Op 30.mrt.2024 om 02:31 schreef olcott:
> 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.
>
If olcott had read the article I referenced,
> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

He would have seen a proof for 0.999... = 1. This article also explains
what it means. With a proper interpretation of these words, there is no
contradiction. He may not like it, but it has been proven. So, either he
continues to talk about his unspecified olcott numbers, or he does not
understand the proof for real numbers, or he changes the meaning of the
words. Of course, if there are details in the proof he does not
understand, he is free to ask for an explanation.

On 30/03/2024 01:11, olcott wrote:
> 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.

Zeno knew better than that some 2400 years ago. He knew that
Achilles caught the tortoise, even though it was an infinite number of
steps; it depends on how fast you complete/encounter the elements of
the sequence. If you travel along the real number line at 1 unit/hour,
starting at 0 at noon, then you reach 0.9 at 12:54, 0.99 at 12:59:24,
0.999 at 12:59:56.4, 0.9999 at 12:59:59.64, 0.99999 at 12:59:59.964,
0.999999 at ..., and this "never ending sequence" is completed by 13:00.
Luckily, none of this matters, as "0.999..." viewed as a real number is
not defined as a "never ending sequence", but as the limit of a sequence
whose terms are defined.

If you switch from the reals to the surreals or hyperreals, then
you may prefer the way "0.999..." is treated, but you have simply moved
the "never-ending" problem to somewhere else, and the solution is the
same -- you need to sub-divide time appropriately to complete many steps
in a short time. Time is inexorable in the Real World, but messwithable
in theoretical physics and in mathematics.

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

On 3/30/24 5:37 AM, Fred. Zwarts wrote:
>>
> If olcott had read the article I referenced,
>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>
> He would have seen a proof for 0.999... = 1. This article also explains
> what it means. With a proper interpretation of these words, there is no
> contradiction. He may not like it, but it has been proven. So, either he
> continues to talk about his unspecified olcott numbers, or he does not
> understand the proof for real numbers, or he changes the meaning of the
> words. Of course, if there are details in the proof he does not
> understand, he is free to ask for an explanation.

His issue is likely with:

With a proper interpretation of these words,

As he has problems with not wanting to let others define the meaning of
the words, but instead uses his "Zeroth Principles" to invent the
meaning that he imagines must be for the meaning of the words, and then
he complains that people don't understand him when he uses is made up
definitions.

Op 30.mrt.2024 om 02:31 schreef olcott:
> 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.
>

It is clear that olcott does not understand limits, because he is
changing the meaning of the words and the symbols. Limits are not
talking about what happens at the end of a sequence. It seems it has to
be spelled out for him, otherwise he will not understand.

0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 = 99/100,
x3 = 999/100, etc. The three dots indicates the limit n→∞. The = symbol
in the context of a limit means in this case:
For each rational ε > 0 (no matter how small) we can find a number N {in
this case 10log(1/ε)}, such that for all n > N the absolute value of the
difference between xn and 1.0 is less than ε.
It is not more and not less. Note that it does not speak of what happens
at the end of the sequence, or about completing the sequence.
If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
system), then he has to specify a rational ε for which no such N can be
found. If he cannot do that, then he is not speaking about real numbers.

On 3/30/2024 4:37 AM, Fred. Zwarts wrote:
> Op 30.mrt.2024 om 02:31 schreef olcott:
>> 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.
>>
> If olcott had read the article I referenced,
>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>
> He would have seen a proof for 0.999... = 1.

Right and another article says that all cats are really a kind of dog.

0.999... means never reaches 1.0.
Anyone saying differently is not telling the truth.

> This article also explains
> what it means. With a proper interpretation of these words, there is no
> contradiction. He may not like it, but it has been proven. So, either he
> continues to talk about his unspecified olcott numbers, or he does not
> understand the proof for real numbers, or he changes the meaning of the
> words. Of course, if there are details in the proof he does not
> understand, he is free to ask for an explanation.

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

On 3/30/2024 5:08 AM, Andy Walker wrote:
> On 30/03/2024 01:11, olcott wrote:
>> 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.
>
>     Zeno knew better than that some 2400 years ago.  He knew that
> Achilles caught the tortoise, even though it was an infinite number of
> steps;  it depends on how fast you complete/encounter the elements of
> the sequence.  If you travel along the real number line at 1 unit/hour,
> starting at 0 at noon, then you reach 0.9 at 12:54, 0.99 at 12:59:24,
> 0.999 at 12:59:56.4, 0.9999 at 12:59:59.64, 0.99999 at 12:59:59.964,
> 0.999999 at ..., and this "never ending sequence" is completed by 13:00.
> Luckily, none of this matters, as "0.999..." viewed as a real number is
> not defined as a "never ending sequence", but as the limit of a sequence
> whose terms are defined.
>

https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes
Zeno essentially "proved" that no one can cross a ten foot wide room
in finite time. He was nuts.

>     If you switch from the reals to the surreals or hyperreals, then
> you may prefer the way "0.999..." is treated, but you have simply moved
> the "never-ending" problem to somewhere else, and the solution is the
> same -- you need to sub-divide time appropriately to complete many steps
> in a short time.  Time is inexorable in the Real World, but messwithable
> in theoretical physics and in mathematics.
>

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

On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
> Op 30.mrt.2024 om 02:31 schreef olcott:
>> 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.
>>
>
> It is clear that olcott does not understand limits, because he is
> changing the meaning of the words and the symbols. Limits are not
> talking about what happens at the end of a sequence. It seems it has to
> be spelled out for him, otherwise he will not understand.
>

0.999... Limits basically pretend that we reach the end of this infinite
sequence even though that it impossible, and says after we reach this
impossible end the value would be 1.0.

> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 = 99/100,
> x3 = 999/100, etc. The three dots indicates the limit n→∞. The = symbol
> in the context of a limit means in this case:
> For each rational ε > 0 (no matter how small) we can find a number N {in
> this case 10log(1/ε)}, such that for all n > N the absolute value of the
> difference between xn and 1.0 is less than ε.
> It is not more and not less. Note that it does not speak of what happens
> at the end of the sequence, or about completing the sequence.
> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
> system), then he has to specify a rational ε for which no such N can be
> found. If he cannot do that, then he is not speaking about 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 3/30/24 9:49 AM, olcott wrote:
> On 3/30/2024 4:37 AM, Fred. Zwarts wrote:
>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>> 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.
>>>
>> If olcott had read the article I referenced,
>>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>>
>> He would have seen a proof for 0.999... = 1.
>
> Right and another article says that all cats are really a kind of dog.
>

So, you can't tell the difference between someone talking truth and
someone lying.

