On 3/26/24 10:45 AM, wij wrote:
> Snipet from https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>
> ...
> Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
> digits may be infinitely long }
>
> Note: This definition implies that repeating decimals are irrational number.
> Let's list a common magic proof in the way as a brief explanation:
> (1) x= 0.999...
> (2) 10x= 9+x // 10x= 9.999...
> (3) 9x=9
> (4) x=1
> Ans: There is no axiom or theorem to prove (1) => (2).
>
> Note: If the steps of converting a number x to <fixed_point_number> is not
> finite, x is not a ratio of two integers, because the following
> statement is always true: ∀x,a∈ℚ, x-a∈ℚ
>
> ---End of quote
>
>

So, if 10 * 0.999... isn't 9.999... what is it?
and if 9 + 0.999... isnt 9.999... what is it?

And why aren't the same numbers the same numbers.

So, either your "wij-Reals" just fail to have the normal mathematical
operations defined or you have a problem with the proof.

Numbers defined with no rules on how to manipulate them are fairly
worthless.

On 03/26/2024 05:40 PM, wij wrote:
> On Tue, 2024-03-26 at 17:28 -0700, Keith Thompson wrote:
>> wij <wyniijj5@gmail.com> writes:

>> Please rephrase that. I don't understand what it means.
>>
>>> You still keep asserting your belief, not a proof but strangely formed statement !!!
>>> You cannot read proof !!! Go home and learn more.
>>
>> I haven't offered a proof because (a) there's no point in doing so if we
>> don't have enough mutual understanding, and (b) I'm not a mathematician,
>> so I'm not confident of my ability to construct a rigorous proof. I
>> might give it a try later, but I see no point in doing so now.
>
>

"Understanding", is a poorly defined word, that means at least
two things. One is to be true together, the other, false together.

So, in mathematics, understanding has that the objects of mathematics
are universal, that people can have different understandings, yet
they must be of the same things.

A proof is in a theory of the objects of the theory,
or a model of the objects of the theory embody the
objects of the theory, proof theory and model theory,
are equi-interpretable.

As fundamental theory, then, is as of where it's fundamental
the objects of the theory, for mathematics, all the entire
universe of the objects of mathematics, and if you don't
believe in infinity, there's still continuity, and geometry,
and the unbounded, and singularities, that there are
fundamental theories of geometry and numbers and
that there are classical theories of these and
then there are super-classical theories of these
and retro-classical theories of these and
most usually for formalism as fundamentalism.

Anyways understanding means the mutually underdefined,
and it sort of works out that that's usual, but doesn't
necessarily have a place in fundamental mathematics, at all.

On Tue, 2024-03-26 at 22:17 -0400, Richard Damon wrote:
> On 3/26/24 10:45 AM, wij wrote:
> > Snipet from https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> >
> > ...
> > Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
> >     digits may be infinitely long }
> >
> >     Note: This definition implies that repeating decimals are irrational number.
> >           Let's list a common magic proof in the way as a brief explanation:
> >             (1) x= 0.999...
> >             (2) 10x= 9+x  // 10x= 9.999...
> >             (3) 9x=9
> >             (4) x=1
> >           Ans: There is no axiom or theorem to prove (1) => (2).
> >
> >     Note: If the steps of converting a number x to <fixed_point_number> is not
> >           finite, x is not a ratio of two integers, because the following
> >           statement is always true: ∀x,a∈ℚ, x-a∈ℚ
> >
> > ---End of quote
> >
> >
>
> So, if 10 * 0.999... isn't 9.999... what is it?
> and if 9 + 0.999... isnt 9.999... what is it?
>
> And why aren't the same numbers the same numbers.
>
> So, either your "wij-Reals" just fail to have the normal mathematical
> operations defined or you have a problem with the proof.
>
> Numbers defined with no rules on how to manipulate them are fairly
> worthless.

Right, but you are actually refering to yourself.
Your math. has big holes you don't realize (and refuse to). You are fooled by
your math.

