Op 30.mrt.2024 om 21:27 schreef olcott:
> 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.

No, olcott is trying to change the meaning of the symbol '='. That *is*
what the '=' means for real numbers, because 'exactly the same' is too
vague. Is 1.0 exactly the same as 1/1? It contains different symbols, so
why should they be exactly the same?
Therefore, in the construction of reals it is defined how to determine
whether two reals are 'exactly' the same. If one real X can be
constructed with a sequence of xn and the other real Y with a sequence
yn, then we can use X = Y if for every rational ε > 0 we can find an N
so that for all n > N |xn - yn| < ε.
The consequence of this is that for each real we can use an infinite
number of Cauchy sequences. E.g. the following sequences
a: 1/1, 1/1, 1/1, 1/1, etc.
b: 9/10, 99/100, 999/1000, etc.
c: 10/9, 100/99, 1000/999, etc.
d: 1/2, 2/3, 3/4, 4/5, etc.
e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
are all sequences that are different representations of the same real
which in decimal notation can be written as 1. So, a=b=c=d=e=1.
Olcott may not like it, but that is how the '=' is defined for reals.
One may try to create another number system with another meaning for
'=', but then we are not talking about reals any more.
If I do not like that 3+4=7, then I can try to create another system for
which 3+4=6 holds, which I like more, but I am no longer speaking of
real numbers (and probably nobody is interested in my number system).

On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
> Op 30.mrt.2024 om 21:27 schreef olcott:
>> 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.
>
> No, olcott is trying to change the meaning of the symbol '='. That *is*
> what the '=' means for real numbers, because 'exactly the same' is too
> vague. Is 1.0 exactly the same as 1/1? It contains different symbols, so
> why should they be exactly the same?

It never means approximately the same value.
It always means exactly the same value.

> Therefore, in the construction of reals it is defined how to determine
> whether two reals are 'exactly' the same. If one real X can be
> constructed with a sequence of xn and the other real Y with a sequence
> yn, then we can use X = Y if for every rational ε > 0 we can find an N
> so that for all n > N |xn - yn| < ε.
> The consequence of this is that for each real we can use an infinite
> number of Cauchy sequences. E.g. the following sequences
> a: 1/1, 1/1, 1/1, 1/1, etc.
> b: 9/10, 99/100, 999/1000, etc.
> c: 10/9, 100/99, 1000/999, etc.
> d: 1/2, 2/3, 3/4, 4/5, etc.
> e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
> are all sequences that are different representations of the same real
> which in decimal notation can be written as 1. So, a=b=c=d=e=1.
> Olcott may not like it, but that is how the '=' is defined for reals.
> One may try to create another number system with another meaning for
> '=', but then we are not talking about reals any more.
> If I do not like that 3+4=7, then I can try to create another system for
> which 3+4=6 holds, which I like more, but I am no longer speaking of
> real numbers (and probably nobody is interested in my number system).
>

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

Op 31.mrt.2024 om 21:02 schreef olcott:
> On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
>> Op 30.mrt.2024 om 21:27 schreef olcott:
>>> 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.
>>
>> No, olcott is trying to change the meaning of the symbol '='. That
>> *is* what the '=' means for real numbers, because 'exactly the same'
>> is too vague. Is 1.0 exactly the same as 1/1? It contains different
>> symbols, so why should they be exactly the same?
>
> It never means approximately the same value.
> It always means exactly the same value.

And what 'exactly the same value' means is explained below. It is a
definition, not an opinion.

>
>> Therefore, in the construction of reals it is defined how to determine
>> whether two reals are 'exactly' the same. If one real X can be
>> constructed with a sequence of xn and the other real Y with a sequence
>> yn, then we can use X = Y if for every rational ε > 0 we can find an N
>> so that for all n > N |xn - yn| < ε.
>> The consequence of this is that for each real we can use an infinite
>> number of Cauchy sequences. E.g. the following sequences
>> a: 1/1, 1/1, 1/1, 1/1, etc.
>> b: 9/10, 99/100, 999/1000, etc.
>> c: 10/9, 100/99, 1000/999, etc.
>> d: 1/2, 2/3, 3/4, 4/5, etc.
>> e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
>> are all sequences that are different representations of the same real
>> which in decimal notation can be written as 1. So, a=b=c=d=e=1.
>> Olcott may not like it, but that is how the '=' is defined for reals.
>> One may try to create another number system with another meaning for
>> '=', but then we are not talking about reals any more.
>> If I do not like that 3+4=7, then I can try to create another system
>> for which 3+4=6 holds, which I like more, but I am no longer speaking
>> of real numbers (and probably nobody is interested in my number system).

For real numbers, a has exactly the same value as b, c, d, e, f and 1.
That is how it is defined. If olcott has another definition of 'exactly
the same value', then he is changing the meaning of the words. The
meaning of '=' is exactly defined for reals.

On 3/31/24 3:02 PM, olcott wrote:
> On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
>> Op 30.mrt.2024 om 21:27 schreef olcott:
>>> 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.
>>
>> No, olcott is trying to change the meaning of the symbol '='. That
>> *is* what the '=' means for real numbers, because 'exactly the same'
>> is too vague. Is 1.0 exactly the same as 1/1? It contains different
>> symbols, so why should they be exactly the same?
>
> It never means approximately the same value.
> It always means exactly the same value.

Which they are, if you read the definition of "the same".

Reals, not necessarily having "finite forms" are defined by rules, and
sequences, and two different sequences can form the exact same real number.

