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Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ]

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  Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] peteolcott <Here@Home> - 2019-09-17 14:33 -0500
    Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] Keith Thompson <kst-u@mib.org> - 2019-09-17 13:28 -0700
      Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] peteolcott <Here@Home> - 2019-09-17 15:31 -0500
    Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2019-09-17 22:16 +0100
      Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ] peteolcott <Here@Home> - 2019-09-17 16:42 -0500

#458596 — Re: Provably unprovable eliminates incompleteness [ Keith and Mike basis for Infinitesimal numbers ]

On 9/17/2019 2:10 PM, David Brown wrote:
> On 17/09/2019 18:48, peteolcott wrote:
>> On 9/17/2019 3:11 AM, Juha Nieminen wrote:
>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>> Even if every other aspect of mathematics disagrees, none-the-less the
>>>> beginning of this interval: (0, 1] stipulates the point immediately after 0.
>>>
>>> There is no point immediately after 0. You might be incapable of
>>> comprehending such a fact of mathematics, but that doesn't mean
>>> it's incorrect.
>>>
>>> An open interval does not mean that there must exist a particular
>>> number at the end of that interval. There is no such number. You cannot
>>> give its value as it's conceptually and mathematically non-existent.
>>> It doesn't exist even in some kind of abstract meta-sense. It doesn't
>>> exist even if you introduce the concept of "infinitesimal" into your
>>> number system (no matter how you define "infinitesimal").
>>>
>>> Positing that there exists such a number leads immediately to a
>>> contradiction, and it's extraordinarily simple to prove that.
>>> Since the assumption was that the number exists, but that assumption
>>> leads to a contradiction, that means that the assumption was incorrect.
>>> "Proof by contradiction" is not just some wishy-washy nebulous term.
>>>
>>
>>
>> On 9/16/2019 11:38 PM, Keith Thompson wrote:
>>  > Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um,
>>  > something or other.  It includes all the real numbers in (0,1],
>>  > and it may also include some other things, each of which is greater
>>  > than 0, less than or equal to 1, and somewhere on the "number line"
>>  > between 0 and 1.  Some of the non-real elements might be points,
>>  > or infinitesimals.  I'll assert that for any two members of *this*
>>  > set, exactly one of a<b, a=b, a>b is true.
>>  >
>>  > I can imagine that *this* set has a least element.  For example
>>  > we can define "iota" as an infinitesimal, something that is not a
>>  > real number but that has a location on the number line, directly
>>  > adjacent to the real number 0.  It is greater than 0 and less than
>>  > any positive real number.
>>
> 
> You can always define extra elements and add them to sets like this. But you can't do so in a way that is consistent in other ways.  So you can make the ordered set of "real in (0, 1] plus iota", but you lose other properties of the reals.  "reals plus 
> iota" is not a field, nor is it complete.
> 
> And it does not make sense to talk about "having a location on the number line" without defining "the number line" - normally that term is used precisely to mean the real numbers.
> 

This is not Keith and Mike's ideas it is their agreement with my ideas:
These are the two key essences of agreement with Infinitesimal Numbers:

On 9/16/2019 11:38 PM, Keith Thompson wrote:
 > we can define "iota" as an infinitesimal, something that is not a
 > real number but that has a location on the number line, directly
 > adjacent to the real number 0.  It is greater than 0 and less than
 > any positive real number.

On 9/16/2019 8:33 PM, Mike Terry wrote:
 > OK, so (A,B] = [A,B] - {A}, and that can be counted as an "operation
in which we get (A,B] from [A,B].

Here are some examples of the next level of elaboration based on the
above two key essences of agreement:
Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]

Infinitesimal Numbers would encode (A, B] as [A[1], B]
We merely extrapolate the very next elaboration
[A[1], B] - {A[1]} = (A[1], B] encoded as [A[2], B]

-- 
Copyright 2019 Pete Olcott All rights reserved

"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein

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#458603

peteolcott <Here@Home> writes:
[...]
> This is not Keith and Mike's ideas it is their agreement with my ideas:
> These are the two key essences of agreement with Infinitesimal Numbers:

Pete's claim that I agree with him is unfounded.  If I don't respond
to similar claims in the future, it's only because I don't want to
waste time and bandwidth.

-- 
Keith Thompson (The_Other_Keith) kst-u@mib.org  <http://www.ghoti.net/~kst>
Will write code for food.
void Void(void) { Void(); } /* The recursive call of the void */

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#458604

On 9/17/2019 3:28 PM, Keith Thompson wrote:
> peteolcott <Here@Home> writes:
> [...]
>> This is not Keith and Mike's ideas it is their agreement with my ideas:
>> These are the two key essences of agreement with Infinitesimal Numbers:
> 
> Pete's claim that I agree with him is unfounded.  If I don't respond
> to similar claims in the future, it's only because I don't want to
> waste time and bandwidth.
> 

THAT YOU REMOVED THIS CONTEXT INDICATES DISHONESTLY.