Seems normal for you.

> 0.999... means never reaches 1.0.

Nope, not in the Real Number system.

> Anyone saying differently is not telling the truth.

Nope. You are just proving that you don't understand what you are
talking about and thus are not telling the truth.

>
>> This article also explains what it means. With a proper interpretation
>> of these words, there is no contradiction. He may not like it, but it
>> has been proven. So, either he continues to talk about his unspecified
>> olcott numbers, or he does not understand the proof for real numbers,
>> or he changes the meaning of the words. Of course, if there are
>> details in the proof he does not understand, he is free to ask for an
>> explanation.
>

On 3/30/24 9:56 AM, olcott wrote:
> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>> 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.
>>>
>>
>> It is clear that olcott does not understand limits, because he is
>> changing the meaning of the words and the symbols. Limits are not
>> talking about what happens at the end of a sequence. It seems it has
>> to be spelled out for him, otherwise he will not understand.
>>
>
> 0.999... Limits basically pretend that we reach the end of this infinite
> sequence even though that it impossible, and says after we reach this
> impossible end the value would be 1.0.

Nope. Shows you don't really understand what limits are.

And are just a pathological liar as you insist that you falsehoods based
on the wrong definitions are the truth.

>
>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
>> = symbol in the context of a limit means in this case:
>> For each rational ε > 0 (no matter how small) we can find a number N
>> {in this case 10log(1/ε)}, such that for all n > N the absolute value
>> of the difference between xn and 1.0 is less than ε.
>> It is not more and not less. Note that it does not speak of what
>> happens at the end of the sequence, or about completing the sequence.
>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>> system), then he has to specify a rational ε for which no such N can
>> be found. If he cannot do that, then he is not speaking about real
>> numbers.
>>
>

On 3/30/24 9:53 AM, olcott wrote:
> On 3/30/2024 5:08 AM, Andy Walker wrote:
>> On 30/03/2024 01:11, olcott wrote:
>>> 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.
>>
>>      Zeno knew better than that some 2400 years ago.  He knew that
>> Achilles caught the tortoise, even though it was an infinite number of
>> steps;  it depends on how fast you complete/encounter the elements of
>> the sequence.  If you travel along the real number line at 1 unit/hour,
>> starting at 0 at noon, then you reach 0.9 at 12:54, 0.99 at 12:59:24,
>> 0.999 at 12:59:56.4, 0.9999 at 12:59:59.64, 0.99999 at 12:59:59.964,
>> 0.999999 at ..., and this "never ending sequence" is completed by 13:00.
>> Luckily, none of this matters, as "0.999..." viewed as a real number is
>> not defined as a "never ending sequence", but as the limit of a sequence
>> whose terms are defined.
>>
>
> https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes
> Zeno essentially "proved" that no one can cross a ten foot wide room
> in finite time. He was nuts.

Nope, he was pointing out that one notion of how mathematics could be
defined couldn't handle such an infinite sequence.

In one sense, not Unlike Russel's showing his set of all sets that don't
contain himself.

>
>>      If you switch from the reals to the surreals or hyperreals, then
>> you may prefer the way "0.999..." is treated, but you have simply moved
>> the "never-ending" problem to somewhere else, and the solution is the
>> same -- you need to sub-divide time appropriately to complete many steps
>> in a short time.  Time is inexorable in the Real World, but messwithable
>> in theoretical physics and in mathematics.
>>
>

On Sat, 2024-03-30 at 10:01 -0400, Richard Damon wrote:
> On 3/30/24 9:56 AM, olcott wrote:
> > On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
> > > Op 30.mrt.2024 om 02:31 schreef olcott:
> > > > 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.
> > > >
> > >
> > > It is clear that olcott does not understand limits, because he is
> > > changing the meaning of the words and the symbols. Limits are not
> > > talking about what happens at the end of a sequence. It seems it has
> > > to be spelled out for him, otherwise he will not understand.
> > >
> >
> > 0.999... Limits basically pretend that we reach the end of this infinite
> > sequence even though that it impossible, and says after we reach this
> > impossible end the value would be 1.0.
>
> Nope. Shows you don't really understand what limits are.
>
> And are just a pathological liar as you insist that you falsehoods based
> on the wrong definitions are the truth.
>

You are nut who always think he is talking B while reading A. (x!=c)
https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits..html

Limit is defined on existing numbers, it cannot define the the number it is using.
Things is very simple: "repeating decimal" means the pattern is infinite.
(you are worse than olcott in this)
If it does not exit, your math (repeating decimal is ...) is garbage talking
about something does not exist and use it as proof of fact.

> >
> > > 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
> > > 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
> > > = symbol in the context of a limit means in this case:
> > > For each rational ε > 0 (no matter how small) we can find a number N
> > > {in this case 10log(1/ε)}, such that for all n > N the absolute value
> > > of the difference between xn and 1.0 is less than ε.
> > > It is not more and not less. Note that it does not speak of what
> > > happens at the end of the sequence, or about completing the sequence.
> > > If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
> > > system), then he has to specify a rational ε for which no such N can
> > > be found. If he cannot do that, then he is not speaking about real
> > > numbers.
> > >
> >
>

On 3/30/24 10:57 AM, wij wrote:
> On Sat, 2024-03-30 at 10:01 -0400, Richard Damon wrote:
>> On 3/30/24 9:56 AM, olcott wrote:
>>> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>>>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>>>> 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.
>>>>>
>>>>
>>>> It is clear that olcott does not understand limits, because he is
>>>> changing the meaning of the words and the symbols. Limits are not
>>>> talking about what happens at the end of a sequence. It seems it has
>>>> to be spelled out for him, otherwise he will not understand.
>>>>
>>>
>>> 0.999... Limits basically pretend that we reach the end of this infinite
>>> sequence even though that it impossible, and says after we reach this
>>> impossible end the value would be 1.0.
>>
>> Nope. Shows you don't really understand what limits are.
>>
>> And are just a pathological liar as you insist that you falsehoods based
>> on the wrong definitions are the truth.
>>
>
> You are nut who always think he is talking B while reading A. (x!=c)
> https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
>
>
> Limit is defined on existing numbers, it cannot define the the number it is using.
> Things is very simple: "repeating decimal" means the pattern is infinite.
> (you are worse than olcott in this)
> If it does not exit, your math (repeating decimal is ...) is garbage talking
> about something does not exist and use it as proof of fact.