>
>> Therefore, in the construction of reals it is defined how to determine
>> whether two reals are 'exactly' the same. If one real X can be
>> constructed with a sequence of xn and the other real Y with a sequence
>> yn, then we can use X = Y if for every rational ε > 0 we can find an N
>> so that for all n > N |xn - yn| < ε.
>> The consequence of this is that for each real we can use an infinite
>> number of Cauchy sequences. E.g. the following sequences
>> a: 1/1, 1/1, 1/1, 1/1, etc.
>> b: 9/10, 99/100, 999/1000, etc.
>> c: 10/9, 100/99, 1000/999, etc.
>> d: 1/2, 2/3, 3/4, 4/5, etc.
>> e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
>> are all sequences that are different representations of the same real
>> which in decimal notation can be written as 1. So, a=b=c=d=e=1.
>> Olcott may not like it, but that is how the '=' is defined for reals.
>> One may try to create another number system with another meaning for
>> '=', but then we are not talking about reals any more.
>> If I do not like that 3+4=7, then I can try to create another system
>> for which 3+4=6 holds, which I like more, but I am no longer speaking
>> of real numbers (and probably nobody is interested in my number system).
>>
>

On 3/31/2024 2:26 PM, Fred. Zwarts wrote:
> Op 31.mrt.2024 om 21:02 schreef olcott:
>> On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
>>> Op 30.mrt.2024 om 21:27 schreef olcott:
>>>> 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.
>>>
>>> No, olcott is trying to change the meaning of the symbol '='. That
>>> *is* what the '=' means for real numbers, because 'exactly the same'
>>> is too vague. Is 1.0 exactly the same as 1/1? It contains different
>>> symbols, so why should they be exactly the same?
>>
>> It never means approximately the same value.
>> It always means exactly the same value.
>
> And what 'exactly the same value' means is explained below. It is a
> definition, not an opinion.
>

No matter what you explain below nothing that anyone can possibly
say can possibly show that 0.000... = 1.0

I use categorically exhaustive reasoning thus eliminating the
possibility of correct rebuttals.

>>
>>> Therefore, in the construction of reals it is defined how to
>>> determine whether two reals are 'exactly' the same. If one real X can
>>> be constructed with a sequence of xn and the other real Y with a
>>> sequence yn, then we can use X = Y if for every rational ε > 0 we can
>>> find an N so that for all n > N |xn - yn| < ε.
>>> The consequence of this is that for each real we can use an infinite
>>> number of Cauchy sequences. E.g. the following sequences
>>> a: 1/1, 1/1, 1/1, 1/1, etc.
>>> b: 9/10, 99/100, 999/1000, etc.
>>> c: 10/9, 100/99, 1000/999, etc.
>>> d: 1/2, 2/3, 3/4, 4/5, etc.
>>> e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
>>> are all sequences that are different representations of the same real
>>> which in decimal notation can be written as 1. So, a=b=c=d=e=1.
>>> Olcott may not like it, but that is how the '=' is defined for reals.
>>> One may try to create another number system with another meaning for
>>> '=', but then we are not talking about reals any more.
>>> If I do not like that 3+4=7, then I can try to create another system
>>> for which 3+4=6 holds, which I like more, but I am no longer speaking
>>> of real numbers (and probably nobody is interested in my number system).
>
> For real numbers, a has exactly the same value as b, c, d, e, f and 1.
> That is how it is defined. If olcott has another definition of 'exactly
> the same value', then he is changing the meaning of the words. The
> meaning of '=' is exactly defined for reals.

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

On 3/31/24 3:42 PM, olcott wrote:
> On 3/31/2024 2:26 PM, Fred. Zwarts wrote:
>> Op 31.mrt.2024 om 21:02 schreef olcott:
>>> On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
>>>> Op 30.mrt.2024 om 21:27 schreef olcott:
>>>>> 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.
>>>>
>>>> No, olcott is trying to change the meaning of the symbol '='. That
>>>> *is* what the '=' means for real numbers, because 'exactly the same'
>>>> is too vague. Is 1.0 exactly the same as 1/1? It contains different
>>>> symbols, so why should they be exactly the same?
>>>
>>> It never means approximately the same value.
>>> It always means exactly the same value.
>>
>> And what 'exactly the same value' means is explained below. It is a
>> definition, not an opinion.
>>
>
> No matter what you explain below nothing that anyone can possibly
> say can possibly show that 0.000... = 1.0

I would HOPE that 0.00000 isn't equial to 1.0000 as they are a full 1.0
apart.

If you mean 0.999... and 1.000, then yes it does, but you are apparently
to stupid to understand.

0.999... isn't "a number" but a Representation for a number, and a
representation that creates an infinite series that approach the number
it describes as a limit.

Unless you can show a finite epsilon for which no N can be created that
all points in the sequence after that N are within that epsilon of the
claimed limit, then you haven't shown anything except that you are just
a stupid liar.

>
> I use categorically exhaustive reasoning thus eliminating the
> possibility of correct rebuttals.

Nope, just establishing the proof that your "Correct Reasoning" isn't
Correct and your "Categorically Exhaustive Reasoning" isn't any better

>
>>>
>>>> Therefore, in the construction of reals it is defined how to
>>>> determine whether two reals are 'exactly' the same. If one real X
>>>> can be constructed with a sequence of xn and the other real Y with a
>>>> sequence yn, then we can use X = Y if for every rational ε > 0 we
>>>> can find an N so that for all n > N |xn - yn| < ε.
>>>> The consequence of this is that for each real we can use an infinite
>>>> number of Cauchy sequences. E.g. the following sequences
>>>> a: 1/1, 1/1, 1/1, 1/1, etc.
>>>> b: 9/10, 99/100, 999/1000, etc.
>>>> c: 10/9, 100/99, 1000/999, etc.
>>>> d: 1/2, 2/3, 3/4, 4/5, etc.
>>>> e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
>>>> are all sequences that are different representations of the same
>>>> real which in decimal notation can be written as 1. So, a=b=c=d=e=1.
>>>> Olcott may not like it, but that is how the '=' is defined for reals.
>>>> One may try to create another number system with another meaning for
>>>> '=', but then we are not talking about reals any more.
>>>> If I do not like that 3+4=7, then I can try to create another system
>>>> for which 3+4=6 holds, which I like more, but I am no longer
>>>> speaking of real numbers (and probably nobody is interested in my
>>>> number system).
>>
>> For real numbers, a has exactly the same value as b, c, d, e, f and 1.
>> That is how it is defined. If olcott has another definition of
>> 'exactly the same value', then he is changing the meaning of the
>> words. The meaning of '=' is exactly defined for reals.
>

