On 28/03/2024 21:25, olcott wrote:
[I wrote:]
>>      But "one geometric point" has measure zero.  Not "never actually
> I just proved otherwise. [0.0, 1.0] has all of the same points
> as [0.0, 1.0) except that it has one more point.

Euclid, "Elements", Book 1, page 1, definition 1:

" A POINT is that which has no parts, or which has no magnitude. "

[Todhunter's translation]. How much more elementary or traditional is it
possible to get? For over 2300 years, that has been the definition.

Furthermore, those two intervals have exactly the same number of
points [in the "one-one correspondence" sense] [hint: "Hilbert's Hotel"
(qv) on the rational numbers of the two intervals]. So they have different
points, but not "one more point. That has been known since the 19thC.

>> zero", but actually and really zero.  Unless, that is, you are using some
>> different and as yet unexplained axioms/definitions.  Which are ...?
> Conventional interval notion proves otherwise.

It proves exactly what I claimed. There are systems in which the
things you and Wij are claiming are nearer to the truth, but it you want to
use one of them you should say what system and meanwhile stop trying to
misuse the traditional real numbers. [Note that "real" in this sense is
not a claim about reality, any more than "imaginary" numbers thereby don't
exist. No-one claims that the names are entirely sensible.]

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Gottschalk

On 3/28/24 11:50 PM, olcott wrote:
> On 3/28/2024 10:36 PM, Keith Thompson wrote:
>> olcott <polcott2@gmail.com> writes:
>> [...]
>>> It seems dead obvious that 0.999... is infinitesimally less than 1.0.
>>
>> Yes, it *seems* dead obvious.  That doesn't make it true, and in fact it
>> isn't.
>>
>
> 0.999... means that is never reaches 1.0.
> and math simply stipulates that it does even though it does not.

0.999... isn't a "number" in the Real Number system, just an alternate
representation for the number 1.

>
>> 0.999... denotes a *limit*.  In particular, it's the limit of the value
>> as the number of 9s increases without bound.  That's what the notation
>
> That is how it has been misinterpreted yet it has always meant
> infinitesimally less than 1.0.

But "infintesimally" doesn't exist in the Real Number System, it deals
onloy with FINITE numbers.

>
>> "0.999..." *means*.  (There are more precise notations for the same
>> thing, such as "0.9̅" (that's a 9 with an overbar, or "vinculum") or
>> "0.(9)".
>>
>
> I already know all that.
>
>> You have a sequence of numbers:
>>
>>      0.9
>>      0.99
>>      0.999
>>      0.9999
>>      0.99999
>>      ...
>>
>> Each member of that sequence is strictly less than 1.0, but the *limit*
>> is exactly 1.0.  The limit of a sequence doesn't have to be a member of
>> the sequence.  The limit is, informally, the value that members of the
>> sequence approach arbitrarily closely.
>>
>
> Yet never reaching.
>
>> <https://en.wikipedia.org/wiki/Limit_of_a_sequence>
>>
>>> That we can say this in English yet not say this in conventional
>>> number systems proves the need for another number system that can
>>> say this.
>>
>> Then I have good news for you.  There are several such systems, for
>> example <https://en.wikipedia.org/wiki/Hyperreal_number>.
>>
>
> Infinitesimally less than 1.0 means one single geometric point
> on the number line less than 1.0.

Nope.

>
>> If your point is that you personally like hyperreals better than you
>> like reals, that's fine, as long as you're clear which number system
>> you're using.
>
> The Infinitesimal number system that I created.

So you are lying about talking about the Reals.

>
>> If you talk about things like "0.999..." without
>> qualification, everyone will assume you're talking about real numbers.
>>
>
> It is already the case that 0.999...
> specifies Infinitesimally less than 1.0.
>
>> And if you're going to play with hyperreal numbers, or surreal numbers,
>> or any of a number of other extensions to the real numbers, I suggest
>> that understanding the real numbers is a necessary prerequisite.  That
>> includes understanding that no real number is either infinitesimal or
>> infinite.
>>
>> Disclaimer: I'm not a mathematician.  I welcome corrections.
>>
>

On 29/03/2024 04:29, olcott wrote:
> x is said to be infinitesimal
> if, and only if, |x| < 1/n for all integers n.
> https://en.wikipedia.org/wiki/Hyperreal_number

That's for the hyperreals; there's a clue in the URL.
There are no such "x" in R, by the Archimedean axiom.