What number does the representation 0.abc represent?

it is BY DEFINITION 0 + a * 10^-1 + b * 10^-2 + c * 10^-3

what number does the representation 0.aaa... represent:

The value of lim(n-> inf) Sum(9 * 10^-i) [for i = 1 to n]

If a = 9, what number is that 0.999.... but also the number 1.0 since
they are the same.

For ANY e > 0, there exists an N that for all values of function/series
witn n >= N the difference between the function and 1 is less then e.

BY THE DEFINITION OF LIMIT, that means that 0.999... IS EQUAL TO 1.000

For Reals

Remember n-ary representations are NOT numbers, but representations of
the number.

the value of repeating n-ary representations are defined by limits

Limits are NOT on "a number" but on a function or series (which is a
sort of function of the number of terms being used).

>
>>>
>>>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>>>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
>>>> = symbol in the context of a limit means in this case:
>>>> For each rational ε > 0 (no matter how small) we can find a number N
>>>> {in this case 10log(1/ε)}, such that for all n > N the absolute value
>>>> of the difference between xn and 1.0 is less than ε.
>>>> It is not more and not less. Note that it does not speak of what
>>>> happens at the end of the sequence, or about completing the sequence.
>>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>>>> system), then he has to specify a rational ε for which no such N can
>>>> be found. If he cannot do that, then he is not speaking about real
>>>> numbers.
>>>>
>>>
>>
>
>

On Sat, 2024-03-30 at 11:34 -0400, Richard Damon wrote:
> On 3/30/24 10:57 AM, wij wrote:
> > On Sat, 2024-03-30 at 10:01 -0400, Richard Damon wrote:
> > > On 3/30/24 9:56 AM, olcott wrote:
> > > > On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
> > > > > Op 30.mrt.2024 om 02:31 schreef olcott:
> > > > > > 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.
> > > > > >
> > > > >
> > > > > It is clear that olcott does not understand limits, because he is
> > > > > changing the meaning of the words and the symbols. Limits are not
> > > > > talking about what happens at the end of a sequence. It seems it has
> > > > > to be spelled out for him, otherwise he will not understand.
> > > > >
> > > >
> > > > 0.999... Limits basically pretend that we reach the end of this infinite
> > > > sequence even though that it impossible, and says after we reach this
> > > > impossible end the value would be 1.0.
> > >
> > > Nope. Shows you don't really understand what limits are.
> > >
> > > And are just a pathological liar as you insist that you falsehoods based
> > > on the wrong definitions are the truth.
> > >
> >
> > You are nut who always think he is talking B while reading A. (x!=c)
> > https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
> >
> >
> > Limit is defined on existing numbers, it cannot define the the number it is using.
> > Things is very simple: "repeating decimal" means the pattern is infinite.
> > (you are worse than olcott in this)
> > If it does not exit, your math (repeating decimal is ...) is garbage talking
> > about something does not exist and use it as proof of fact.
>
>
> What number does the representation 0.abc represent?
>
> it is BY DEFINITION 0 + a * 10^-1 + b * 10^-2 + c * 10^-3
>
> what number does the representation 0.aaa... represent:
>
> The value of lim(n-> inf) Sum(9 * 10^-i) [for i = 1 to n]
>
> If a = 9, what number is that 0.999.... but also the number 1.0 since
> they are the same.
>

I should have provided all that can explain your doubt, but you still
keep insisting 0.999...=1 with no proof, like POOP.

> For ANY e > 0, there exists an N that for all values of function/series
> witn n >= N the difference between the function and 1 is less then e.
>
> BY THE DEFINITION OF LIMIT, that means that 0.999... IS EQUAL TO 1.000
>
See the link above. limit says the limit of 0.999... is 1, not 0.999... is 1.
You keep talking RD's POOP.

> For Reals
>
For your POO Real (not even the obsolete real)

> Remember n-ary representations are NOT numbers, but representations of
> the number.
>
> the value of repeating n-ary representations are defined by limits
>
I already said, limit is defined on existing number system, it cannot define
numbers it talks about.

> Limits are NOT on "a number" but on a function or series (which is a
> sort of function of the number of terms being used).
>
Ok, now you changed to another excuse. So, you are not really talking
about numbers, right.

> >
> > > >
> > > > > 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 > > > > > 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
> > > > > = symbol in the context of a limit means in this case:
> > > > > For each rational ε > 0 (no matter how small) we can find a number N
> > > > > {in this case 10log(1/ε)}, such that for all n > N the absolute value
> > > > > of the difference between xn and 1.0 is less than ε.
> > > > > It is not more and not less. Note that it does not speak of what
> > > > > happens at the end of the sequence, or about completing the sequence.
> > > > > If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
> > > > > system), then he has to specify a rational ε for which no such N can
> > > > > be found. If he cannot do that, then he is not speaking about real
> > > > > numbers.
> > > > >
> > > >
> > >
> >
> >
>

On 03/29/2024 09:47 PM, Ross Finlayson wrote:
> 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 .)
>
>

What you do is introduce numbers and trichotomy,
then a number line starting just one way,
demonstrate addition and multiplication,
and that addition can go on forever and
multiplication can go on forever, then
show additive partitions and multiplicative factors,
that subtraction works back down to zero and
that divsion works back down to one,
with even sums and even quotients and
thusly the whole modularity of the thing,
at some point there's the number line both ways,
then all students know arithmetic is closed,
then that equalities and inequalities are transitive,
and they know that each number has partitions and factors,
so really then there's the modular,
then get into powers then roots,
geometry by now being a usual thing,
and algebra of course following the rules
because they know everything there is to know about the numbers,
including that a continuous process from A to B goes through all points
A to B and in order,
then when reaching rules about the continuous and infinitely-divisible,
providing delta-epsilonics and the infinite limit,
then they know both that continuity has no jumps,
and, that the infinite limit makes a rule to start calculus.

Then also by then they should know significant digits
and scientific numbers and approximations and rounding.

Counting is sort of on the side of that, but it's very useful,
relating numbers to things in the world, and of course is
already part of the sense of orders and individuals of things.

Then anyways, the continuum limit, of functions,
works out also very useful, and uses the same sort of rules,
so that all the rules are one big rule,
and everybody knows everything about all of it.

I understand this very well, it makes perfect sense.