On Tue, 2024-03-26 at 22:17 -0400, Richard Damon wrote:
> On 3/26/24 10:45 AM, wij wrote:
> > Snipet from https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> >
> > ...
> > Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
> >     digits may be infinitely long }
> >
> >     Note: This definition implies that repeating decimals are irrational number.
> >           Let's list a common magic proof in the way as a brief explanation:
> >             (1) x= 0.999...
> >             (2) 10x= 9+x  // 10x= 9.999...
> >             (3) 9x=9
> >             (4) x=1
> >           Ans: There is no axiom or theorem to prove (1) => (2).
> >
> >     Note: If the steps of converting a number x to <fixed_point_number> is not
> >           finite, x is not a ratio of two integers, because the following
> >           statement is always true: ∀x,a∈ℚ, x-a∈ℚ
> >
> > ---End of quote
> >
> >
>
> So, if 10 * 0.999... isn't 9.999... what is it?
> and if 9 + 0.999... isnt 9.999... what is it?
>
> And why aren't the same numbers the same numbers.
>
> So, either your "wij-Reals" just fail to have the normal mathematical
> operations defined or you have a problem with the proof.
>
> Numbers defined with no rules on how to manipulate them are fairly
> worthless.

The update was available:
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download

Hope, it can solve your doubt.

On 3/27/24 8:56 PM, wij wrote:
> On Tue, 2024-03-26 at 22:17 -0400, Richard Damon wrote:
>> On 3/26/24 10:45 AM, wij wrote:
>>> Snipet from https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>
>>> ...
>>> Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
>>>     digits may be infinitely long }
>>>
>>>     Note: This definition implies that repeating decimals are irrational number.
>>>           Let's list a common magic proof in the way as a brief explanation:
>>>             (1) x= 0.999...
>>>             (2) 10x= 9+x  // 10x= 9.999...
>>>             (3) 9x=9
>>>             (4) x=1
>>>           Ans: There is no axiom or theorem to prove (1) => (2).
>>>
>>>     Note: If the steps of converting a number x to <fixed_point_number> is not
>>>           finite, x is not a ratio of two integers, because the following
>>>           statement is always true: ∀x,a∈ℚ, x-a∈ℚ
>>>
>>> ---End of quote
>>>
>>>
>>
>> So, if 10 * 0.999... isn't 9.999... what is it?
>> and if 9 + 0.999... isnt 9.999... what is it?
>>
>> And why aren't the same numbers the same numbers.
>>
>> So, either your "wij-Reals" just fail to have the normal mathematical
>> operations defined or you have a problem with the proof.
>>
>> Numbers defined with no rules on how to manipulate them are fairly
>> worthless.
>
> The update was available:
> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>
> Hope, it can solve your doubt.
>

But the name "Real" is still very bad.

Particularly since you seem to say that any number that can't be
expressed in a finite number of digits in SOME base, is not a number in
your system, since they can not be explicitly defined, OR HAVE MATH DONE
ON THEM, since

0.9999.... * 10 = 9. and somethnig not defined after it. (it isn't even
..999...)

So, your system seems more to be just the rationals. and you don't seem
to provide a clear set of axioms of what you allow to be done with these
numbers.

On Wed, 2024-03-27 at 21:05 -0400, Richard Damon wrote:
> On 3/27/24 8:56 PM, wij wrote:
> > On Tue, 2024-03-26 at 22:17 -0400, Richard Damon wrote:
> > > On 3/26/24 10:45 AM, wij wrote:
> > > > Snipet from https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> > > >
> > > > ...
> > > > Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
> > > >      digits may be infinitely long }
> > > >
> > > >      Note: This definition implies that repeating decimals are irrational number.
> > > >            Let's list a common magic proof in the way as a brief explanation:
> > > >              (1) x= 0.999...
> > > >              (2) 10x= 9+x  // 10x= 9.999...
> > > >              (3) 9x=9
> > > >              (4) x=1
> > > >            Ans: There is no axiom or theorem to prove (1) => (2).
> > > >
> > > >      Note: If the steps of converting a number x to <fixed_point_number> is not
> > > >            finite, x is not a ratio of two integers, because the following
> > > >            statement is always true: ∀x,a∈ℚ, x-a∈ℚ
> > > >
> > > > ---End of quote
> > > >
> > > >
> > >
> > > So, if 10 * 0.999... isn't 9.999... what is it?
> > > and if 9 + 0.999... isnt 9.999... what is it?
> > >
> > > And why aren't the same numbers the same numbers.
> > >
> > > So, either your "wij-Reals" just fail to have the normal mathematical
> > > operations defined or you have a problem with the proof.
> > >
> > > Numbers defined with no rules on how to manipulate them are fairly
> > > worthless.
> >
> > The update was available:
> > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> >
> > Hope, it can solve your doubt.
> >
>
> But the name "Real" is still very bad.
>
> Particularly since you seem to say that any number that can't be
> expressed in a finite number of digits in SOME base, is not a number in
> your system, 

I did not say that. ℝ just numbers expressible by <fixed_point_number>.