> 0.999... specifies infinitesimally < 1.0

No it doesn't. It specifies different things in different
number systems, which is why mathematicians don't use that notation
in contexts where there could be ambiguity.

> and math guys have no way to say that so they
> simply round up to 1.0

You've just referred to some "math guys" -- the proponents of
the hyperreals -- who say exactly what they mean by "infinitesimal".
You could equally have referred to the surreals [qv] where similar
statements are made and explained by "math guys". Maths has moved
on over the past few centuries. You and Wij need to move on with
the "math guys".

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Gottschalk

On 3/29/2024 8:23 AM, Andy Walker wrote:
> On 29/03/2024 04:29, olcott wrote:
>> x is said to be infinitesimal
>> if, and only if, |x| < 1/n for all integers n.
>> https://en.wikipedia.org/wiki/Hyperreal_number
>
>     That's for the hyperreals;  there's a clue in the URL.
> There are no such "x" in R, by the Archimedean axiom.
>
>> 0.999... specifies infinitesimally < 1.0
>
>     No it doesn't.  It specifies different things in different
> number systems, which is why mathematicians don't use that notation
> in contexts where there could be ambiguity.
>
>> and math guys have no way to say that so they
>> simply round up to 1.0
>
>     You've just referred to some "math guys" -- the proponents of
> the hyperreals -- who say exactly what they mean by "infinitesimal".
> You could equally have referred to the surreals [qv] where similar
> statements are made and explained by "math guys".  Maths has moved
> on over the past few centuries.  You and Wij need to move on with
> the "math guys".
>

Yet my system seems to make more sense.
[0.0, 1.0] - [0.0, 1.0) Only the last point on the number line
of the first interval is not contained in the second interval.

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

Op 29.mrt.2024 om 15:21 schreef olcott:
> On 3/29/2024 8:23 AM, Andy Walker wrote:
>> On 29/03/2024 04:29, olcott wrote:
>>> x is said to be infinitesimal
>>> if, and only if, |x| < 1/n for all integers n.
>>> https://en.wikipedia.org/wiki/Hyperreal_number
>>
>>      That's for the hyperreals;  there's a clue in the URL.
>> There are no such "x" in R, by the Archimedean axiom.
>>
>>> 0.999... specifies infinitesimally < 1.0
>>
>>      No it doesn't.  It specifies different things in different
>> number systems, which is why mathematicians don't use that notation
>> in contexts where there could be ambiguity.
>>
>>> and math guys have no way to say that so they
>>> simply round up to 1.0
>>
>>      You've just referred to some "math guys" -- the proponents of
>> the hyperreals -- who say exactly what they mean by "infinitesimal".
>> You could equally have referred to the surreals [qv] where similar
>> statements are made and explained by "math guys".  Maths has moved
>> on over the past few centuries.  You and Wij need to move on with
>> the "math guys".
>>
>
> Yet my system seems to make more sense.
> [0.0, 1.0] - [0.0, 1.0) Only the last point on the number line
> of the first interval is not contained in the second interval.
>

Olcott's system does not (yet) make sense, because olcott has not
defined his system. E.g., what is a point in this context? What is a
number line?
This is how Reals are defined:
> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

If olcott wants to discuss his system, he should define his system at
least as detailed as reals are in this article and indicate where he is
deviating from reals, otherwise it is unclear where he is talking about
and discussion do not make sense.

On Fri, 2024-03-29 at 15:55 +0100, Fred. Zwarts wrote:
> Op 29.mrt.2024 om 15:21 schreef olcott:
> > On 3/29/2024 8:23 AM, Andy Walker wrote:
> > > On 29/03/2024 04:29, olcott wrote:
> > > > x is said to be infinitesimal
> > > > if, and only if, |x| < 1/n for all integers n.
> > > > https://en.wikipedia.org/wiki/Hyperreal_number
> > >
> > >      That's for the hyperreals;  there's a clue in the URL.
> > > There are no such "x" in R, by the Archimedean axiom.
> > >
> > > > 0.999... specifies infinitesimally < 1.0
> > >
> > >      No it doesn't.  It specifies different things in different
> > > number systems, which is why mathematicians don't use that notation
> > > in contexts where there could be ambiguity.
> > >
> > > > and math guys have no way to say that so they
> > > > simply round up to 1.0
> > >
> > >      You've just referred to some "math guys" -- the proponents of
> > > the hyperreals -- who say exactly what they mean by "infinitesimal".
> > > You could equally have referred to the surreals [qv] where similar
> > > statements are made and explained by "math guys".  Maths has moved
> > > on over the past few centuries.  You and Wij need to move on with
> > > the "math guys".
> > >
> >
> > Yet my system seems to make more sense.
> > [0.0, 1.0] - [0.0, 1.0) Only the last point on the number line
> > of the first interval is not contained in the second interval.
> >
>
> Olcott's system does not (yet) make sense, because olcott has not
> defined his system. E.g., what is a point in this context? What is a
> number line?
> This is how Reals are defined:
> > https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>
> If olcott wants to discuss his system, he should define his system at
> least as detailed as reals are in this article and indicate where he is
> deviating from reals, otherwise it is unclear where he is talking about
> and discussion do not make sense.