Op 31.mrt.2024 om 21:42 schreef olcott:
> On 3/31/2024 2:26 PM, Fred. Zwarts wrote:
>> Op 31.mrt.2024 om 21:02 schreef olcott:
>>> On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
>>>> Op 30.mrt.2024 om 21:27 schreef olcott:
>>>>> 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.
>>>>
>>>> No, olcott is trying to change the meaning of the symbol '='. That
>>>> *is* what the '=' means for real numbers, because 'exactly the same'
>>>> is too vague. Is 1.0 exactly the same as 1/1? It contains different
>>>> symbols, so why should they be exactly the same?
>>>
>>> It never means approximately the same value.
>>> It always means exactly the same value.
>>
>> And what 'exactly the same value' means is explained below. It is a
>> definition, not an opinion.
>>
>
> No matter what you explain below nothing that anyone can possibly
> say can possibly show that 0.000... = 1.0
>
> I use categorically exhaustive reasoning thus eliminating the
> possibility of correct rebuttals.

OK, then it is clear that olcott is not talking about real numbers,
because for reals categorically exhaustive reasoning proved that
0.999... = 1 and olcott could not point to an error in the proof.
It would have been less confusiong when he had mentioned that explicitly.

>
>>>
>>>> Therefore, in the construction of reals it is defined how to
>>>> determine whether two reals are 'exactly' the same. If one real X
>>>> can be constructed with a sequence of xn and the other real Y with a
>>>> sequence yn, then we can use X = Y if for every rational ε > 0 we
>>>> can find an N so that for all n > N |xn - yn| < ε.
>>>> The consequence of this is that for each real we can use an infinite
>>>> number of Cauchy sequences. E.g. the following sequences
>>>> a: 1/1, 1/1, 1/1, 1/1, etc.
>>>> b: 9/10, 99/100, 999/1000, etc.
>>>> c: 10/9, 100/99, 1000/999, etc.
>>>> d: 1/2, 2/3, 3/4, 4/5, etc.
>>>> e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
>>>> are all sequences that are different representations of the same
>>>> real which in decimal notation can be written as 1. So, a=b=c=d=e=1.
>>>> Olcott may not like it, but that is how the '=' is defined for reals.
>>>> One may try to create another number system with another meaning for
>>>> '=', but then we are not talking about reals any more.
>>>> If I do not like that 3+4=7, then I can try to create another system
>>>> for which 3+4=6 holds, which I like more, but I am no longer
>>>> speaking of real numbers (and probably nobody is interested in my
>>>> number system).
>>
>> For real numbers, a has exactly the same value as b, c, d, e, f and 1.
>> That is how it is defined. If olcott has another definition of
>> 'exactly the same value', then he is changing the meaning of the
>> words. The meaning of '=' is exactly defined for reals.
>

On 4/1/2024 2:31 AM, Fred. Zwarts wrote:
> Op 31.mrt.2024 om 21:42 schreef olcott:
>> On 3/31/2024 2:26 PM, Fred. Zwarts wrote:
>>> Op 31.mrt.2024 om 21:02 schreef olcott:
>>>> On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
>>>>> Op 30.mrt.2024 om 21:27 schreef olcott:
>>>>>> 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.
>>>>>
>>>>> No, olcott is trying to change the meaning of the symbol '='. That
>>>>> *is* what the '=' means for real numbers, because 'exactly the
>>>>> same' is too vague. Is 1.0 exactly the same as 1/1? It contains
>>>>> different symbols, so why should they be exactly the same?
>>>>
>>>> It never means approximately the same value.
>>>> It always means exactly the same value.
>>>
>>> And what 'exactly the same value' means is explained below. It is a
>>> definition, not an opinion.
>>>
>>
>> No matter what you explain below nothing that anyone can possibly
>> say can possibly show that 1.000... = 1.0
>>
>> I use categorically exhaustive reasoning thus eliminating the
>> possibility of correct rebuttals.
>
> OK, then it is clear that olcott is not talking about real numbers,
> because for reals categorically exhaustive reasoning proved that
> 0.999... = 1 and olcott could not point to an error in the proof.
> It would have been less confusiong when he had mentioned that explicitly.
>

Typo corrected
No matter what you explain below nothing that anyone can possibly
say can possibly show that 0.999... = 1.0

0.999...
Means an infinite never ending sequence that never reaches 1.0

If biology "proved" that cats are a kind of dog then no matter
what this "proof" contains we know in advance that it must be
incorrect.