On 9/16/2019 11:38 PM, Keith Thompson wrote:
 > we can define "iota" as an infinitesimal, something that is not a
 > real number but that has a location on the number line, directly
 > adjacent to the real number 0.  It is greater than 0 and less than
 > any positive real number.

-- 
Copyright 2019 Pete Olcott All rights reserved

"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein

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#458607

On 17/09/2019 20:33, peteolcott wrote:
> On 9/17/2019 2:10 PM, David Brown wrote:
>> On 17/09/2019 18:48, peteolcott wrote:
>>> On 9/17/2019 3:11 AM, Juha Nieminen wrote:
>>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>>> Even if every other aspect of mathematics disagrees, none-the-less the
>>>>> beginning of this interval: (0, 1] stipulates the point immediately
>>>>> after 0.
>>>>
>>>> There is no point immediately after 0. You might be incapable of
>>>> comprehending such a fact of mathematics, but that doesn't mean
>>>> it's incorrect.
>>>>
>>>> An open interval does not mean that there must exist a particular
>>>> number at the end of that interval. There is no such number. You cannot
>>>> give its value as it's conceptually and mathematically non-existent.
>>>> It doesn't exist even in some kind of abstract meta-sense. It doesn't
>>>> exist even if you introduce the concept of "infinitesimal" into your
>>>> number system (no matter how you define "infinitesimal").
>>>>
>>>> Positing that there exists such a number leads immediately to a
>>>> contradiction, and it's extraordinarily simple to prove that.
>>>> Since the assumption was that the number exists, but that assumption
>>>> leads to a contradiction, that means that the assumption was incorrect.
>>>> "Proof by contradiction" is not just some wishy-washy nebulous term.
>>>>
>>>
>>>
>>> On 9/16/2019 11:38 PM, Keith Thompson wrote:
>>>  > Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um,
>>>  > something or other.  It includes all the real numbers in (0,1],
>>>  > and it may also include some other things, each of which is greater
>>>  > than 0, less than or equal to 1, and somewhere on the "number line"
>>>  > between 0 and 1.  Some of the non-real elements might be points,
>>>  > or infinitesimals.  I'll assert that for any two members of *this*
>>>  > set, exactly one of a<b, a=b, a>b is true.
>>>  >
>>>  > I can imagine that *this* set has a least element.  For example
>>>  > we can define "iota" as an infinitesimal, something that is not a
>>>  > real number but that has a location on the number line, directly
>>>  > adjacent to the real number 0.  It is greater than 0 and less than
>>>  > any positive real number.
>>>
>>
>> You can always define extra elements and add them to sets like this.
>> But you can't do so in a way that is consistent in other ways.  So you
>> can make the ordered set of "real in (0, 1] plus iota", but you lose
>> other properties of the reals.  "reals plus iota" is not a field, nor
>> is it complete.
>>
>> And it does not make sense to talk about "having a location on the
>> number line" without defining "the number line" - normally that term
>> is used precisely to mean the real numbers.
>>
>
> This is not Keith and Mike's ideas it is their agreement with my ideas:
> These are the two key essences of agreement with Infinitesimal Numbers:
>
> On 9/16/2019 11:38 PM, Keith Thompson wrote:
>> we can define "iota" as an infinitesimal, something that is not a
>> real number but that has a location on the number line, directly
>> adjacent to the real number 0.  It is greater than 0 and less than
>> any positive real number.
>
> On 9/16/2019 8:33 PM, Mike Terry wrote:
>> OK, so (A,B] = [A,B] - {A}, and that can be counted as an "operation
> in which we get (A,B] from [A,B].
>
> Here are some examples of the next level of elaboration based on the
> above two key essences of agreement:
> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]

That's actually funny - you want to use my words as a way of enhancing 
your own reputation, but what you find is just a trivial identity, 
obvious to anyone who's done junior level "introduction to sets"!

This shows that to you, [A, B] - {A} = (A, B] is some kind of 
sophisticated "idea" that you've had and I'm "agreeing with you're idea. 
  What if I agreed that 2 + 5 = 7 ?  I guess that would be me agreeing 
with your ideas about arithmetic!

It sort of confirms the DUMBO bit in the DELUDED DUMBO label you've 
acquired.   :)

Regards,
Mike.