> since they can not be explicitly defined, OR HAVE MATH DONE
> ON THEM, since
>
> 0.9999.... * 10 = 9. and somethnig not defined after it. (it isn't even
> .999...)
>

What are you referring to?

> So, your system seems more to be just the rationals. and you don't seem
> to provide a clear set of axioms of what you allow to be done with these
> numbers.

On 3/27/24 10:01 PM, wij wrote:
> On Wed, 2024-03-27 at 21:05 -0400, Richard Damon wrote:
>> On 3/27/24 8:56 PM, wij wrote:
>>> On Tue, 2024-03-26 at 22:17 -0400, Richard Damon wrote:
>>>> On 3/26/24 10:45 AM, wij wrote:
>>>>> Snipet from https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>>>
>>>>> ...
>>>>> Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
>>>>>      digits may be infinitely long }
>>>>>
>>>>>      Note: This definition implies that repeating decimals are irrational number.
>>>>>            Let's list a common magic proof in the way as a brief explanation:
>>>>>              (1) x= 0.999...
>>>>>              (2) 10x= 9+x  // 10x= 9.999...
>>>>>              (3) 9x=9
>>>>>              (4) x=1
>>>>>            Ans: There is no axiom or theorem to prove (1) => (2).
>>>>>
>>>>>      Note: If the steps of converting a number x to <fixed_point_number> is not
>>>>>            finite, x is not a ratio of two integers, because the following
>>>>>            statement is always true: ∀x,a∈ℚ, x-a∈ℚ
>>>>>
>>>>> ---End of quote
>>>>>
>>>>>
>>>>
>>>> So, if 10 * 0.999... isn't 9.999... what is it?
>>>> and if 9 + 0.999... isnt 9.999... what is it?
>>>>
>>>> And why aren't the same numbers the same numbers.
>>>>
>>>> So, either your "wij-Reals" just fail to have the normal mathematical
>>>> operations defined or you have a problem with the proof.
>>>>
>>>> Numbers defined with no rules on how to manipulate them are fairly
>>>> worthless.
>>>
>>> The update was available:
>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>
>>> Hope, it can solve your doubt.
>>>
>>
>> But the name "Real" is still very bad.
>>
>> Particularly since you seem to say that any number that can't be
>> expressed in a finite number of digits in SOME base, is not a number in
>> your system,
>
> I did not say that. ℝ just numbers expressible by <fixed_point_number>.
Near the top of the paper is:

+-------------+
| Real Number |
+-------------+

>
>> since they can not be explicitly defined, OR HAVE MATH DONE
>> ON THEM, since
>>
>> 0.9999.... * 10 = 9. and somethnig not defined after it. (it isn't even
>> .999...)
>>
>
> What are you referring to?
>

IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
the structure of the rear end of 0.999... is never seen.

>> So, your system seems more to be just the rationals. and you don't seem
>> to provide a clear set of axioms of what you allow to be done with these
>> numbers.
>
>
>