On 3/30/24 12:00 PM, wij wrote:
> On Sat, 2024-03-30 at 11:34 -0400, Richard Damon wrote:
>> On 3/30/24 10:57 AM, wij wrote:
>>> On Sat, 2024-03-30 at 10:01 -0400, Richard Damon wrote:
>>>> On 3/30/24 9:56 AM, olcott wrote:
>>>>> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>>>>>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>>>>>> 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.
>>>>>>>
>>>>>>
>>>>>> It is clear that olcott does not understand limits, because he is
>>>>>> changing the meaning of the words and the symbols. Limits are not
>>>>>> talking about what happens at the end of a sequence. It seems it has
>>>>>> to be spelled out for him, otherwise he will not understand.
>>>>>>
>>>>>
>>>>> 0.999... Limits basically pretend that we reach the end of this infinite
>>>>> sequence even though that it impossible, and says after we reach this
>>>>> impossible end the value would be 1.0.
>>>>
>>>> Nope. Shows you don't really understand what limits are.
>>>>
>>>> And are just a pathological liar as you insist that you falsehoods based
>>>> on the wrong definitions are the truth.
>>>>
>>>
>>> You are nut who always think he is talking B while reading A. (x!=c)
>>> https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
>>>
>>>
>>> Limit is defined on existing numbers, it cannot define the the number it is using.
>>> Things is very simple: "repeating decimal" means the pattern is infinite.
>>> (you are worse than olcott in this)
>>> If it does not exit, your math (repeating decimal is ...) is garbage talking
>>> about something does not exist and use it as proof of fact.
>>
>>
>> What number does the representation 0.abc represent?
>>
>> it is BY DEFINITION 0 + a * 10^-1 + b * 10^-2 + c * 10^-3
>>
>> what number does the representation 0.aaa... represent:
>>
>> The value of lim(n-> inf) Sum(9 * 10^-i) [for i = 1 to n]
>>
>> If a = 9, what number is that 0.999.... but also the number 1.0 since
>> they are the same.
>>
>
> I should have provided all that can explain your doubt, but you still
> keep insisting 0.999...=1 with no proof, like POOP.

But I did, see below

Since there is no finite e that doesn't have an N, the limit of the
function, which is the MEANING of the representation 0.999... is 1

>
>> For ANY e > 0, there exists an N that for all values of function/series
>> witn n >= N the difference between the function and 1 is less then e.
>>
>> BY THE DEFINITION OF LIMIT, that means that 0.999... IS EQUAL TO 1.000
>>
> See the link above. limit says the limit of 0.999... is 1, not 0.999... is 1.
> You keep talking RD's POOP.

Where?

It defines what a limit is. Is says NOTHING about what "0.999..." is.

That comes out of the DEFINITION of the REPRESENTATION of repeating
decimal literals.

>
>> For Reals
>>
> For your POO Real (not even the obsolete real)

Nope, I am quoting from the classical (which you want to call obsolete,
but can't show how it is obsolete) theory.

>
>> Remember n-ary representations are NOT numbers, but representations of
>> the number.
>>
>> the value of repeating n-ary representations are defined by limits
>>
> I already said, limit is defined on existing number system, it cannot define
> numbers it talks about.

But 0.999... ISN'T a "NUMBER" but the REPRESENTATION of a Number, and
for numbers in the Real Number System, the number it represents is 1

>
>> Limits are NOT on "a number" but on a function or series (which is a
>> sort of function of the number of terms being used).
>>
> Ok, now you changed to another excuse. So, you are not really talking
> about numbers, right.

Since there is no such thing as the limit of a "Number", YOUR
explanation is just illogical

>
>>>
>>>>>
>>>>>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>>>>>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
>>>>>> = symbol in the context of a limit means in this case:
>>>>>> For each rational ε > 0 (no matter how small) we can find a number N
>>>>>> {in this case 10log(1/ε)}, such that for all n > N the absolute value
>>>>>> of the difference between xn and 1.0 is less than ε.
>>>>>> It is not more and not less. Note that it does not speak of what
>>>>>> happens at the end of the sequence, or about completing the sequence.
>>>>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>>>>>> system), then he has to specify a rational ε for which no such N can
>>>>>> be found. If he cannot do that, then he is not speaking about real
>>>>>> numbers.
>>>>>>
>>>>>
>>>>
>>>
>>>
>>
>
>

Op 30.mrt.2024 om 14:56 schreef olcott:
> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>> 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.
>>>
>>
>> It is clear that olcott does not understand limits, because he is
>> changing the meaning of the words and the symbols. Limits are not
>> talking about what happens at the end of a sequence. It seems it has
>> to be spelled out for him, otherwise he will not understand.
>>
>
> 0.999... Limits basically pretend that we reach the end of this infinite
> sequence even though that it impossible, and says after we reach this
> impossible end the value would be 1.0.

So, olcott did not understand the explanation (below) and continues to
claim that limits talk about reaching the end of the sequence. Since for
real numbers this is not true, he must be talking about is unspecified
olcott numbers.

>
>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
>> = symbol in the context of a limit means in this case:
>> For each rational ε > 0 (no matter how small) we can find a number N
>> {in this case 10log(1/ε)}, such that for all n > N the absolute value
>> of the difference between xn and 1.0 is less than ε.
>> It is not more and not less. Note that it does not speak of what
>> happens at the end of the sequence, or about completing the sequence.
>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>> system), then he has to specify a rational ε for which no such N can
>> be found. If he cannot do that, then he is not speaking about real
>> numbers.
>>
>

I see olcott did not attempt to specify a rational ε, so, he had no
rebuttal against the claim that 0.999... = 1 using the correct meaning
of the words and symbols for reals.

Op 30.mrt.2024 om 14:56 schreef olcott:
> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>> 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.
>>>
>>
>> It is clear that olcott does not understand limits, because he is
>> changing the meaning of the words and the symbols. Limits are not
>> talking about what happens at the end of a sequence. It seems it has
>> to be spelled out for him, otherwise he will not understand.
>>
>
> 0.999... Limits basically pretend that we reach the end of this infinite
> sequence even though that it impossible, and says after we reach this
> impossible end the value would be 1.0.

No, if olcott had paid attention to the text below, or the article I
referenced:
> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

he would have noted that limits do not pretend to reach the end. They
only tell us that we don't need to go further than needed and that this
is reachable for any given rational ε > 0. It is interesting that this
is sufficient to construct reals.

>
>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
>> = symbol in the context of a limit means in this case:
>> For each rational ε > 0 (no matter how small) we can find a number N
>> {in this case 10log(1/ε)}, such that for all n > N the absolute value
>> of the difference between xn and 1.0 is less than ε.
>> It is not more and not less. Note that it does not speak of what
>> happens at the end of the sequence, or about completing the sequence.
>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>> system), then he has to specify a rational ε for which no such N can
>> be found. If he cannot do that, then he is not speaking about real
>> numbers.
>>

Good to see that there is no objection against this proof.