Why do you keep quoting something you don't even understand?
You should know that you cannot event proof "1+2+3" !

On Fri, 2024-03-29 at 23:20 +0800, wij wrote:
> On Fri, 2024-03-29 at 15:55 +0100, Fred. Zwarts wrote:
> > Op 29.mrt.2024 om 15:21 schreef olcott:
> > > On 3/29/2024 8:23 AM, Andy Walker wrote:
> > > > On 29/03/2024 04:29, olcott wrote:
> > > > > x is said to be infinitesimal
> > > > > if, and only if, |x| < 1/n for all integers n.
> > > > > https://en.wikipedia.org/wiki/Hyperreal_number
> > > >
> > > >      That's for the hyperreals;  there's a clue in the URL.
> > > > There are no such "x" in R, by the Archimedean axiom.
> > > >
> > > > > 0.999... specifies infinitesimally < 1.0
> > > >
> > > >      No it doesn't.  It specifies different things in different
> > > > number systems, which is why mathematicians don't use that notation
> > > > in contexts where there could be ambiguity.
> > > >
> > > > > and math guys have no way to say that so they
> > > > > simply round up to 1.0
> > > >
> > > >      You've just referred to some "math guys" -- the proponents of
> > > > the hyperreals -- who say exactly what they mean by "infinitesimal"..
> > > > You could equally have referred to the surreals [qv] where similar
> > > > statements are made and explained by "math guys".  Maths has moved
> > > > on over the past few centuries.  You and Wij need to move on with
> > > > the "math guys".
> > > >
> > >
> > > Yet my system seems to make more sense.
> > > [0.0, 1.0] - [0.0, 1.0) Only the last point on the number line
> > > of the first interval is not contained in the second interval.
> > >
> >
> > Olcott's system does not (yet) make sense, because olcott has not
> > defined his system. E.g., what is a point in this context? What is a
> > number line?
> > This is how Reals are defined:
> > > https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
> >
> > If olcott wants to discuss his system, he should define his system at
> > least as detailed as reals are in this article and indicate where he is
> > deviating from reals, otherwise it is unclear where he is talking about
> > and discussion do not make sense.
>
> Why do you keep quoting something you don't even understand?
> You should know that you cannot event proof "1+2+3" !
>
typo: You should know that you cannot even prove "1+2=3" !

On 3/29/2024 8:13 AM, Richard Damon wrote:
> On 3/28/24 11:50 PM, olcott wrote:
>> On 3/28/2024 10:36 PM, Keith Thompson wrote:
>>> olcott <polcott2@gmail.com> writes:
>>> [...]
>>>> It seems dead obvious that 0.999... is infinitesimally less than 1.0.
>>>
>>> Yes, it *seems* dead obvious.  That doesn't make it true, and in fact it
>>> isn't.
>>>
>>
>> 0.999... means that is never reaches 1.0.
>> and math simply stipulates that it does even though it does not.
>
>
> 0.999... isn't a "number" in the Real Number system, just an alternate
> representation for the number 1.
>