On Wed, 2024-03-27 at 22:09 -0400, Richard Damon wrote:
> On 3/27/24 10:01 PM, wij wrote:
> > On Wed, 2024-03-27 at 21:05 -0400, Richard Damon wrote:
> > > On 3/27/24 8:56 PM, wij wrote:
> > > > On Tue, 2024-03-26 at 22:17 -0400, Richard Damon wrote:
> > > > > On 3/26/24 10:45 AM, wij wrote:
> > > > > > Snipet from
> > > > > > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> > > > > >
> > > > > > ...
> > > > > > Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
> > > > > >       digits may be infinitely long }
> > > > > >
> > > > > >       Note: This definition implies that repeating decimals are irrational number.
> > > > > >             Let's list a common magic proof in the way as a brief explanation:
> > > > > >               (1) x= 0.999...
> > > > > >               (2) 10x= 9+x  // 10x= 9.999...
> > > > > >               (3) 9x=9
> > > > > >               (4) x=1
> > > > > >             Ans: There is no axiom or theorem to prove (1) => (2).
> > > > > >
> > > > > >       Note: If the steps of converting a number x to <fixed_point_number> is not
> > > > > >             finite, x is not a ratio of two integers, because the following
> > > > > >             statement is always true: ∀x,a∈ℚ, x-a∈ℚ
> > > > > >
> > > > > > ---End of quote
> > > > > >
> > > > > >
> > > > >
> > > > > So, if 10 * 0.999... isn't 9.999... what is it?
> > > > > and if 9 + 0.999... isnt 9.999... what is it?
> > > > >
> > > > > And why aren't the same numbers the same numbers.
> > > > >
> > > > > So, either your "wij-Reals" just fail to have the normal mathematical
> > > > > operations defined or you have a problem with the proof.
> > > > >
> > > > > Numbers defined with no rules on how to manipulate them are fairly
> > > > > worthless.
> > > >
> > > > The update was available:
> > > > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> > > >
> > > > Hope, it can solve your doubt.
> > > >
> > >
> > > But the name "Real" is still very bad.
> > >
> > > Particularly since you seem to say that any number that can't be
> > > expressed in a finite number of digits in SOME base, is not a number in
> > > your system,
> >
> > I did not say that. ℝ just numbers expressible by <fixed_point_number>.
> Near the top of the paper is:
>
>
> +-------------+
> > Real Number |
> +-------------+
>
>
>
>
> >
> > > since they can not be explicitly defined, OR HAVE MATH DONE
> > > ON THEM, since
> > >
> > > 0.9999.... * 10 = 9. and somethnig not defined after it. (it isn't even
> > > .999...)
> > >
> >
> > What are you referring to?
> >
>
>
>
>      IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
>      the structure of the rear end of 0.999... is never seen.
>
>
Will you explain more specific? I did not mention anything "0.9999.... * 10 = 9. and somethnig not
defined after it. (it isn't even
..999...)"

>
> > > So, your system seems more to be just the rationals. and you don't seem
> > > to provide a clear set of axioms of what you allow to be done with these
> > > numbers.
> >
> >
> >
>

On 3/27/24 10:18 PM, wij wrote:
> On Wed, 2024-03-27 at 22:09 -0400, Richard Damon wrote:
>> On 3/27/24 10:01 PM, wij wrote:
>>> On Wed, 2024-03-27 at 21:05 -0400, Richard Damon wrote:
>>>> On 3/27/24 8:56 PM, wij wrote:
>>>>> On Tue, 2024-03-26 at 22:17 -0400, Richard Damon wrote:
>>>>>> On 3/26/24 10:45 AM, wij wrote:
>>>>>>> Snipet from
>>>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>>>>>
>>>>>>> ...
>>>>>>> Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
>>>>>>>       digits may be infinitely long }
>>>>>>>
>>>>>>>       Note: This definition implies that repeating decimals are irrational number.
>>>>>>>             Let's list a common magic proof in the way as a brief explanation:
>>>>>>>               (1) x= 0.999...
>>>>>>>               (2) 10x= 9+x  // 10x= 9.999...
>>>>>>>               (3) 9x=9
>>>>>>>               (4) x=1
>>>>>>>             Ans: There is no axiom or theorem to prove (1) => (2).
>>>>>>>
>>>>>>>       Note: If the steps of converting a number x to <fixed_point_number> is not
>>>>>>>             finite, x is not a ratio of two integers, because the following
>>>>>>>             statement is always true: ∀x,a∈ℚ, x-a∈ℚ
>>>>>>>
>>>>>>> ---End of quote
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> So, if 10 * 0.999... isn't 9.999... what is it?
>>>>>> and if 9 + 0.999... isnt 9.999... what is it?
>>>>>>
>>>>>> And why aren't the same numbers the same numbers.
>>>>>>
>>>>>> So, either your "wij-Reals" just fail to have the normal mathematical
>>>>>> operations defined or you have a problem with the proof.
>>>>>>
>>>>>> Numbers defined with no rules on how to manipulate them are fairly
>>>>>> worthless.
>>>>>
>>>>> The update was available:
>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>>>
>>>>> Hope, it can solve your doubt.
>>>>>
>>>>
>>>> But the name "Real" is still very bad.
>>>>
>>>> Particularly since you seem to say that any number that can't be
>>>> expressed in a finite number of digits in SOME base, is not a number in
>>>> your system,
>>>
>>> I did not say that. ℝ just numbers expressible by <fixed_point_number>.
>> Near the top of the paper is:
>>
>>
>> +-------------+
>>> Real Number |
>> +-------------+
>>
>>
>>
>>
>>>
>>>> since they can not be explicitly defined, OR HAVE MATH DONE
>>>> ON THEM, since
>>>>
>>>> 0.9999.... * 10 = 9. and somethnig not defined after it. (it isn't even
>>>> .999...)
>>>>
>>>
>>> What are you referring to?
>>>
>>
>>
>>
>>      IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
>>      the structure of the rear end of 0.999... is never seen.
>>
>>
> Will you explain more specific? I did not mention anything "0.9999.... * 10 = 9. and somethnig not
> defined after it. (it isn't even
> .999...)"