>>
>>>>
>>>>> Therefore, in the construction of reals it is defined how to
>>>>> determine whether two reals are 'exactly' the same. If one real X
>>>>> can be constructed with a sequence of xn and the other real Y with
>>>>> a sequence yn, then we can use X = Y if for every rational ε > 0 we
>>>>> can find an N so that for all n > N |xn - yn| < ε.
>>>>> The consequence of this is that for each real we can use an
>>>>> infinite number of Cauchy sequences. E.g. the following sequences
>>>>> a: 1/1, 1/1, 1/1, 1/1, etc.
>>>>> b: 9/10, 99/100, 999/1000, etc.
>>>>> c: 10/9, 100/99, 1000/999, etc.
>>>>> d: 1/2, 2/3, 3/4, 4/5, etc.
>>>>> e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
>>>>> are all sequences that are different representations of the same
>>>>> real which in decimal notation can be written as 1. So, a=b=c=d=e=1.
>>>>> Olcott may not like it, but that is how the '=' is defined for reals.
>>>>> One may try to create another number system with another meaning
>>>>> for '=', but then we are not talking about reals any more.
>>>>> If I do not like that 3+4=7, then I can try to create another
>>>>> system for which 3+4=6 holds, which I like more, but I am no longer
>>>>> speaking of real numbers (and probably nobody is interested in my
>>>>> number system).
>>>
>>> For real numbers, a has exactly the same value as b, c, d, e, f and
>>> 1. That is how it is defined. If olcott has another definition of
>>> 'exactly the same value', then he is changing the meaning of the
>>> words. The meaning of '=' is exactly defined for reals.
>>
>
>

Op 01.apr.2024 om 16:33 schreef olcott:
> On 4/1/2024 2:31 AM, Fred. Zwarts wrote:
>> Op 31.mrt.2024 om 21:42 schreef olcott:
>>> On 3/31/2024 2:26 PM, Fred. Zwarts wrote:
>>>> Op 31.mrt.2024 om 21:02 schreef olcott:
>>>>> On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
>>>>>> Op 30.mrt.2024 om 21:27 schreef olcott:
>>>>>>> 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.
>>>>>>
>>>>>> No, olcott is trying to change the meaning of the symbol '='. That
>>>>>> *is* what the '=' means for real numbers, because 'exactly the
>>>>>> same' is too vague. Is 1.0 exactly the same as 1/1? It contains
>>>>>> different symbols, so why should they be exactly the same?
>>>>>
>>>>> It never means approximately the same value.
>>>>> It always means exactly the same value.
>>>>
>>>> And what 'exactly the same value' means is explained below. It is a
>>>> definition, not an opinion.
>>>>
>>>
>>> No matter what you explain below nothing that anyone can possibly
>>> say can possibly show that 1.000... = 1.0
>>>
>>> I use categorically exhaustive reasoning thus eliminating the
>>> possibility of correct rebuttals.
>>
>> OK, then it is clear that olcott is not talking about real numbers,
>> because for reals categorically exhaustive reasoning proved that
>> 0.999... = 1 and olcott could not point to an error in the proof.
>> It would have been less confusiong when he had mentioned that explicitly.
>>
>
> Typo corrected
> No matter what you explain below nothing that anyone can possibly
> say can possibly show that 0.999... = 1.0
>
> 0.999...
> Means an infinite never ending sequence that never reaches 1.0

Which nobody denied.
Olcott again changes the question.
The question is not does this sequence end, or does it reach 1.0, but:
which real is represented with this sequence?
The answer is: This sequence represents one real: 1.
Therefore we can say 0.999... = 1.0. It follows directly from the
construction of reals.

>
> If biology "proved" that cats are a kind of dog then no matter
> what this "proof" contains we know in advance that it must be
> incorrect.

Similarly, if olcott 'proved' that 0.999... ≠ 1 then, no matter what
this "proof" contains, we know that it must be incorrect. Most probably
he is changing the question, changing the meaning of the words or the
symbols, or is talking about olcott numbers instead of reals.

>
>>>
>>>>>
>>>>>> Therefore, in the construction of reals it is defined how to
>>>>>> determine whether two reals are 'exactly' the same. If one real X
>>>>>> can be constructed with a sequence of xn and the other real Y with
>>>>>> a sequence yn, then we can use X = Y if for every rational ε > 0
>>>>>> we can find an N so that for all n > N |xn - yn| < ε.
>>>>>> The consequence of this is that for each real we can use an
>>>>>> infinite number of Cauchy sequences. E.g. the following sequences
>>>>>> a: 1/1, 1/1, 1/1, 1/1, etc.
>>>>>> b: 9/10, 99/100, 999/1000, etc.
>>>>>> c: 10/9, 100/99, 1000/999, etc.
>>>>>> d: 1/2, 2/3, 3/4, 4/5, etc.
>>>>>> e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
>>>>>> are all sequences that are different representations of the same
>>>>>> real which in decimal notation can be written as 1. So, a=b=c=d=e=1.
>>>>>> Olcott may not like it, but that is how the '=' is defined for reals.
>>>>>> One may try to create another number system with another meaning
>>>>>> for '=', but then we are not talking about reals any more.
>>>>>> If I do not like that 3+4=7, then I can try to create another
>>>>>> system for which 3+4=6 holds, which I like more, but I am no
>>>>>> longer speaking of real numbers (and probably nobody is interested
>>>>>> in my number system).
>>>>
>>>> For real numbers, a has exactly the same value as b, c, d, e, f and
>>>> 1. That is how it is defined. If olcott has another definition of
>>>> 'exactly the same value', then he is changing the meaning of the
>>>> words. The meaning of '=' is exactly defined for reals.
>>>
>>
>>
>