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#458608

On 9/17/2019 4:16 PM, Mike Terry wrote:
> On 17/09/2019 20:33, peteolcott wrote:
>> On 9/17/2019 2:10 PM, David Brown wrote:
>>> On 17/09/2019 18:48, peteolcott wrote:
>>>> On 9/17/2019 3:11 AM, Juha Nieminen wrote:
>>>>> In comp.lang.c++ peteolcott <Here@home> wrote:
>>>>>> Even if every other aspect of mathematics disagrees, none-the-less the
>>>>>> beginning of this interval: (0, 1] stipulates the point immediately
>>>>>> after 0.
>>>>>
>>>>> There is no point immediately after 0. You might be incapable of
>>>>> comprehending such a fact of mathematics, but that doesn't mean
>>>>> it's incorrect.
>>>>>
>>>>> An open interval does not mean that there must exist a particular
>>>>> number at the end of that interval. There is no such number. You cannot
>>>>> give its value as it's conceptually and mathematically non-existent.
>>>>> It doesn't exist even in some kind of abstract meta-sense. It doesn't
>>>>> exist even if you introduce the concept of "infinitesimal" into your
>>>>> number system (no matter how you define "infinitesimal").
>>>>>
>>>>> Positing that there exists such a number leads immediately to a
>>>>> contradiction, and it's extraordinarily simple to prove that.
>>>>> Since the assumption was that the number exists, but that assumption
>>>>> leads to a contradiction, that means that the assumption was incorrect.
>>>>> "Proof by contradiction" is not just some wishy-washy nebulous term.
>>>>>
>>>>
>>>>
>>>> On 9/16/2019 11:38 PM, Keith Thompson wrote:
>>>>  > Arbitrarily inventing a new notation, let _(0,1]_ be a set of, um,
>>>>  > something or other.  It includes all the real numbers in (0,1],
>>>>  > and it may also include some other things, each of which is greater
>>>>  > than 0, less than or equal to 1, and somewhere on the "number line"
>>>>  > between 0 and 1.  Some of the non-real elements might be points,
>>>>  > or infinitesimals.  I'll assert that for any two members of *this*
>>>>  > set, exactly one of a<b, a=b, a>b is true.
>>>>  >
>>>>  > I can imagine that *this* set has a least element.  For example
>>>>  > we can define "iota" as an infinitesimal, something that is not a
>>>>  > real number but that has a location on the number line, directly
>>>>  > adjacent to the real number 0.  It is greater than 0 and less than
>>>>  > any positive real number.
>>>>
>>>
>>> You can always define extra elements and add them to sets like this.
>>> But you can't do so in a way that is consistent in other ways.  So you
>>> can make the ordered set of "real in (0, 1] plus iota", but you lose
>>> other properties of the reals.  "reals plus iota" is not a field, nor
>>> is it complete.
>>>
>>> And it does not make sense to talk about "having a location on the
>>> number line" without defining "the number line" - normally that term
>>> is used precisely to mean the real numbers.
>>>
>>
>> This is not Keith and Mike's ideas it is their agreement with my ideas:
>> These are the two key essences of agreement with Infinitesimal Numbers:
>>
>> On 9/16/2019 11:38 PM, Keith Thompson wrote:
>>> we can define "iota" as an infinitesimal, something that is not a
>>> real number but that has a location on the number line, directly
>>> adjacent to the real number 0.  It is greater than 0 and less than
>>> any positive real number.
>>
>> On 9/16/2019 8:33 PM, Mike Terry wrote:
>>> OK, so (A,B] = [A,B] - {A}, and that can be counted as an "operation
>> in which we get (A,B] from [A,B].
>>
>> Here are some examples of the next level of elaboration based on the
>> above two key essences of agreement:
>> Mike Terry agreed with this verbatim: [A, B] - {A} = (A, B]
> 
> That's actually funny - you want to use my words as a way of enhancing your own reputation, but what you find is just a trivial identity, obvious to anyone who's done junior level "introduction to sets"!
> 
> This shows that to you, [A, B] - {A} = (A, B] is some kind of sophisticated "idea" that you've had and I'm "agreeing with you're idea.  What if I agreed that 2 + 5 = 7 ?  I guess that would be me agreeing with your ideas about arithmetic!
> 
> It sort of confirms the DUMBO bit in the DELUDED DUMBO label you've acquired.   :)
> 
> Regards,
> Mike.
> 

The verifiable facts indicate your agreement with a key basis of
Infinitesimal Numbers. When you append ad hominem to this the factual
basis ALL OF THE AGREEMENT REMAINS.

[A, B] - {A} = (A, B] agrees that a single point can be subtracted
from an interval producing an interval that lacks this point.

Keith's agreement was more substantial:
On 9/16/2019 11:38 PM, Keith Thompson wrote:
 > we can define "iota" as an infinitesimal, something that is not a
 > real number but that has a location on the number line, directly
 > adjacent to the real number 0.  It is greater than 0 and less than
 > any positive real number.

When we provide notational conventions to encode (A, B] as [A[1], B]
Then ALL of the essence of the Infinitesimal Number system has been specified
and we can do this:

[A[1], B] - {A[1]} = (A[1], B] encoded as [A[2], B]

Every point on the number line indicating a positive number would be
encoded as: {0.0[1], ∞[0]]



-- 
Copyright 2019 Pete Olcott All rights reserved

"Great spirits have always encountered violent
opposition from mediocre minds." Albert Einstein

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