The line above was taken directly from the paper that I downloaded by
clicking on the link.

You say, and I quote:

0.999.... * 10 = 9. and somthing not defined after it. (it isn't even
..999...)

>
>>
>>>> So, your system seems more to be just the rationals. and you don't seem
>>>> to provide a clear set of axioms of what you allow to be done with these
>>>> numbers.
>>>
>>>
>>>
>>
>
>

On Wed, 2024-03-27 at 22:38 -0400, Richard Damon wrote:
> On 3/27/24 10:18 PM, wij wrote:
> > On Wed, 2024-03-27 at 22:09 -0400, Richard Damon wrote:
> > > On 3/27/24 10:01 PM, wij wrote:
> > > > On Wed, 2024-03-27 at 21:05 -0400, Richard Damon wrote:
> > > > > On 3/27/24 8:56 PM, wij wrote:
> > > > > > On Tue, 2024-03-26 at 22:17 -0400, Richard Damon wrote:
> > > > > > > On 3/26/24 10:45 AM, wij wrote:
> > > > > > > > Snipet from
> > > > > > > > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> > > > > > > >
> > > > > > > > ...
> > > > > > > > Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
> > > > > > > >        digits may be infinitely long }
> > > > > > > >
> > > > > > > >        Note: This definition implies that repeating decimals are irrational number.
> > > > > > > >              Let's list a common magic proof in the way as a brief explanation:
> > > > > > > >                (1) x= 0.999...
> > > > > > > >                (2) 10x= 9+x  // 10x= 9.999...
> > > > > > > >                (3) 9x=9
> > > > > > > >                (4) x=1
> > > > > > > >              Ans: There is no axiom or theorem to prove (1) => (2).
> > > > > > > >
> > > > > > > >        Note: If the steps of converting a number x to <fixed_point_number> is not
> > > > > > > >              finite, x is not a ratio of two integers, because the following
> > > > > > > >              statement is always true: ∀x,a∈ℚ, x-a∈ℚ
> > > > > > > >
> > > > > > > > ---End of quote
> > > > > > > >
> > > > > > > >
> > > > > > >
> > > > > > > So, if 10 * 0.999... isn't 9.999... what is it?
> > > > > > > and if 9 + 0.999... isnt 9.999... what is it?
> > > > > > >
> > > > > > > And why aren't the same numbers the same numbers.
> > > > > > >
> > > > > > > So, either your "wij-Reals" just fail to have the normal mathematical
> > > > > > > operations defined or you have a problem with the proof.
> > > > > > >
> > > > > > > Numbers defined with no rules on how to manipulate them are fairly
> > > > > > > worthless.
> > > > > >
> > > > > > The update was available:
> > > > > > https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
> > > > > >
> > > > > > Hope, it can solve your doubt.
> > > > > >
> > > > >
> > > > > But the name "Real" is still very bad.
> > > > >
> > > > > Particularly since you seem to say that any number that can't be
> > > > > expressed in a finite number of digits in SOME base, is not a number in
> > > > > your system,
> > > >
> > > > I did not say that. ℝ just numbers expressible by <fixed_point_number>.
> > > Near the top of the paper is:
> > >
> > >
> > > +-------------+
> > > > Real Number |
> > > +-------------+
> > >
> > >
> > >
> > >
> > > >
> > > > > since they can not be explicitly defined, OR HAVE MATH DONE
> > > > > ON THEM, since
> > > > >
> > > > > 0.9999.... * 10 = 9. and somethnig not defined after it. (it isn't even
> > > > > .999...)
> > > > >
> > > >
> > > > What are you referring to?
> > > >
> > >
> > >
> > >
> > >       IOW, by repeatedly multiplying 0.999.... with 10, you can only see 9,
> > >       the structure of the rear end of 0.999.... is never seen.
> > >
> > >
> > Will you explain more specific? I did not mention anything "0.9999.... * 10 = 9. and somethnig not
> > defined after it. (it isn't even
> > .999...)"
>
> The line above was taken directly from the paper that I downloaded by
> clicking on the link.
>
> You say, and I quote:
>
> 0.999.... * 10 = 9. and somthing not defined after it. (it isn't even
> .999...)
>
>
I uploaded again: What/where were you referring to?
--------------------------