That is not true. 0.999... means never reaches 1.0

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/29/2024 9:55 AM, Fred. Zwarts wrote:
> Op 29.mrt.2024 om 15:21 schreef olcott:
>> On 3/29/2024 8:23 AM, Andy Walker wrote:
>>> On 29/03/2024 04:29, olcott wrote:
>>>> x is said to be infinitesimal
>>>> if, and only if, |x| < 1/n for all integers n.
>>>> https://en.wikipedia.org/wiki/Hyperreal_number
>>>
>>>      That's for the hyperreals;  there's a clue in the URL.
>>> There are no such "x" in R, by the Archimedean axiom.
>>>
>>>> 0.999... specifies infinitesimally < 1.0
>>>
>>>      No it doesn't.  It specifies different things in different
>>> number systems, which is why mathematicians don't use that notation
>>> in contexts where there could be ambiguity.
>>>
>>>> and math guys have no way to say that so they
>>>> simply round up to 1.0
>>>
>>>      You've just referred to some "math guys" -- the proponents of
>>> the hyperreals -- who say exactly what they mean by "infinitesimal".
>>> You could equally have referred to the surreals [qv] where similar
>>> statements are made and explained by "math guys".  Maths has moved
>>> on over the past few centuries.  You and Wij need to move on with
>>> the "math guys".
>>>
>>
>> Yet my system seems to make more sense.
>> [0.0, 1.0] - [0.0, 1.0) Only the last point on the number line
>> of the first interval is not contained in the second interval.
>>
>
> Olcott's system does not (yet) make sense, because olcott has not
> defined his system. E.g., what is a point in this context? What is a
> number line?
> This is how Reals are defined:
>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>
> If olcott wants to discuss his system, he should define his system at
> least as detailed as reals are in this article and indicate where he is
> deviating from reals, otherwise it is unclear where he is talking about
> and discussion do not make sense.

[0.0, 1.0] and [0.0, 1.0) it is already commonly understood
that both of these intervals have all geometric points on
the number line in common except for the last geometric point
of the first interval.

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/29/24 11:20 AM, wij wrote:
> On Fri, 2024-03-29 at 15:55 +0100, Fred. Zwarts wrote:
>> Op 29.mrt.2024 om 15:21 schreef olcott:
>>> On 3/29/2024 8:23 AM, Andy Walker wrote:
>>>> On 29/03/2024 04:29, olcott wrote:
>>>>> x is said to be infinitesimal
>>>>> if, and only if, |x| < 1/n for all integers n.
>>>>> https://en.wikipedia.org/wiki/Hyperreal_number
>>>>
>>>>      That's for the hyperreals;  there's a clue in the URL.
>>>> There are no such "x" in R, by the Archimedean axiom.
>>>>
>>>>> 0.999... specifies infinitesimally < 1.0
>>>>
>>>>      No it doesn't.  It specifies different things in different
>>>> number systems, which is why mathematicians don't use that notation
>>>> in contexts where there could be ambiguity.
>>>>
>>>>> and math guys have no way to say that so they
>>>>> simply round up to 1.0
>>>>
>>>>      You've just referred to some "math guys" -- the proponents of
>>>> the hyperreals -- who say exactly what they mean by "infinitesimal".
>>>> You could equally have referred to the surreals [qv] where similar
>>>> statements are made and explained by "math guys".  Maths has moved
>>>> on over the past few centuries.  You and Wij need to move on with
>>>> the "math guys".
>>>>
>>>
>>> Yet my system seems to make more sense.
>>> [0.0, 1.0] - [0.0, 1.0) Only the last point on the number line
>>> of the first interval is not contained in the second interval.
>>>
>>
>> Olcott's system does not (yet) make sense, because olcott has not
>> defined his system. E.g., what is a point in this context? What is a
>> number line?
>> This is how Reals are defined:
>>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>>
>> If olcott wants to discuss his system, he should define his system at
>> least as detailed as reals are in this article and indicate where he is
>> deviating from reals, otherwise it is unclear where he is talking about
>> and discussion do not make sense.
>
> Why do you keep quoting something you don't even understand?
> You should know that you cannot event proof "1+2+3" !
>

You DO know that this is actually provable from a couple of basic
axioms. (I presume you mean 1 + 2 = 3)

It does get into somewhat uncommon notations for the definitons of numbers.