On 4/1/2024 1:39 PM, Fred. Zwarts wrote:
> Op 01.apr.2024 om 16:33 schreef olcott:
>> On 4/1/2024 2:31 AM, Fred. Zwarts wrote:
>>> Op 31.mrt.2024 om 21:42 schreef olcott:
>>>> On 3/31/2024 2:26 PM, Fred. Zwarts wrote:
>>>>> Op 31.mrt.2024 om 21:02 schreef olcott:
>>>>>> On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
>>>>>>> Op 30.mrt.2024 om 21:27 schreef olcott:
>>>>>>>> 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.
>>>>>>>
>>>>>>> No, olcott is trying to change the meaning of the symbol '='.
>>>>>>> That *is* what the '=' means for real numbers, because 'exactly
>>>>>>> the same' is too vague. Is 1.0 exactly the same as 1/1? It
>>>>>>> contains different symbols, so why should they be exactly the same?
>>>>>>
>>>>>> It never means approximately the same value.
>>>>>> It always means exactly the same value.
>>>>>
>>>>> And what 'exactly the same value' means is explained below. It is a
>>>>> definition, not an opinion.
>>>>>
>>>>
>>>> No matter what you explain below nothing that anyone can possibly
>>>> say can possibly show that 1.000... = 1.0
>>>>
>>>> I use categorically exhaustive reasoning thus eliminating the
>>>> possibility of correct rebuttals.
>>>
>>> OK, then it is clear that olcott is not talking about real numbers,
>>> because for reals categorically exhaustive reasoning proved that
>>> 0.999... = 1 and olcott could not point to an error in the proof.
>>> It would have been less confusiong when he had mentioned that
>>> explicitly.
>>>
>>
>> Typo corrected
>> No matter what you explain below nothing that anyone can possibly
>> say can possibly show that 0.999... = 1.0
>>
>> 0.999...
>> Means an infinite never ending sequence that never reaches 1.0
>
> Which nobody denied.
> Olcott again changes the question.
> The question is not does this sequence end, or does it reach 1.0, but:
> which real is represented with this sequence?

Since PI is represented by a single geometric point on the number line
then 0.999... would be correctly represented by the geometric point
immediately to the left of 1.0 on the number line or the RHS of this
interval [0,0, 1.0). If there is no Real number at that point then
there is no Real number that exactly represents 0.999...

> The answer is: This sequence represents one real: 1.
> Therefore we can say 0.999... = 1.0. It follows directly from the
> construction of reals.
>
>>
>> If biology "proved" that cats are a kind of dog then no matter
>> what this "proof" contains we know in advance that it must be
>> incorrect.
>
> Similarly, if olcott 'proved' that 0.999... ≠ 1 then, no matter what
> this "proof" contains, we know that it must be incorrect. Most probably
> he is changing the question, changing the meaning of the words or the
> symbols, or is talking about olcott numbers instead of reals.
>
>>
>>>>
>>>>>>
>>>>>>> Therefore, in the construction of reals it is defined how to
>>>>>>> determine whether two reals are 'exactly' the same. If one real X
>>>>>>> can be constructed with a sequence of xn and the other real Y
>>>>>>> with a sequence yn, then we can use X = Y if for every rational ε
>>>>>>> > 0 we can find an N so that for all n > N |xn - yn| < ε.
>>>>>>> The consequence of this is that for each real we can use an
>>>>>>> infinite number of Cauchy sequences. E.g. the following sequences
>>>>>>> a: 1/1, 1/1, 1/1, 1/1, etc.
>>>>>>> b: 9/10, 99/100, 999/1000, etc.
>>>>>>> c: 10/9, 100/99, 1000/999, etc.
>>>>>>> d: 1/2, 2/3, 3/4, 4/5, etc.
>>>>>>> e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
>>>>>>> are all sequences that are different representations of the same
>>>>>>> real which in decimal notation can be written as 1. So, a=b=c=d=e=1.
>>>>>>> Olcott may not like it, but that is how the '=' is defined for
>>>>>>> reals.
>>>>>>> One may try to create another number system with another meaning
>>>>>>> for '=', but then we are not talking about reals any more.
>>>>>>> If I do not like that 3+4=7, then I can try to create another
>>>>>>> system for which 3+4=6 holds, which I like more, but I am no
>>>>>>> longer speaking of real numbers (and probably nobody is
>>>>>>> interested in my number system).
>>>>>
>>>>> For real numbers, a has exactly the same value as b, c, d, e, f and
>>>>> 1. That is how it is defined. If olcott has another definition of
>>>>> 'exactly the same value', then he is changing the meaning of the
>>>>> words. The meaning of '=' is exactly defined for reals.
>>>>
>>>
>>>
>>
>

Op 01.apr.2024 om 20:54 schreef olcott:
> On 4/1/2024 1:39 PM, Fred. Zwarts wrote:
>> Op 01.apr.2024 om 16:33 schreef olcott:
>>> On 4/1/2024 2:31 AM, Fred. Zwarts wrote:
>>>> Op 31.mrt.2024 om 21:42 schreef olcott:
>>>>> On 3/31/2024 2:26 PM, Fred. Zwarts wrote:
>>>>>> Op 31.mrt.2024 om 21:02 schreef olcott:
>>>>>>> On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
>>>>>>>> Op 30.mrt.2024 om 21:27 schreef olcott:
>>>>>>>>> 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.
>>>>>>>>
>>>>>>>> No, olcott is trying to change the meaning of the symbol '='.
>>>>>>>> That *is* what the '=' means for real numbers, because 'exactly
>>>>>>>> the same' is too vague. Is 1.0 exactly the same as 1/1? It
>>>>>>>> contains different symbols, so why should they be exactly the same?
>>>>>>>
>>>>>>> It never means approximately the same value.
>>>>>>> It always means exactly the same value.
>>>>>>
>>>>>> And what 'exactly the same value' means is explained below. It is
>>>>>> a definition, not an opinion.
>>>>>>
>>>>>
>>>>> No matter what you explain below nothing that anyone can possibly
>>>>> say can possibly show that 1.000... = 1.0
>>>>>
>>>>> I use categorically exhaustive reasoning thus eliminating the
>>>>> possibility of correct rebuttals.
>>>>
>>>> OK, then it is clear that olcott is not talking about real numbers,
>>>> because for reals categorically exhaustive reasoning proved that
>>>> 0.999... = 1 and olcott could not point to an error in the proof.
>>>> It would have been less confusiong when he had mentioned that
>>>> explicitly.
>>>>
>>>
>>> Typo corrected
>>> No matter what you explain below nothing that anyone can possibly
>>> say can possibly show that 0.999... = 1.0
>>>
>>> 0.999...
>>> Means an infinite never ending sequence that never reaches 1.0
>>
>> Which nobody denied.
>> Olcott again changes the question.
>> The question is not does this sequence end, or does it reach 1.0, but:
>> which real is represented with this sequence?
>
> Since PI is represented by a single geometric point on the number line
> then 0.999... would be correctly represented by the geometric point
> immediately to the left of 1.0 on the number line or the RHS of this
> interval [0,0, 1.0).

