

On the other hand I just showed how infinitesimal does map
to the LHS of (0,1] and
∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))

> If at some point in the future you demonstrate that you understand what
> an interval is, I *might* consider rejoining this discussion. Barring
> that, I'm done for now. (I suspect some readers here will be relieved
> to hear that.)
>

You cannot refute my reasoning you can only show that my reasoning is
unconventional. In other words you cannot show that I am wrong you can
only show that others don't usually look at these things this way.

--
Copyright 2022 Pete Olcott

"Talent hits a target no one else can hit;
Genius hits a target no one else can see."
Arthur Schopenhauer

On 3/25/22 9:15 AM, olcott wrote:
> On 3/25/2022 12:23 AM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
>> [...]
>>> I am asserting that the length of the line segment specified by the
>>> interval [0,1] is one geometric point longer than the length of the
>>> line segment specified by the interval [0,1).
>>
>> The length of a line segment is not measured in geometric points.
>>
>
> Not usually because no one ever noticed that the line segment specified
> by the interval [0,1] is infinitesimally longer than the line segment
> specified by the interval [0,1)

You mean immeasurably longer. The Reals have no infinitesimals.

>
>>> I am also asserting that this is self-evidently true on the basis of
>>> the meaning of intervals.
>>
>> You keep saying that, but you haven't told us what you think
>> the "meaning of intervals" is.  Can you provide a mathematically
>> rigorous definition?
>>
>
> [0,1] as one more point that [0,1) has appended to the end of its
> ordered set of points.
>
>> The intervals [0,1] and [0,1) are sets of real numbers, and can be
>
> Which is bijective to points on a number line.

Which means you are talking of the REALS, and thus there are no
infinitesimals.

>
>> defined and understood without reference to geometry.  If [0,1)
>> had a largest element, it would be possible to demonstrate it
>> without reference to geometry.
>
> That is a false assumption.
>
>> In fact it's possible to prove
>> that it has no largest element without reference to geometry.
>
> This proof would be incorrect.
>
>> Treating real numbers as points on a number line can be useful for
>> informal understanding, but is not necessary when reasoning about
>> real numbers or sets of real numbers.
>>
>
> [0,1] - [0,1) = a single point at 1.0
>
>> Again, in my posts on these threads I've been talking about sets
>> of real numbers -- and until recently I thought you were too.
>>
>
> They are bijective to points on a number line.
>

Which contains NO infinitesimals, and thus the lengths that differ by a
finite number of points are the same.

On 3/25/22 1:42 PM, olcott wrote:
> On 3/25/2022 12:25 PM, Keith Thompson wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>> On 3/25/2022 12:23 AM, Keith Thompson wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>> [...]
>>>>> I am also asserting that this is self-evidently true on the basis of
>>>>> the meaning of intervals.
>>>> You keep saying that, but you haven't told us what you think
>>>> the "meaning of intervals" is.  Can you provide a mathematically
>>>> rigorous definition?
>>>
>>> [0,1] as one more point that [0,1) has appended to the end of its
>>> ordered set of points.
>>
>> Clearly you cannot provide a mathematically rigorous definition of
>> "interval".  If you could, you would have done so by now.
>>
>
> The smallest number is the point on the number line
> specified by LHS of this interval (0,1].
>
> Alternatively it is element s of the set of Real numbers:
> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>
>> Your claim relies on your faulty intuition about how intervals work,
>> but you can't even say what an interval is.
>
> As long as we know there is a bijection between points on a number line
> and reals as soon as we have uniquely identified a point this point
> specifies its corresponding real.
>
>

Right, and there is NO 'smallest real' > 0 or 'largest real' < 1 by the
density property. So since you agree that you are talking about a
geometry that bijects with the reals, there is no leftmost or rightmost
point in the open intervals.

On 3/25/22 1:50 PM, olcott wrote:
> On 3/25/2022 12:45 PM, Jeff Barnett wrote:
>> On 3/25/2022 11:25 AM, Keith Thompson wrote:
>>> olcott <NoOne@NoWhere.com> writes:
>>>> On 3/25/2022 12:23 AM, Keith Thompson wrote:
>>>>> olcott <NoOne@NoWhere.com> writes:
>>> [...]
>>>>>> I am also asserting that this is self-evidently true on the basis of
>>>>>> the meaning of intervals.
>>>>> You keep saying that, but you haven't told us what you think
>>>>> the "meaning of intervals" is.  Can you provide a mathematically
>>>>> rigorous definition?
>>>>
>>>> [0,1] as one more point that [0,1) has appended to the end of its
>>>> ordered set of points.
>>>
>>> Clearly you cannot provide a mathematically rigorous definition of
>>> "interval".  If you could, you would have done so by now.
>>>
>>> Your claim relies on your faulty intuition about how intervals work,
>>> but you can't even say what an interval is.
>> Word of caution: Never issue a challenge to the dunce; his single
>> working capability is to find and regurgitate things using Google. Of
>> course he might try to reword it to serve trolling purposes whether he
>> understands what he's found or not. Than you are off to the races.
>
>
> The smallest number is the point on the number line
> specified by LHS of this interval (0,1].
>
> Alternatively it is element s of the set of Real numbers:
> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))
>
> If you find the above anywhere on Google it violates my copyright.
>
>

The 'smallest postive real' doesn't exist, as there is always one
smaller than any given one.

So, your statement is FALSE.

Second, your comment about Google is incorrect, by posting to usenet,
you have implicitly given permission for a vast array of automatic
copies to be made.

Crank Peter Olcott wrote:
....
> On the other hand I just showed how infinitesimal does map
> to the LHS of (0,1] and

You've shown nothing at all, as this proposition, below:

> ∃s ∈ ℝ ((s > 0) ∧ ∀n ∈ ℝ ((r > 0) → (s <= n)))

is FALSE (it is not even well-formed, r is unbound)

You are a clown, Peter.