As a Hint, we rewrite that stetment to be S0 + SS0 = SSS0

Op 29.mrt.2024 om 16:46 schreef olcott:
> On 3/29/2024 8:13 AM, Richard Damon wrote:
>> On 3/28/24 11:50 PM, olcott wrote:
>>> On 3/28/2024 10:36 PM, Keith Thompson wrote:
>>>> olcott <polcott2@gmail.com> writes:
>>>> [...]
>>>>> It seems dead obvious that 0.999... is infinitesimally less than 1.0.
>>>>
>>>> Yes, it *seems* dead obvious.  That doesn't make it true, and in
>>>> fact it
>>>> isn't.
>>>>
>>>
>>> 0.999... means that is never reaches 1.0.
>>> and math simply stipulates that it does even though it does not.
>>
>>
>> 0.999... isn't a "number" in the Real Number system, just an alternate
>> representation for the number 1.
>>
>
> That is not true. 0.999... means never reaches 1.0

Maybe for olcott's unspecified olcott numbers. For real numbers 0.999...
equals 1.0. There are many proofs. See e.g.
> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

On 3/29/2024 3:47 PM, Fred. Zwarts wrote:
> Op 29.mrt.2024 om 16:46 schreef olcott:
>> On 3/29/2024 8:13 AM, Richard Damon wrote:
>>> On 3/28/24 11:50 PM, olcott wrote:
>>>> On 3/28/2024 10:36 PM, Keith Thompson wrote:
>>>>> olcott <polcott2@gmail.com> writes:
>>>>> [...]
>>>>>> It seems dead obvious that 0.999... is infinitesimally less than 1.0.
>>>>>
>>>>> Yes, it *seems* dead obvious.  That doesn't make it true, and in
>>>>> fact it
>>>>> isn't.
>>>>>
>>>>
>>>> 0.999... means that is never reaches 1.0.
>>>> and math simply stipulates that it does even though it does not.
>>>
>>>
>>> 0.999... isn't a "number" in the Real Number system, just an
>>> alternate representation for the number 1.
>>>
>>
>> That is not true. 0.999... means never reaches 1.0
>

0.999... means never reaches 1.0.
That math rounds this up to 1.0 because they don't have any
other number system to say the actual truth is still an error.

> Maybe for olcott's unspecified olcott numbers. For real numbers 0.999...
> equals 1.0. There are many proofs. See e.g.
>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>
>

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 03/29/2024 06:23 AM, Andy Walker wrote:
> On 29/03/2024 04:29, olcott wrote:
>> x is said to be infinitesimal
>> if, and only if, |x| < 1/n for all integers n.
>> https://en.wikipedia.org/wiki/Hyperreal_number
>
> That's for the hyperreals; there's a clue in the URL.
> There are no such "x" in R, by the Archimedean axiom.
>
>> 0.999... specifies infinitesimally < 1.0
>
> No it doesn't. It specifies different things in different
> number systems, which is why mathematicians don't use that notation
> in contexts where there could be ambiguity.
>
>> and math guys have no way to say that so they
>> simply round up to 1.0
>
> You've just referred to some "math guys" -- the proponents of
> the hyperreals -- who say exactly what they mean by "infinitesimal".
> You could equally have referred to the surreals [qv] where similar
> statements are made and explained by "math guys". Maths has moved
> on over the past few centuries. You and Wij need to move on with
> the "math guys".
>

Hyperreals and sur-reals are the most usually referenced modern
infinitesimals, they don't really say anything except as being
sort of "conservative extensions" of the standard field, though
there's that hyper-integers are as like an extension and the
sur-reals are not an Archimedean field.

The iota-values or raw differentials match most people's
ideas of what infinitesimals, constant infinitesimals, are.

Here it's iota-values first then completeness in the
complete ordered field is after that.

"Maths has moved on ...".

"Fred. Zwarts" <F.Zwarts@HetNet.nl> writes:
> Op 29.mrt.2024 om 16:46 schreef olcott:
>> On 3/29/2024 8:13 AM, Richard Damon wrote:
>>> On 3/28/24 11:50 PM, olcott wrote:
>>>> On 3/28/2024 10:36 PM, Keith Thompson wrote:
>>>>> olcott <polcott2@gmail.com> writes:
>>>>> [...]
>>>>>> It seems dead obvious that 0.999... is infinitesimally less than 1.0.
>>>>>
>>>>> Yes, it *seems* dead obvious.  That doesn't make it true, and in
>>>>> fact it
>>>>> isn't.
>>>>>
>>>>
>>>> 0.999... means that is never reaches 1.0.
>>>> and math simply stipulates that it does even though it does not.
>>>
>>>
>>> 0.999... isn't a "number" in the Real Number system, just an
>>> alternate representation for the number 1.
>>>
>> That is not true. 0.999... means never reaches 1.0
>
> Maybe for olcott's unspecified olcott numbers. For real numbers
> 0.999... equals 1.0. There are many proofs. See e.g.
>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

olcott almost has a point, in that the sequence of values 0.9, 0.99,
0.999, 0.9999, ... (continuing in the obvious manner) never reaches
1.0. No element of that unending sequence of real numbers is exactly
equal to 1.0.