On 4/1/2024 2:37 PM, Fred. Zwarts wrote:
> Op 01.apr.2024 om 20:54 schreef olcott:
>> On 4/1/2024 1:39 PM, Fred. Zwarts wrote:
>>> Op 01.apr.2024 om 16:33 schreef olcott:
>>>> On 4/1/2024 2:31 AM, Fred. Zwarts wrote:
>>>>> Op 31.mrt.2024 om 21:42 schreef olcott:
>>>>>> On 3/31/2024 2:26 PM, Fred. Zwarts wrote:
>>>>>>> Op 31.mrt.2024 om 21:02 schreef olcott:
>>>>>>>> On 3/31/2024 1:52 PM, Fred. Zwarts wrote:
>>>>>>>>> Op 30.mrt.2024 om 21:27 schreef olcott:
>>>>>>>>>> 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.
>>>>>>>>>
>>>>>>>>> No, olcott is trying to change the meaning of the symbol '='.
>>>>>>>>> That *is* what the '=' means for real numbers, because 'exactly
>>>>>>>>> the same' is too vague. Is 1.0 exactly the same as 1/1? It
>>>>>>>>> contains different symbols, so why should they be exactly the
>>>>>>>>> same?
>>>>>>>>
>>>>>>>> It never means approximately the same value.
>>>>>>>> It always means exactly the same value.
>>>>>>>
>>>>>>> And what 'exactly the same value' means is explained below. It is
>>>>>>> a definition, not an opinion.
>>>>>>>
>>>>>>
>>>>>> No matter what you explain below nothing that anyone can possibly
>>>>>> say can possibly show that 1.000... = 1.0
>>>>>>
>>>>>> I use categorically exhaustive reasoning thus eliminating the
>>>>>> possibility of correct rebuttals.
>>>>>
>>>>> OK, then it is clear that olcott is not talking about real numbers,
>>>>> because for reals categorically exhaustive reasoning proved that
>>>>> 0.999... = 1 and olcott could not point to an error in the proof.
>>>>> It would have been less confusiong when he had mentioned that
>>>>> explicitly.
>>>>>
>>>>
>>>> Typo corrected
>>>> No matter what you explain below nothing that anyone can possibly
>>>> say can possibly show that 0.999... = 1.0
>>>>
>>>> 0.999...
>>>> Means an infinite never ending sequence that never reaches 1.0
>>>
>>> Which nobody denied.
>>> Olcott again changes the question.
>>> The question is not does this sequence end, or does it reach 1.0,
>>> but: which real is represented with this sequence?
>>
>> Since PI is represented by a single geometric point on the number line
>> then 0.999... would be correctly represented by the geometric point
>> immediately to the left of 1.0 on the number line or the RHS of this
>> interval [0,0, 1.0).
>
> In the real number system it is incorrect to talk about a number
> immediately next to another number. So, this is not about real numbers.
>

On 2024-04-01 14:30, olcott wrote:
> On 4/1/2024 2:37 PM, Fred. Zwarts wrote:
>> Op 01.apr.2024 om 20:54 schreef olcott:

>>> Since PI is represented by a single geometric point on the number line
>>> then 0.999... would be correctly represented by the geometric point
>>> immediately to the left of 1.0 on the number line or the RHS of this
>>> interval [0,0, 1.0).
>>
>> In the real number system it is incorrect to talk about a number
>> immediately next to another number. So, this is not about real numbers.
>>
>
> PI is a real number.
> If there is no real number that represents 0.999...
> that does not provide a reason to say 0.999... = 1.0.

I'm a bit unclear why you keep bringing pi into this. pi isn't a
repeating decimal, unlike 0.999... which is.

But if you want to talk about pi, that also can be construed as the
limit of an infinite series:

π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...

For any *finite* number of terms, the above series never quite reaches
pi, but the LIMIT of this series is exactly equal to pi, not to some
value one 'geometric point' (which has a length of exactly zero) away
from that limit And for this series your peculiar notion that it is a
geometric point away is particularly absurd since it isn't clear whether
you'd want it to be one 'geometric point' greater or less than this
limit since the series doesn't converge on its limit from a single
direction.

Similarly, the value of the series 9/10 + 99/100 + 999/1000... is
exactly equal to the LIMIT of that series. That's what the notation
0.999... means, by definition.

André

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

On 2024-04-01 14:59, André G. Isaak wrote:

> π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
>
> For any *finite* number of terms, the above series never quite reaches
> pi, but the LIMIT of this series is exactly equal to pi.
Obviously, I mean the limit is exactly equal to π/4.

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

On Mon, 2024-04-01 at 15:03 -0600, André G. Isaak wrote:
> On 2024-04-01 14:59, André G. Isaak wrote:
>
>
> > π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
> >
> > For any *finite* number of terms, the above series never quite reaches
> > pi, but the LIMIT of this series is exactly equal to pi.
> Obviously, I mean the limit is exactly equal to π/4.
>

Note that the limit of sequence about is π/4. But none of any number in the sequence is π/4

On Tue, 2024-04-02 at 05:46 +0800, wij wrote:
> On Mon, 2024-04-01 at 15:03 -0600, André G. Isaak wrote:
> > On 2024-04-01 14:59, André G. Isaak wrote:
> >
> >
> > > π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
> > >
> > > For any *finite* number of terms, the above series never quite reaches
> > > pi, but the LIMIT of this series is exactly equal to pi.
> > Obviously, I mean the limit is exactly equal to π/4.
> >
>
> Note that the limit of sequence about is π/4. But none of any number in the sequence is π/4
>

To be more correctly:

π/4 = lim 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...

or π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11... +non_zero_remainder

or π/4 ≒ 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...