On Sat, 2024-03-30 at 13:23 -0400, Richard Damon wrote:
> On 3/30/24 12:00 PM, wij wrote:
> > On Sat, 2024-03-30 at 11:34 -0400, Richard Damon wrote:
> > > On 3/30/24 10:57 AM, wij wrote:
> > > > On Sat, 2024-03-30 at 10:01 -0400, Richard Damon wrote:
> > > > > On 3/30/24 9:56 AM, olcott wrote:
> > > > > > On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
> > > > > > > Op 30.mrt.2024 om 02:31 schreef olcott:
> > > > > > > > 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.
> > > > > > > >
> > > > > > >
> > > > > > > It is clear that olcott does not understand limits, because he is
> > > > > > > changing the meaning of the words and the symbols. Limits are not
> > > > > > > talking about what happens at the end of a sequence. It seems it has
> > > > > > > to be spelled out for him, otherwise he will not understand.
> > > > > > >
> > > > > >
> > > > > > 0.999... Limits basically pretend that we reach the end of this infinite
> > > > > > sequence even though that it impossible, and says after we reach this
> > > > > > impossible end the value would be 1.0.
> > > > >
> > > > > Nope. Shows you don't really understand what limits are.
> > > > >
> > > > > And are just a pathological liar as you insist that you falsehoods based
> > > > > on the wrong definitions are the truth.
> > > > >
> > > >
> > > > You are nut who always think he is talking B while reading A. (x!=c)
> > > > https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
> > > >
> > > >
> > > > Limit is defined on existing numbers, it cannot define the the number it is using.
> > > > Things is very simple: "repeating decimal" means the pattern is infinite.
> > > > (you are worse than olcott in this)
> > > > If it does not exit, your math (repeating decimal is ...) is garbage talking
> > > > about something does not exist and use it as proof of fact.
> > >
> > >
> > > What number does the representation 0.abc represent?
> > >
> > > it is BY DEFINITION 0 + a * 10^-1 + b * 10^-2 + c * 10^-3
> > >
> > > what number does the representation 0.aaa... represent:
> > >
> > > The value of lim(n-> inf) Sum(9 * 10^-i) [for i = 1 to n]
> > >
> > > If a = 9, what number is that 0.999.... but also the number 1.0 since
> > > they are the same.
> > >
> >
> > I should have provided all that can explain your doubt, but you still
> > keep insisting 0.999...=1 with no proof, like POOP.
>
> But I did, see below
>
> Since there is no finite e that doesn't have an N, the limit of the
> function, which is the MEANING of the representation 0.999... is 1
>

I guess you were talking about delta-epsilon method. But, firstly, we are
not talking about the number system here.

In lim(x->c) f(x), x is the number you choose. The numbers in the sequence
chosen are those expressible. But you would agree there exits numbers not
expressible. And, you seemed to choose "finite e".

> >
> > > For ANY e > 0, there exists an N that for all values of function/series
> > > witn n >= N the difference between the function and 1 is less then e.
> > >
> > > BY THE DEFINITION OF LIMIT, that means that 0.999... IS EQUAL TO 1.000
> > >
> > See the link above. limit says the limit of 0.999... is 1, not 0.999... is 1.
> > You keep talking RD's POOP.
>
> Where?
>
4.1 Limits of Functions
https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits..html
> It defines what a limit is. Is says NOTHING about what "0.999..." is.
>
> That comes out of the DEFINITION of the REPRESENTATION of repeating
> decimal literals.
>
> >
> > > For Reals
> > >
> > For your POO Real (not even the obsolete real)
>
> Nope, I am quoting from the classical (which you want to call obsolete,
> but can't show how it is obsolete) theory.
>
> >
> > > Remember n-ary representations are NOT numbers, but representations of
> > > the number.
> > >
> > > the value of repeating n-ary representations are defined by limits
> > >
> > I already said, limit is defined on existing number system, it cannot define
> > numbers it talks about.
>
> But 0.999... ISN'T a "NUMBER" but the REPRESENTATION of a Number, and
> for numbers in the Real Number System, the number it represents is 1
>

If RD's limt of a "number" is not a number, RD's limit should say so explicitly,
E.g. lim(x->1) x // x is 0.999.. but not infinitely long because it is not a
number. You contradict yourself.

> >
> > > Limits are NOT on "a number" but on a function or series (which is a
> > > sort of function of the number of terms being used).
> > >
> > Ok, now you changed to another excuse. So, you are not really talking
> > about numbers, right.
>
> Since there is no such thing as the limit of a "Number", YOUR
> explanation is just illogical
>
> >
> > > >
> > > > > >
> > > > > > > 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 > > > > > > > 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
> > > > > > > = symbol in the context of a limit means in this case:
> > > > > > > For each rational ε > 0 (no matter how small) we can find a number N
> > > > > > > {in this case 10log(1/ε)}, such that for all n > N the absolute value
> > > > > > > of the difference between xn and 1.0 is less than ε.
> > > > > > > It is not more and not less. Note that it does not speak of what
> > > > > > > happens at the end of the sequence, or about completing the sequence.
> > > > > > > If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
> > > > > > > system), then he has to specify a rational ε for which no such N can
> > > > > > > be found. If he cannot do that, then he is not speaking about real
> > > > > > > numbers.
> > > > > > >
> > > > > >
> > > > >
> > > >
> > > >
> > >
> >
> >
>

On 3/30/2024 2:45 PM, Fred. Zwarts wrote:
> Op 30.mrt.2024 om 14:56 schreef olcott:
>> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>>> 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.
>>>>
>>>
>>> It is clear that olcott does not understand limits, because he is
>>> changing the meaning of the words and the symbols. Limits are not
>>> talking about what happens at the end of a sequence. It seems it has
>>> to be spelled out for him, otherwise he will not understand.
>>>
>>
>> 0.999... Limits basically pretend that we reach the end of this
>> infinite sequence even though that it impossible, and says after we
>> reach this
>> impossible end the value would be 1.0.
>
> No, if olcott had paid attention to the text below, or the article I
> referenced:
>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>
> he would have noted that limits do not pretend to reach the end. They