What he either doesn't understand, or pretends not to understand, is
that the notation "0.999..." does not refer either to any element of
that sequence or to the entire sequence. It refers to the *limit* of
the sequence. The limit of the sequence happens not to be an element of
the sequence, and it's exactly equal to 1.0.

This is all stated in terms of the real numbers, which are a well
defined set. There are other systems with different properties. If we
were talking about the hyperreals, for example, olcott's statement might
be correct (though I'm not sure of that). But olcott seems to be
insisting, quite incorrectly, that his statements apply to the reals.

If he's talking about the reals, he's wrong. If he's talking about
something other than the reals, he's boring. Either way, he will not
change his mind. Attempts to explain limits and real numbers to him
will fail.

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

On 3/29/2024 7:25 PM, Keith Thompson wrote:
> "Fred. Zwarts" <F.Zwarts@HetNet.nl> writes:
>> Op 29.mrt.2024 om 16:46 schreef olcott:
>>> On 3/29/2024 8:13 AM, Richard Damon wrote:
>>>> On 3/28/24 11:50 PM, olcott wrote:
>>>>> On 3/28/2024 10:36 PM, Keith Thompson wrote:
>>>>>> olcott <polcott2@gmail.com> writes:
>>>>>> [...]
>>>>>>> It seems dead obvious that 0.999... is infinitesimally less than 1.0.
>>>>>>
>>>>>> Yes, it *seems* dead obvious.  That doesn't make it true, and in
>>>>>> fact it
>>>>>> isn't.
>>>>>>
>>>>>
>>>>> 0.999... means that is never reaches 1.0.
>>>>> and math simply stipulates that it does even though it does not.
>>>>
>>>>
>>>> 0.999... isn't a "number" in the Real Number system, just an
>>>> alternate representation for the number 1.
>>>>
>>> That is not true. 0.999... means never reaches 1.0
>>
>> Maybe for olcott's unspecified olcott numbers. For real numbers
>> 0.999... equals 1.0. There are many proofs. See e.g.
>>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>
> olcott almost has a point, in that the sequence of values 0.9, 0.99,
> 0.999, 0.9999, ... (continuing in the obvious manner) never reaches
> 1.0. No element of that unending sequence of real numbers is exactly
> equal to 1.0.
>
> What he either doesn't understand, or pretends not to understand, is
> that the notation "0.999..." does not refer either to any element of
> that sequence or to the entire sequence. It refers to the *limit* of
> the sequence. The limit of the sequence happens not to be an element of
> the sequence, and it's exactly equal to 1.0.
>
In other words when one gets to the end of a never ending sequence
(a contradiction) thenn (then and only then) they reach 1.0.

> This is all stated in terms of the real numbers, which are a well
> defined set. There are other systems with different properties. If we
> were talking about the hyperreals, for example, olcott's statement might
> be correct (though I'm not sure of that). But olcott seems to be
> insisting, quite incorrectly, that his statements apply to the reals.
>

Pi exists at a single geometric point on the number line.

> If he's talking about the reals, he's wrong. If he's talking about
> something other than the reals, he's boring. Either way, he will not
> change his mind. Attempts to explain limits and real numbers to him
> will fail.
>

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

olcott <polcott2@gmail.com> writes:
> On 3/29/2024 7:25 PM, Keith Thompson wrote:
[...]
>> What he either doesn't understand, or pretends not to understand, is
>> that the notation "0.999..." does not refer either to any element of
>> that sequence or to the entire sequence. It refers to the *limit* of
>> the sequence. The limit of the sequence happens not to be an element of
>> the sequence, and it's exactly equal to 1.0.
>>
> In other words when one gets to the end of a never ending sequence
> (a contradiction) thenn (then and only then) they reach 1.0.

No.

You either don't understand, or are pretending not to understand, what
the limit of sequence is. I'm not offering to explain it to you.

>> This is all stated in terms of the real numbers, which are a well
>> defined set. There are other systems with different properties. If we
>> were talking about the hyperreals, for example, olcott's statement might
>> be correct (though I'm not sure of that). But olcott seems to be
>> insisting, quite incorrectly, that his statements apply to the reals.
>
> Pi exists at a single geometric point on the number line.