+-------------+
| Real Number |
+-------------+

n-ary Fixed-Point Number::= Number represented by a string of digits, the
string may contain a minus sign or a point:

<fixed_point_number>::= [-] <dstr1> [ . <dstr2> ]
<dstr1>::= 0 | <nzd> { 0, <nzd> }
<dstr2>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary

Two n-ary fixed-point number (same n-ary) x,y are equal iff their
<fixed_point_number> representation are identical.

Real Nunmber(ℝ)::= {x| x is finitely represented by n-ary <fixed_point_number>
and those that cannot be finitely represented }

Note: Numbers that is not finitely representable cannot all be explicitly
defined, this is the property of real number based on discrete symbols
(like quantum?). E.g.

A= lim(n->∞) 1-3/10^n = 0.999...
B= lim(n->∞) 1-2/2^n = 0.999...
C= lim(n->∞) 1-1/n = 0.999...
...

IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
the structure of the rear end of 0.999... is not seen.

Note: This definition implies that repeating decimals are irrational number.
Let's list a common magic proof in the way as a brief explanation:
(1) x= 0.999...
(2) 10x= 9+x // 10x= 9.999...
(3) 9x=9
(4) x=1
Ans: There is no axiom or theorem to prove (1) => (2).

Note: To determine whether a repeating decimal x is rational or not, we can
repeatedly subtract the repeating pattern p(i) from x.
If x-p(1)-p(2)-...=0 can be verified in finite steps, then x is
rational. Otherwise, x is irrational, because, if x is rational, the
last remaining piece r(i)= x-p(1)-p(2)-... must exactly be the
repeating pattern p(i). But, by definition of 'repeating', r(i) cannot
be pattern p(i). Therefore, repeating decimal is irrational.

>
> >
> > >
> > > > > So, your system seems more to be just the rationals. and you don't seem
> > > > > to provide a clear set of axioms of what you allow to be done with these
> > > > > numbers.
> > > >
> > > >
> > > >
> > >
> >
> >
>