Other people were saying that math says 0.999... = 1.0

> only tell us that we don't need to go further than needed and that this
> is reachable for any given rational ε > 0. It is interesting that this
> is sufficient to construct reals.
>
>>
>>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞.
>>> The = symbol in the context of a limit means in this case:
>>> For each rational ε > 0 (no matter how small) we can find a number N
>>> {in this case 10log(1/ε)}, such that for all n > N the absolute value
>>> of the difference between xn and 1.0 is less than ε.
>>> It is not more and not less. Note that it does not speak of what
>>> happens at the end of the sequence, or about completing the sequence.
>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>>> system), then he has to specify a rational ε for which no such N can
>>> be found. If he cannot do that, then he is not speaking about real
>>> numbers.
>>>
>
> Good to see that there is no objection against this proof.
>

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

Op 30.mrt.2024 om 20:57 schreef olcott:
> On 3/30/2024 2:45 PM, Fred. Zwarts wrote:
>> Op 30.mrt.2024 om 14:56 schreef olcott:
>>> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>>>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>>>> 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.
>>>>>
>>>>
>>>> It is clear that olcott does not understand limits, because he is
>>>> changing the meaning of the words and the symbols. Limits are not
>>>> talking about what happens at the end of a sequence. It seems it has
>>>> to be spelled out for him, otherwise he will not understand.
>>>>
>>>
>>> 0.999... Limits basically pretend that we reach the end of this
>>> infinite sequence even though that it impossible, and says after we
>>> reach this
>>> impossible end the value would be 1.0.
>>
>> No, if olcott had paid attention to the text below, or the article I
>> referenced:
>>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>>
>> he would have noted that limits do not pretend to reach the end. They
>
> Other people were saying that math says 0.999... = 1.0

Indeed and they were right. Olcott's problem seems to be that he thinks
that he has to go to the end to prove it, but that is not needed. We
only have to go as far as needed for any given ε. Going to the end is
his problem, not that of math in the real number system.
0.999... = 1.0 means that with this sequence we can come as close to 1.0
as needed. It does not say (nor deny) that 1.0 will be reached. That is
the meaning of the = symbol in the context of limits. It is olcott's
problem that he changes the meaning of the = symbol.

>
>> only tell us that we don't need to go further than needed and that
>> this is reachable for any given rational ε > 0. It is interesting that
>> this is sufficient to construct reals.
>>
>>>
>>>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>>>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞.
>>>> The = symbol in the context of a limit means in this case:
>>>> For each rational ε > 0 (no matter how small) we can find a number N
>>>> {in this case 10log(1/ε)}, such that for all n > N the absolute
>>>> value of the difference between xn and 1.0 is less than ε.
>>>> It is not more and not less. Note that it does not speak of what
>>>> happens at the end of the sequence, or about completing the sequence.
>>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>>>> system), then he has to specify a rational ε for which no such N can
>>>> be found. If he cannot do that, then he is not speaking about real
>>>> numbers.
>>>>
>>
>> Good to see that there is no objection against this proof.
>>
>

--
Paradoxes in the relation between Creator and creature.
<http://www.wirholt.nl/English>.

On 3/30/2024 3:18 PM, Fred. Zwarts wrote:
> Op 30.mrt.2024 om 20:57 schreef olcott:
>> On 3/30/2024 2:45 PM, Fred. Zwarts wrote:
>>> Op 30.mrt.2024 om 14:56 schreef olcott:
>>>> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>>>>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>>>>> 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.
>>>>>>
>>>>>
>>>>> It is clear that olcott does not understand limits, because he is
>>>>> changing the meaning of the words and the symbols. Limits are not
>>>>> talking about what happens at the end of a sequence. It seems it
>>>>> has to be spelled out for him, otherwise he will not understand.
>>>>>
>>>>
>>>> 0.999... Limits basically pretend that we reach the end of this
>>>> infinite sequence even though that it impossible, and says after we
>>>> reach this
>>>> impossible end the value would be 1.0.
>>>
>>> No, if olcott had paid attention to the text below, or the article I
>>> referenced:
>>>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>>>
>>> he would have noted that limits do not pretend to reach the end. They
>>
>> Other people were saying that math says 0.999... = 1.0
>
> Indeed and they were right. Olcott's problem seems to be that he thinks
> that he has to go to the end to prove it, but that is not needed. We
> only have to go as far as needed for any given ε. Going to the end is
> his problem, not that of math in the real number system.
> 0.999... = 1.0 means that with this sequence we can come as close to 1.0
> as needed.

That is not what the "=" sign means. It means exactly the same as.

> It does not say (nor deny) that 1.0 will be reached. That is
> the meaning of the = symbol in the context of limits. It is olcott's
> problem that he changes the meaning of the = symbol.
>
>>
>>> only tell us that we don't need to go further than needed and that
>>> this is reachable for any given rational ε > 0. It is interesting
>>> that this is sufficient to construct reals.
>>>
>>>>
>>>>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>>>>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞.
>>>>> The = symbol in the context of a limit means in this case:
>>>>> For each rational ε > 0 (no matter how small) we can find a number
>>>>> N {in this case 10log(1/ε)}, such that for all n > N the absolute
>>>>> value of the difference between xn and 1.0 is less than ε.
>>>>> It is not more and not less. Note that it does not speak of what
>>>>> happens at the end of the sequence, or about completing the sequence.
>>>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>>>>> system), then he has to specify a rational ε for which no such N
>>>>> can be found. If he cannot do that, then he is not speaking about
>>>>> real numbers.
>>>>>
>>>
>>> Good to see that there is no objection against this proof.
>>>
>>
>