Irrelevant.

[...]

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

Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
[...]
> You either don't understand, or are pretending not to understand, what
> the limit of sequence is. I'm not offering to explain it to you.

I accidentally omitted a word. What I meant to write was :

You either don't understand, or are pretending not to understand, what
the limit of a sequence is. I'm not offering to explain it to you.

(The original was probably clear enough.)

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

On 3/29/2024 8:21 PM, Keith Thompson wrote:
> olcott <polcott2@gmail.com> writes:
>> On 3/29/2024 7:25 PM, Keith Thompson wrote:
> [...]
>>> What he either doesn't understand, or pretends not to understand, is
>>> that the notation "0.999..." does not refer either to any element of
>>> that sequence or to the entire sequence. It refers to the *limit* of
>>> the sequence. The limit of the sequence happens not to be an element of
>>> the sequence, and it's exactly equal to 1.0.
>>>
>> In other words when one gets to the end of a never ending sequence
>> (a contradiction) thenn (then and only then) they reach 1.0.
>
> No.
>
> You either don't understand, or are pretending not to understand, what
> the limit of sequence is. I'm not offering to explain it to you.
>

I know (or at least knew) what limits are from my college calculus 40
years ago. If anyone or anything in any way says that 0.999... equals
1.0 then they <are> saying what happens at the end of a never ending
sequence and this is a contradiction.

>>> This is all stated in terms of the real numbers, which are a well
>>> defined set. There are other systems with different properties. If we
>>> were talking about the hyperreals, for example, olcott's statement might
>>> be correct (though I'm not sure of that). But olcott seems to be
>>> insisting, quite incorrectly, that his statements apply to the reals.
>>
>> Pi exists at a single geometric point on the number line.
>
> Irrelevant.

One geometric point to the left or to the right is incorrect.

>
> [...]
>

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

olcott <polcott2@gmail.com> writes:
> On 3/29/2024 8:21 PM, Keith Thompson wrote:
>> olcott <polcott2@gmail.com> writes:
>>> On 3/29/2024 7:25 PM, Keith Thompson wrote:
>> [...]
>>>> What he either doesn't understand, or pretends not to understand, is
>>>> that the notation "0.999..." does not refer either to any element of
>>>> that sequence or to the entire sequence. It refers to the *limit* of
>>>> the sequence. The limit of the sequence happens not to be an element of
>>>> the sequence, and it's exactly equal to 1.0.
>>>>
>>> In other words when one gets to the end of a never ending sequence
>>> (a contradiction) thenn (then and only then) they reach 1.0.
>> No.
>>
>> You either don't understand, or are pretending not to understand,
>> what the limit of sequence is. I'm not offering to explain it to
>> you.
>
> I know (or at least knew) what limits are from my college calculus 40
> years ago. If anyone or anything in any way says that 0.999... equals
> 1.0 then they <are> saying what happens at the end of a never ending
> sequence and this is a contradiction.

Apparently you've forgotten what limits are. I'm still not offering to
explain them.

>>>> This is all stated in terms of the real numbers, which are a well
>>>> defined set. There are other systems with different properties. If we
>>>> were talking about the hyperreals, for example, olcott's statement might
>>>> be correct (though I'm not sure of that). But olcott seems to be
>>>> insisting, quite incorrectly, that his statements apply to the reals.
>>>
>>> Pi exists at a single geometric point on the number line.
>> Irrelevant.
>
> One geometric point to the left or to the right is incorrect.

You apparently think there's a geometric point immediately to the left
of pi. Real numbers don't work that way.

Say the numeric value corresponding to the geometric point immediately
to the left of pi on the real number line is x. What is the real value
of (pi+x)/2? Is it greater than x? Is it less than pi?

I'm going to drop out of this discussion unless someone says something
sufficiently interesting.