On 4/1/2024 3:59 PM, André G. Isaak wrote:
> On 2024-04-01 14:30, olcott wrote:
>> On 4/1/2024 2:37 PM, Fred. Zwarts wrote:
>>> Op 01.apr.2024 om 20:54 schreef olcott:
>
>>>> Since PI is represented by a single geometric point on the number line
>>>> then 0.999... would be correctly represented by the geometric point
>>>> immediately to the left of 1.0 on the number line or the RHS of this
>>>> interval [0,0, 1.0).
>>>
>>> In the real number system it is incorrect to talk about a number
>>> immediately next to another number. So, this is not about real numbers.
>>>
>>
>> PI is a real number.
>> If there is no real number that represents 0.999...
>> that does not provide a reason to say 0.999... = 1.0.
>
> I'm a bit unclear why you keep bringing pi into this. pi isn't a
> repeating decimal, unlike 0.999... which is.
>
> But if you want to talk about pi, that also can be construed as the
> limit of an infinite series:
>
> π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
>
> For any *finite* number of terms, the above series never quite reaches
> pi, but the LIMIT of this series is exactly equal to pi, not to some

To says that 0.999... = 1.0 means that after the never ending
sequence ends (a contradiction) then we reach exactly 1.0.

> value one 'geometric point' (which has a length of exactly zero) away
> from that limit And for this series your peculiar notion that it is a
> geometric point away is particularly absurd since it isn't clear whether
> you'd want it to be one 'geometric point' greater or less than this
> limit since the series doesn't converge on its limit from a single
> direction.
>
> Similarly, the value of the series 9/10 + 99/100 + 999/1000... is
> exactly equal to the LIMIT of that series. That's what the notation
> 0.999... means, by definition.
>
> André
>

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

On 4/1/2024 4:46 PM, wij wrote:
> On Mon, 2024-04-01 at 15:03 -0600, André G. Isaak wrote:
>> On 2024-04-01 14:59, André G. Isaak wrote:
>>
>>
>>> π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
>>>
>>> For any *finite* number of terms, the above series never quite reaches
>>> pi, but the LIMIT of this series is exactly equal to pi.
>> Obviously, I mean the limit is exactly equal to π/4.
>>
>
> Note that the limit of sequence about is π/4. But none of any number in the sequence is π/4
>

Yes

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

wij <wyniijj5@gmail.com> writes:

> On Tue, 2024-04-02 at 05:46 +0800, wij wrote:
>> On Mon, 2024-04-01 at 15:03 -0600, André G. Isaak wrote:
>> > On 2024-04-01 14:59, André G. Isaak wrote:
>> >
>> >
>> > > π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
>> > >
>> > > For any *finite* number of terms, the above series never quite reaches
>> > > pi, but the LIMIT of this series is exactly equal to pi.
>> > Obviously, I mean the limit is exactly equal to π/4.
>> >
>>
>> Note that the limit of sequence about is π/4. But none of any number
>> in the sequence is π/4
>>
>
> To be more correctly:
>
> π/4 = lim 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...

No, that's no more correct because the limit is already implied and,
more importantly, the /correct/ limit is implied. The limit is that of
a sequence of partial sums, so to write it without the implied limit one
should write either

π/4 = Sigma_{n=0}^oo -1^n * 1/(2n+1)

or

π/4 = lim_{k->oo} Sigma_{n=0}^k -1^n * 1/(2n+1).

Just sticking "lim" in front gains you nothing.

> or π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11... +non_zero_remainder

No, because the limit is implied. That's what the informal "..." means.
The only correct equation using the rather informal ... is

π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11... + 0.

You can say what you probably mean using the proper notation as above,
but I doubt you want to do that.

> or π/4 ≒ 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...

No. The ... implies the limit of the partial sums and, as you know,
that limit is π/4. Exactly π/4.

André already said the no finite partial sum is equal to π/4. What do
you gain by trying to say that again?

Have you got any further in defining the operations on your "numbers as
strings or TMs" so that (a+b)/2 is neither a nor b when a=1 and
b=0.999...? That's a fun project, but I don't think you are able to do
it.

--
Ben.

olcott <polcott333@gmail.com> writes:
[...]
> Since PI is represented by a single geometric point on the number line
> then 0.999... would be correctly represented by the geometric point
> immediately to the left of 1.0 on the number line or the RHS of this
> interval [0,0, 1.0). If there is no Real number at that point then
> there is no Real number that exactly represents 0.999...
[...]

In the following I'm talking about real numbers, and only real
numbers -- not hyperreals, or surreals, or any other extension to the
real numbers.

You assert that there is a geometric point immediately to the left of
1.0 on the number line. (I disagree, but let's go with it for now.)

Am I correct in assuming that this means that that point corresponds to
a real number that is distinct from, and less than, 1.0?

More generally, does each real number correspond to a point on the
number line, and does each point on the number line correspond to a real
number? (The real numbers can be formally defined without reference to
geometry, but let's go with your geometric model for now.)

If so, let's call that real number (immediately to the left of 1.0) x.

Consider ((x + 1.0)/2.0). Let's call that number y. (The intent is to
construct a real number that is exactly halfway between x and 1.0.)

Is y a real number? (If not, the real numbers are, unexpectedly, not
closed under common arithmetic operations.)

Is y less than, equal to, or greater than x?

Is y less than, equal to, or greater than 1.0?

Again, I am talking *only* about real numbers.

Given your past history, I do not expect straight answers to these
questions, but I'm prepared to be pleasantly surprised.