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

On 3/30/24 3:52 PM, wij wrote:
> On Sat, 2024-03-30 at 13:23 -0400, Richard Damon wrote:
>> On 3/30/24 12:00 PM, wij wrote:
>>> On Sat, 2024-03-30 at 11:34 -0400, Richard Damon wrote:
>>>> On 3/30/24 10:57 AM, wij wrote:
>>>>> On Sat, 2024-03-30 at 10:01 -0400, Richard Damon wrote:
>>>>>> On 3/30/24 9:56 AM, olcott wrote:
>>>>>>> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>>>>>>>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>>>>>>>> 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.
>>>>>>>>>
>>>>>>>>
>>>>>>>> It is clear that olcott does not understand limits, because he is
>>>>>>>> changing the meaning of the words and the symbols. Limits are not
>>>>>>>> talking about what happens at the end of a sequence. It seems it has
>>>>>>>> to be spelled out for him, otherwise he will not understand.
>>>>>>>>
>>>>>>>
>>>>>>> 0.999... Limits basically pretend that we reach the end of this infinite
>>>>>>> sequence even though that it impossible, and says after we reach this
>>>>>>> impossible end the value would be 1.0.
>>>>>>
>>>>>> Nope. Shows you don't really understand what limits are.
>>>>>>
>>>>>> And are just a pathological liar as you insist that you falsehoods based
>>>>>> on the wrong definitions are the truth.
>>>>>>
>>>>>
>>>>> You are nut who always think he is talking B while reading A. (x!=c)
>>>>> https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
>>>>>
>>>>>
>>>>> Limit is defined on existing numbers, it cannot define the the number it is using.
>>>>> Things is very simple: "repeating decimal" means the pattern is infinite.
>>>>> (you are worse than olcott in this)
>>>>> If it does not exit, your math (repeating decimal is ...) is garbage talking
>>>>> about something does not exist and use it as proof of fact.
>>>>
>>>>
>>>> What number does the representation 0.abc represent?
>>>>
>>>> it is BY DEFINITION 0 + a * 10^-1 + b * 10^-2 + c * 10^-3
>>>>
>>>> what number does the representation 0.aaa... represent:
>>>>
>>>> The value of lim(n-> inf) Sum(9 * 10^-i) [for i = 1 to n]
>>>>
>>>> If a = 9, what number is that 0.999.... but also the number 1.0 since
>>>> they are the same.
>>>>
>>>
>>> I should have provided all that can explain your doubt, but you still
>>> keep insisting 0.999...=1 with no proof, like POOP.
>>
>> But I did, see below
>>
>> Since there is no finite e that doesn't have an N, the limit of the
>> function, which is the MEANING of the representation 0.999... is 1
>>
>
> I guess you were talking about delta-epsilon method. But, firstly, we are
> not talking about the number system here.
>
> In lim(x->c) f(x), x is the number you choose. The numbers in the sequence
> chosen are those expressible. But you would agree there exits numbers not
> expressible. And, you seemed to choose "finite e".
>
>>>
>>>> For ANY e > 0, there exists an N that for all values of function/series
>>>> witn n >= N the difference between the function and 1 is less then e.
>>>>
>>>> BY THE DEFINITION OF LIMIT, that means that 0.999... IS EQUAL TO 1.000
>>>>
>>> See the link above. limit says the limit of 0.999... is 1, not 0.999... is 1.
>>> You keep talking RD's POOP.
>>
>> Where?
>>
> 4.1 Limits of Functions
> https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
>> It defines what a limit is. Is says NOTHING about what "0.999..." is.
>>
>> That comes out of the DEFINITION of the REPRESENTATION of repeating
>> decimal literals.
>>
>>>
>>>> For Reals
>>>>
>>> For your POO Real (not even the obsolete real)
>>
>> Nope, I am quoting from the classical (which you want to call obsolete,
>> but can't show how it is obsolete) theory.
>>
>>>
>>>> Remember n-ary representations are NOT numbers, but representations of
>>>> the number.
>>>>
>>>> the value of repeating n-ary representations are defined by limits
>>>>
>>> I already said, limit is defined on existing number system, it cannot define
>>> numbers it talks about.
>>
>> But 0.999... ISN'T a "NUMBER" but the REPRESENTATION of a Number, and
>> for numbers in the Real Number System, the number it represents is 1
>>
>
> If RD's limt of a "number" is not a number, RD's limit should say so explicitly,
> E.g. lim(x->1) x // x is 0.999.. but not infinitely long because it is not a
> number. You contradict yourself.

But for looking at something like 0.9999...
The term in the limit was as n (the number of digits used) when to
infinity. Not as the "number" go to 1.

We can show that the error between the partial expression of the number
with only a finite number of digits can be made arbitrary close to 1 (by
a choice of a small enough e) be choosing a suitably large enough N.

So *ANY* finite error can be acheived with a finite number of digits, so
the LIMIT when we have the infinite number of digits, would be at zero
error.

It can't be any other number, as any other number would have a finite
difference between it and the limit, and thus making e smaller than that
size would not be satisified, so the only number the limit can be is
what we were testing with the e (epsilon).

0.999... is not "A Number", but just a representation of the number, and
all the ways to interpret that representation end up using limits (since
the pattern is infinite in length).

>
>>>
>>>> Limits are NOT on "a number" but on a function or series (which is a
>>>> sort of function of the number of terms being used).
>>>>
>>> Ok, now you changed to another excuse. So, you are not really talking
>>> about numbers, right.
>>
>> Since there is no such thing as the limit of a "Number", YOUR
>> explanation is just illogical
>>
>>>
>>>>>
>>>>>>>
>>>>>>>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>>>>>>>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞. The
>>>>>>>> = symbol in the context of a limit means in this case:
>>>>>>>> For each rational ε > 0 (no matter how small) we can find a number N
>>>>>>>> {in this case 10log(1/ε)}, such that for all n > N the absolute value
>>>>>>>> of the difference between xn and 1.0 is less than ε.
>>>>>>>> It is not more and not less. Note that it does not speak of what
>>>>>>>> happens at the end of the sequence, or about completing the sequence.
>>>>>>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>>>>>>>> system), then he has to specify a rational ε for which no such N can
>>>>>>>> be found. If he cannot do that, then he is not speaking about real
>>>>>>>> numbers.
>>>>>>>>
>>>>>>>
>>>>>>
>>>>>
>>>>>
>>>>
>>>
>>>
>>
>
>

On 3/30/2024 3:18 PM, Fred. Zwarts wrote:
> Op 30.mrt.2024 om 20:57 schreef olcott:
>> On 3/30/2024 2:45 PM, Fred. Zwarts wrote:
>>> Op 30.mrt.2024 om 14:56 schreef olcott:
>>>> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>>>>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>>>>> 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.
>>>>>>
>>>>>
>>>>> It is clear that olcott does not understand limits, because he is
>>>>> changing the meaning of the words and the symbols. Limits are not
>>>>> talking about what happens at the end of a sequence. It seems it
>>>>> has to be spelled out for him, otherwise he will not understand.
>>>>>
>>>>
>>>> 0.999... Limits basically pretend that we reach the end of this
>>>> infinite sequence even though that it impossible, and says after we
>>>> reach this
>>>> impossible end the value would be 1.0.
>>>
>>> No, if olcott had paid attention to the text below, or the article I
>>> referenced:
>>>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>>>
>>> he would have noted that limits do not pretend to reach the end. They
>>
>> Other people were saying that math says 0.999... = 1.0
>
> Indeed and they were right. Olcott's problem seems to be that he thinks
> that he has to go to the end to prove it, but that is not needed. We
> only have to go as far as needed for any given ε. Going to the end is
> his problem, not that of math in the real number system.