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

On 3/29/2024 8:43 PM, Keith Thompson wrote:
> olcott <polcott2@gmail.com> writes:
>> On 3/29/2024 8:21 PM, Keith Thompson wrote:
>>> olcott <polcott2@gmail.com> writes:
>>>> On 3/29/2024 7:25 PM, Keith Thompson wrote:
>>> [...]
>>>>> What he either doesn't understand, or pretends not to understand, is
>>>>> that the notation "0.999..." does not refer either to any element of
>>>>> that sequence or to the entire sequence. It refers to the *limit* of
>>>>> the sequence. The limit of the sequence happens not to be an element of
>>>>> the sequence, and it's exactly equal to 1.0.
>>>>>
>>>> In other words when one gets to the end of a never ending sequence
>>>> (a contradiction) thenn (then and only then) they reach 1.0.
>>> No.
>>>
>>> You either don't understand, or are pretending not to understand,
>>> what the limit of sequence is. I'm not offering to explain it to
>>> you.
>>
>> I know (or at least knew) what limits are from my college calculus 40
>> years ago. If anyone or anything in any way says that 0.999... equals
>> 1.0 then they <are> saying what happens at the end of a never ending
>> sequence and this is a contradiction.
>
> Apparently you've forgotten what limits are. I'm still not offering to
> explain them.
>
>>>>> This is all stated in terms of the real numbers, which are a well
>>>>> defined set. There are other systems with different properties. If we
>>>>> were talking about the hyperreals, for example, olcott's statement might
>>>>> be correct (though I'm not sure of that). But olcott seems to be
>>>>> insisting, quite incorrectly, that his statements apply to the reals.
>>>>
>>>> Pi exists at a single geometric point on the number line.
>>> Irrelevant.
>>
>> One geometric point to the left or to the right is incorrect.
>
> You apparently think there's a geometric point immediately to the left
> of pi. Real numbers don't work that way.
>
> Say the numeric value corresponding to the geometric point immediately
> to the left of pi on the real number line is x. What is the real value
> of (pi+x)/2? Is it greater than x? Is it less than pi?
>
> I'm going to drop out of this discussion unless someone says something
> sufficiently interesting.
>

Can you quit publishing my email address?

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 2024-03-29 20:18, olcott wrote:

>
> Can you quit publishing my email address?

He's not. You are. His newsreader is just quoting you.

André

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

On 3/29/2024 9:28 PM, André G. Isaak wrote:
> On 2024-03-29 20:18, olcott wrote:
>
>>
>> Can you quit publishing my email address?
>
> He's not. You are. His newsreader is just quoting you.
>
> André
>

Somehow the newsgroup provider started publishing it.
It didn't do this initially.

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 3/29/24 10:44 PM, olcott wrote:
> On 3/29/2024 9:28 PM, André G. Isaak wrote:
>> On 2024-03-29 20:18, olcott wrote:
>>
>>>
>>> Can you quit publishing my email address?
>>
>> He's not. You are. His newsreader is just quoting you.
>>
>> André
>>
>
> Somehow the newsgroup provider started publishing it.
> It didn't do this initially.
>

It is normally controlled by your newsreader, and the format technically
REQUIRES an email address.

If your config still has a fake address configured, your NewsProvider
might have disabled anonymous postings for anti-spam purposes, though I
thought Eternal-September still allowed them (if not abused).

Richard Damon <Richard@Damon-Family.org> writes:
> On 3/29/24 10:44 PM, olcott wrote:
>> On 3/29/2024 9:28 PM, André G. Isaak wrote:
>>> On 2024-03-29 20:18, olcott wrote:
>>>
>>>>
>>>> Can you quit publishing my email address?
>>>
>>> He's not. You are. His newsreader is just quoting you.
>>>
>>> André
>>>
>> Somehow the newsgroup provider started publishing it.
>> It didn't do this initially.
>>
>
> It is normally controlled by your newsreader, and the format
> technically REQUIRES an email address.
>
> If your config still has a fake address configured, your NewsProvider
> might have disabled anonymous postings for anti-spam purposes, though
> I thought Eternal-September still allowed them (if not abused).

Eternal-September doesn't check the validity of a poster's email address
(see the headers of this message).

--
sig omitted for this message only

On 03/29/2024 06:24 PM, Keith Thompson wrote:
> Keith Thompson <Keith.S.Thompson+u@gmail.com> writes:
> [...]
>> You either don't understand, or are pretending not to understand, what
>> the limit of sequence is. I'm not offering to explain it to you.
>
> I accidentally omitted a word. What I meant to write was :
>
> You either don't understand, or are pretending not to understand, what
> the limit of a sequence is. I'm not offering to explain it to you.
>
> (The original was probably clear enough.)
>

How about the limit of a function?

(Specifically the function of the ratio n/d of
natural integers, only as d -> oo .)