--
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 4/1/24 10:33 AM, olcott wrote:
>
> Typo corrected
> No matter what you explain below nothing that anyone can possibly
> say can possibly show that 0.999... = 1.0

WRONG, and proof that you are a stupid liar.

>
> 0.999...
> Means an infinite never ending sequence that never reaches 1.0

Right, but the REAL NUMBER that it represents is DEFINED by the limit of
the sequence, which is 1.0

>
> If biology "proved" that cats are a kind of dog then no matter
> what this "proof" contains we know in advance that it must be
> incorrect.

Right, so your "PROOF" must be wrong, since the FACTS of the Real
Numbers DEFINE that 0.999.... == 1

DEFINITIONS matter.

On 4/1/24 2:54 PM, olcott wrote:
>
> Since PI is represented by a single geometric point on the number line
> then 0.999... would be correctly represented by the geometric point
> immediately to the left of 1.0 on the number line or the RHS of this
> interval [0,0, 1.0). If there is no Real number at that point then
> there is no Real number that exactly represents 0.999...

But there is no such thing as the "Geometric point immediately to the
left" of another point, as geometric points, like Real Numbers are
"Dense" so between any two distinct points, is always an infinite number
of other points.

You brain just doesn't seem to be able to handle infinities like this.

The Real Number system perfectly handles the representation 0.999....
and shows that its VALUE is 1.0

On 4/1/24 4:30 PM, olcott wrote:
>
> PI is a real number.
> If there is no real number that represents 0.999...
> that does not provide a reason to say 0.999... = 1.0.

But there IS a Real Number that represents the value of 0.999...

and that number is 1.000

Your disagreement, without showing cause, doesn't change the fact.

If it makes no sense to you, that just means your common sense is broken
for these sorts of things.

You keep on trying to argue with false statements and lies, which don't
actually prove anything, except that you are just a stupid liar.

On Tue, 2024-04-02 at 00:00 +0100, Ben Bacarisse wrote:
> wij <wyniijj5@gmail.com> writes:
>
> > On Tue, 2024-04-02 at 05:46 +0800, wij wrote:
> > > On Mon, 2024-04-01 at 15:03 -0600, André G. Isaak wrote:
> > > > On 2024-04-01 14:59, André G. Isaak wrote:
> > > >
> > > >
> > > > > π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
> > > > >
> > > > > For any *finite* number of terms, the above series never quite reaches
> > > > > pi, but the LIMIT of this series is exactly equal to pi.
> > > > Obviously, I mean the limit is exactly equal to π/4.
> > > >
> > >
> > > Note that the limit of sequence about is π/4. But none of any number
> > > in the sequence is π/4
> > >
> >
> > To be more correctly:
> >
> >   π/4 = lim 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
>
> No, that's no more correct because the limit is already implied and,
> more importantly, the /correct/ limit is implied.  The limit is that of
> a sequence of partial sums, so to write it without the implied limit one
> should write either
>
>   π/4 = Sigma_{n=0}^oo -1^n * 1/(2n+1)
>
> or
>
>   π/4 = lim_{k->oo} Sigma_{n=0}^k -1^n * 1/(2n+1).
>
> Just sticking "lim" in front gains you nothing.
>
> > or π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11... +non_zero_remainder
>
> No, because the limit is implied.  That's what the informal "..." means.
> The only correct equation using the rather informal ... is
>
>   π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11... + 0.
>
> You can say what you probably mean using the proper notation as above,
> but I doubt you want to do that.
>
> > or π/4 ≒ 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
>
> No.  The ... implies the limit of the partial sums and, as you know,
> that limit is π/4.  Exactly π/4.
>
> André already said the no finite partial sum is equal to π/4.  What do
> you gain by trying to say that again?
>

It seems you don't understand what the limit means.

> Have you got any further in defining the operations on your "numbers as
> strings or TMs" so that (a+b)/2 is neither a nor b when a=1 and
> b=0.999...?  That's a fun project, but I don't think you are able to do
> it.
>

Please stop these sort of rebuttal, or you need to go home and learn more
Or, your kid will teach you in several years.

On 4/1/2024 6:11 PM, Keith Thompson wrote:
> olcott <polcott333@gmail.com> writes:
> [...]
>> Since PI is represented by a single geometric point on the number line
>> then 0.999... would be correctly represented by the geometric point
>> immediately to the left of 1.0 on the number line or the RHS of this
>> interval [0,0, 1.0). If there is no Real number at that point then
>> there is no Real number that exactly represents 0.999...
> [...]
>
> In the following I'm talking about real numbers, and only real
> numbers -- not hyperreals, or surreals, or any other extension to the
> real numbers.
>
> You assert that there is a geometric point immediately to the left of
> 1.0 on the number line. (I disagree, but let's go with it for now.)
>
> Am I correct in assuming that this means that that point corresponds to
> a real number that is distinct from, and less than, 1.0?
>

IDK, probably not. I am saying that 0.999... exactly equals this number.

> More generally, does each real number correspond to a point on the
> number line, and does each point on the number line correspond to a real
> number? (The real numbers can be formally defined without reference to
> geometry, but let's go with your geometric model for now.)
>

The line segment [0.0, 1.0] is exactly one geometric point longer than
[0.0, 1.10), having all points in common besides the last point.

> If so, let's call that real number (immediately to the left of 1.0) x.
>
> Consider ((x + 1.0)/2.0). Let's call that number y. (The intent is to
> construct a real number that is exactly halfway between x and 1.0.)
>
> Is y a real number? (If not, the real numbers are, unexpectedly, not
> closed under common arithmetic operations.)
>
> Is y less than, equal to, or greater than x?
>
> Is y less than, equal to, or greater than 1.0?
>
> Again, I am talking *only* about real numbers.
>
> Given your past history, I do not expect straight answers to these
> questions, but I'm prepared to be pleasantly surprised.
>

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