0.999... is the length of this line segment [0.0, 1.0)

> 0.999... = 1.0 means that with this sequence we can come as close to 1.0
> as needed. It does not say (nor deny) that 1.0 will be reached. That is
> the meaning of the = symbol in the context of limits. It is olcott's
> problem that he changes the meaning of the = symbol.
>
>>
>>> only tell us that we don't need to go further than needed and that
>>> this is reachable for any given rational ε > 0. It is interesting
>>> that this is sufficient to construct reals.
>>>
>>>>
>>>>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>>>>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞.
>>>>> The = symbol in the context of a limit means in this case:
>>>>> For each rational ε > 0 (no matter how small) we can find a number
>>>>> N {in this case 10log(1/ε)}, such that for all n > N the absolute
>>>>> value of the difference between xn and 1.0 is less than ε.
>>>>> It is not more and not less. Note that it does not speak of what
>>>>> happens at the end of the sequence, or about completing the sequence.
>>>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>>>>> system), then he has to specify a rational ε for which no such N
>>>>> can be found. If he cannot do that, then he is not speaking about
>>>>> real numbers.
>>>>>
>>>
>>> Good to see that there is no objection against this proof.
>>>
>>
>

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

On 3/30/24 5:02 PM, olcott wrote:
> On 3/30/2024 3:18 PM, Fred. Zwarts wrote:
>> Op 30.mrt.2024 om 20:57 schreef olcott:
>>> On 3/30/2024 2:45 PM, Fred. Zwarts wrote:
>>>> Op 30.mrt.2024 om 14:56 schreef olcott:
>>>>> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>>>>>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>>>>>> 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.
>>>>>>>
>>>>>>
>>>>>> It is clear that olcott does not understand limits, because he is
>>>>>> changing the meaning of the words and the symbols. Limits are not
>>>>>> talking about what happens at the end of a sequence. It seems it
>>>>>> has to be spelled out for him, otherwise he will not understand.
>>>>>>
>>>>>
>>>>> 0.999... Limits basically pretend that we reach the end of this
>>>>> infinite sequence even though that it impossible, and says after we
>>>>> reach this
>>>>> impossible end the value would be 1.0.
>>>>
>>>> No, if olcott had paid attention to the text below, or the article I
>>>> referenced:
>>>>> https://en.wikipedia.org/wiki/Construction_of_the_real_numbers
>>>>
>>>> he would have noted that limits do not pretend to reach the end. They
>>>
>>> Other people were saying that math says 0.999... = 1.0
>>
>> Indeed and they were right. Olcott's problem seems to be that he
>> thinks that he has to go to the end to prove it, but that is not
>> needed. We only have to go as far as needed for any given ε. Going to
>> the end is his problem, not that of math in the real number system.
>
> 0.999... is the length of this line segment [0.0, 1.0)

Which is of Length 1.000, with or without the end points, as points have
no length.

>
>> 0.999... = 1.0 means that with this sequence we can come as close to
>> 1.0 as needed. It does not say (nor deny) that 1.0 will be reached.
>> That is the meaning of the = symbol in the context of limits. It is
>> olcott's problem that he changes the meaning of the = symbol.
>>
>>>
>>>> only tell us that we don't need to go further than needed and that
>>>> this is reachable for any given rational ε > 0. It is interesting
>>>> that this is sufficient to construct reals.
>>>>
>>>>>
>>>>>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>>>>>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞.
>>>>>> The = symbol in the context of a limit means in this case:
>>>>>> For each rational ε > 0 (no matter how small) we can find a number
>>>>>> N {in this case 10log(1/ε)}, such that for all n > N the absolute
>>>>>> value of the difference between xn and 1.0 is less than ε.
>>>>>> It is not more and not less. Note that it does not speak of what
>>>>>> happens at the end of the sequence, or about completing the sequence.
>>>>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>>>>>> system), then he has to specify a rational ε for which no such N
>>>>>> can be found. If he cannot do that, then he is not speaking about
>>>>>> real numbers.
>>>>>>
>>>>
>>>> Good to see that there is no objection against this proof.
>>>>
>>>
>>
>

On 03/30/2024 10:38 AM, Fred. Zwarts wrote:
> Op 30.mrt.2024 om 14:56 schreef olcott:
>> On 3/30/2024 7:10 AM, Fred. Zwarts wrote:
>>> Op 30.mrt.2024 om 02:31 schreef olcott:
>>>> 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.
>>>>
>>>
>>> It is clear that olcott does not understand limits, because he is
>>> changing the meaning of the words and the symbols. Limits are not
>>> talking about what happens at the end of a sequence. It seems it has
>>> to be spelled out for him, otherwise he will not understand.
>>>
>>
>> 0.999... Limits basically pretend that we reach the end of this
>> infinite sequence even though that it impossible, and says after we
>> reach this
>> impossible end the value would be 1.0.
>
> So, olcott did not understand the explanation (below) and continues to
> claim that limits talk about reaching the end of the sequence. Since for
> real numbers this is not true, he must be talking about is unspecified
> olcott numbers.
>
>>
>>> 0.999... indicates the Cauchy sequence xn, where x1 = 9/10, x2 =
>>> 99/100, x3 = 999/100, etc. The three dots indicates the limit n→∞.
>>> The = symbol in the context of a limit means in this case:
>>> For each rational ε > 0 (no matter how small) we can find a number N
>>> {in this case 10log(1/ε)}, such that for all n > N the absolute value
>>> of the difference between xn and 1.0 is less than ε.
>>> It is not more and not less. Note that it does not speak of what
>>> happens at the end of the sequence, or about completing the sequence.
>>> If olcott wants to prove that 0.999... ≠ 1.0 (in the real number
>>> system), then he has to specify a rational ε for which no such N can
>>> be found. If he cannot do that, then he is not speaking about real
>>> numbers.
>>>
>>
>
> I see olcott did not attempt to specify a rational ε, so, he had no
> rebuttal against the claim that 0.999... = 1 using the correct meaning
> of the words and symbols for reals.

"Correct" here can mean various things, "no options",
"no alternatives", "no other explanations", yet when
there's an option, and alternative, an other explanation,
then it just means "usual".

Here, take the limit of the function, naturals n, d,
f(n) = n/d, 0 <= n <= d, d -> oo, f(0) = 1, lim n->oo f(n) = 1.

You see, 0 <= n <= d.

It's a different "set" of numbers in [0,1], but its values
lay in the extent [0,1], are dense in [0,1], complete in
[0,1], have measure 1.0 in [0,1], it's a continuous domain.

It's "standard infinitesimals of nonstandard analysis: iota-values".

It _is_ what it _is_.