On 3/27/24 10:45 PM, wij wrote:
> On Wed, 2024-03-27 at 22:38 -0400, Richard Damon wrote:
>> On 3/27/24 10:18 PM, wij wrote:
>>> On Wed, 2024-03-27 at 22:09 -0400, Richard Damon wrote:
>>>> On 3/27/24 10:01 PM, wij wrote:
>>>>> On Wed, 2024-03-27 at 21:05 -0400, Richard Damon wrote:
>>>>>> On 3/27/24 8:56 PM, wij wrote:
>>>>>>> On Tue, 2024-03-26 at 22:17 -0400, Richard Damon wrote:
>>>>>>>> On 3/26/24 10:45 AM, wij wrote:
>>>>>>>>> Snipet from
>>>>>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>>>>>>>
>>>>>>>>> ...
>>>>>>>>> Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
>>>>>>>>>        digits may be infinitely long }
>>>>>>>>>
>>>>>>>>>        Note: This definition implies that repeating decimals are irrational number.
>>>>>>>>>              Let's list a common magic proof in the way as a brief explanation:
>>>>>>>>>                (1) x= 0.999...
>>>>>>>>>                (2) 10x= 9+x  // 10x= 9.999...
>>>>>>>>>                (3) 9x=9
>>>>>>>>>                (4) x=1
>>>>>>>>>              Ans: There is no axiom or theorem to prove (1) => (2).
>>>>>>>>>
>>>>>>>>>        Note: If the steps of converting a number x to <fixed_point_number> is not
>>>>>>>>>              finite, x is not a ratio of two integers, because the following
>>>>>>>>>              statement is always true: ∀x,a∈ℚ, x-a∈ℚ
>>>>>>>>>
>>>>>>>>> ---End of quote
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>> So, if 10 * 0.999... isn't 9.999... what is it?
>>>>>>>> and if 9 + 0.999... isnt 9.999... what is it?
>>>>>>>>
>>>>>>>> And why aren't the same numbers the same numbers.
>>>>>>>>
>>>>>>>> So, either your "wij-Reals" just fail to have the normal mathematical
>>>>>>>> operations defined or you have a problem with the proof.
>>>>>>>>
>>>>>>>> Numbers defined with no rules on how to manipulate them are fairly
>>>>>>>> worthless.
>>>>>>>
>>>>>>> The update was available:
>>>>>>> https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
>>>>>>>
>>>>>>> Hope, it can solve your doubt.
>>>>>>>
>>>>>>
>>>>>> But the name "Real" is still very bad.
>>>>>>
>>>>>> Particularly since you seem to say that any number that can't be
>>>>>> expressed in a finite number of digits in SOME base, is not a number in
>>>>>> your system,
>>>>>
>>>>> I did not say that. ℝ just numbers expressible by <fixed_point_number>.
>>>> Near the top of the paper is:
>>>>
>>>>
>>>> +-------------+
>>>>> Real Number |
>>>> +-------------+
>>>>
>>>>
>>>>
>>>>
>>>>>
>>>>>> since they can not be explicitly defined, OR HAVE MATH DONE
>>>>>> ON THEM, since
>>>>>>
>>>>>> 0.9999.... * 10 = 9. and somethnig not defined after it. (it isn't even
>>>>>> .999...)
>>>>>>
>>>>>
>>>>> What are you referring to?
>>>>>
>>>>
>>>>
>>>>
>>>>       IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
>>>>       the structure of the rear end of 0.999... is never seen.
>>>>
>>>>
>>> Will you explain more specific? I did not mention anything "0.9999.... * 10 = 9. and somethnig not
>>> defined after it. (it isn't even
>>> .999...)"
>>
>> The line above was taken directly from the paper that I downloaded by
>> clicking on the link.
>>
>> You say, and I quote:
>>
>> 0.999.... * 10 = 9. and somthing not defined after it. (it isn't even
>> .999...)
>>
>>
> I uploaded again: What/where were you referring to?
> --------------------------
>
> +-------------+
> | Real Number |
> +-------------+
>
> n-ary Fixed-Point Number::= Number represented by a string of digits, the
> string may contain a minus sign or a point:
>
> <fixed_point_number>::= [-] <dstr1> [ . <dstr2> ]
> <dstr1>::= 0 | <nzd> { 0, <nzd> }
> <dstr2>::= { 0, <nzd> } <nzd>
> <nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // 'digit' varys depending on n-ary
>
> Two n-ary fixed-point number (same n-ary) x,y are equal iff their
> <fixed_point_number> representation are identical.
>
> Real Nunmber(ℝ)::= {x| x is finitely represented by n-ary <fixed_point_number>
> and those that cannot be finitely represented }
>
> Note: Numbers that is not finitely representable cannot all be explicitly
> defined, this is the property of real number based on discrete symbols
> (like quantum?). E.g.
>
> A= lim(n->∞) 1-3/10^n = 0.999...
> B= lim(n->∞) 1-2/2^n = 0.999...
> C= lim(n->∞) 1-1/n = 0.999...
> ...
>
> IOW, by repeatedly multiplying 0.999... with 10, you can only see 9,
> the structure of the rear end of 0.999... is not seen.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

>
> Note: This definition implies that repeating decimals are irrational number.
> Let's list a common magic proof in the way as a brief explanation:
> (1) x= 0.999...
> (2) 10x= 9+x // 10x= 9.999...
> (3) 9x=9
> (4) x=1
> Ans: There is no axiom or theorem to prove (1) => (2).
>
> Note: To determine whether a repeating decimal x is rational or not, we can
> repeatedly subtract the repeating pattern p(i) from x.
> If x-p(1)-p(2)-...=0 can be verified in finite steps, then x is
> rational. Otherwise, x is irrational, because, if x is rational, the
> last remaining piece r(i)= x-p(1)-p(2)-... must exactly be the
> repeating pattern p(i). But, by definition of 'repeating', r(i) cannot
> be pattern p(i). Therefore, repeating decimal is irrational.
>
>
>>
>>>
>>>>
>>>>>> So, your system seems more to be just the rationals. and you don't seem
>>>>>> to provide a clear set of axioms of what you allow to be done with these
>>>>>> numbers.
>>>>>
>>>>>
>>>>>
>>>>
>>>
>>>
>>
>
>